Subgroups — C₃₁×#34203 #21490962

Order 7936.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2362	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2360	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2344	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2342	C₂	1	4	yes	maximal
8	C₃₁×#2323	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2316	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₂²⋊Q₈	C₄	1	16	yes	complemented
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃₁×#519	—	2	8	no	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
23	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
24	C₂×C₃₁×#117	—	2	8	no	—
25	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂×C₃₁×#117	C₂²	1	4	yes	—
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₂×C₃₁×#117	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
39	C₃₁×#391	C₂²	1	4	yes	complemented
40	C₃₁×#391	C₂²	1	4	yes	complemented
41	C₃₁×#389	C₂²	1	4	yes	complemented
42	C₃₁×#389	C₂²	1	4	yes	complemented
43	C₃₁×#389	C₂²	1	8	yes	complemented
44	C₃₁×#388	C₂²	1	8	yes	—
45	C₃₁×#388	C₂²	1	8	yes	—
46	C₃₁×#387	C₂²	1	8	yes	complemented
47	C₃₁×#387	C₂²	1	8	yes	complemented
48	C₃₁×#385	C₂²	1	8	yes	—
49	C₃₁×#381	C₂²	1	4	yes	—
50	C₃₁×#381	C₂²	1	2	yes	—
51	C₃₁×#380	C₂²	1	4	yes	—
52	C₃₁×#380	C₂²	1	2	yes	—
53	C₃₁×#380	C₂²	1	2	yes	—
54	C₃₁×#379	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
58	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
60	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
61	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
62	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₃₁	—	2	4	no	—
64	C₂²×Q₈×C₃₁	—	2	4	no	—
65	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
66	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
67	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	—	2	4	no	—
69	C₂³×C₄×C₃₁	—	2	4	no	—
70	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₂²⋊Q₈	C₂×C₄	1	16	yes	—
81	C₃₁×C₂²⋊Q₈	C₂×C₄	1	16	yes	—
82	C₃₁×C₂²⋊Q₈	C₂×C₄	1	16	yes	complemented
83	C₃₁×C₂²⋊Q₈	C₂×C₄	1	16	yes	complemented
84	C₄×Q₈×C₃₁	—	2	16	no	—
85	C₄×Q₈×C₃₁	—	2	16	no	—
86	C₃₁×#117	—	2	4	no	—
87	C₃₁×#117	—	2	4	no	—
88	C₃₁×#117	—	2	4	no	—
89	C₃₁×#117	—	2	8	no	—
90	C₃₁×#117	—	2	8	no	—
91	C₃₁×#117	—	2	16	no	—
92	C₃₁×#117	—	2	8	no	—
93	C₃₁×#117	D₄	1	8	yes	—
94	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
95	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
96	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
109	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄²×C₃₁	—	2	8	no	—
120	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₃₁×#95	C₂³	1	4	yes	—
128	C₃₁×#95	C₂³	1	8	yes	—
129	C₃₁×#95	C₂³	1	4	yes	—
130	C₃₁×#95	C₂³	1	8	yes	—
131	C₃₁×#95	C₂³	1	8	yes	—
132	C₃₁×#95	C₂³	1	8	yes	—
133	C₂⁴×C₃₁	—	2	2	no	—
134	C₂⁴×C₃₁	—	2	2	no	—
135	C₂⁴×C₃₁	—	2	4	no	—
136	C₂⁴×C₃₁	—	4	2	no	—
137	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
138	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
139	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
140	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
141	C₂×Q₈×C₃₁	—	2	32	no	—
142	C₂×Q₈×C₃₁	—	2	4	no	—
143	C₂×Q₈×C₃₁	—	2	4	no	—
144	C₂×Q₈×C₃₁	—	2	4	no	—
145	C₂×Q₈×C₃₁	—	2	4	no	—
146	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
147	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	32	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	8	no	—
162	C₂²×C₄×C₃₁	—	2	8	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	—	4	4	no	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
172	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₃₁	#41	1	4	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	—	2	8	no	—
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	16	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	4	8	no	—
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
194	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
195	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	4	4	no	—
202	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
203	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
204	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
205	C₄²×C₃₁	—	2	8	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	16	no	—
208	C₄²×C₃₁	—	2	4	no	—
209	C₄²×C₃₁	—	2	4	no	—
210	C₄²×C₃₁	—	4	8	no	—
211	C₄²×C₃₁	C₂×D₄	1	8	yes	—
212	#34203	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	1	no	—
214	C₂³×C₃₁	—	2	1	no	—
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	8	no	—
219	C₂³×C₃₁	—	2	8	no	—
220	C₂³×C₃₁	—	2	1	no	—
221	C₂³×C₃₁	—	2	1	no	—
222	C₂³×C₃₁	—	2	2	no	—
223	C₂³×C₃₁	—	2	2	no	—
224	C₂³×C₃₁	—	4	4	no	—
225	C₂³×C₃₁	—	4	8	no	—
226	C₂³×C₃₁	—	4	8	no	—
227	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
228	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
229	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
230	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
231	C₂³×C₃₁	C₂×#41	1	2	yes	—
232	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
233	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
234	Q₈×C₃₁	—	2	16	no	—
235	Q₈×C₃₁	—	2	16	no	—
236	C₂×C₄×C₃₁	—	2	8	no	—
237	C₂×C₄×C₃₁	—	2	16	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	8	no	—
247	C₂×C₄×C₃₁	—	2	8	no	—
248	C₂×C₄×C₃₁	—	2	8	no	—
249	C₂×C₄×C₃₁	—	2	8	no	—
250	C₂×C₄×C₃₁	—	2	8	no	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	4	8	no	—
261	C₂×C₄×C₃₁	—	4	16	no	—
262	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
264	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
265	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
270	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
273	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
274	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
275	C₂×#519	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic
277	#2362	C₂×C₃₁	1	4	yes	complemented
278	#2360	C₂×C₃₁	1	4	yes	complemented
279	#2344	C₂×C₃₁	1	4	yes	complemented
280	#2342	C₂×C₃₁	1	4	yes	—
281	#2323	C₂×C₃₁	1	4	yes	complemented
282	#2316	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#2310	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₄×C₂²⋊Q₈	C₂×C₃₁	1	4	yes	complemented
285	#2280	C₂×C₃₁	1	1	yes	characteristic
286	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	2	no	—
291	C₂²×C₃₁	—	2	4	no	—
292	C₂²×C₃₁	—	2	4	no	—
293	C₂²×C₃₁	—	2	4	no	—
294	C₂²×C₃₁	—	2	4	no	—
295	C₂²×C₃₁	—	2	2	no	—
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	4	16	no	—
299	C₂²×C₃₁	—	4	2	no	—
300	C₂²×C₃₁	—	4	2	no	—
301	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
302	C₂²×C₃₁	#519	1	4	yes	complemented
303	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
304	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
305	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
306	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
307	C₂²×C₃₁	#585	1	2	yes	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	2	2	no	—
310	C₄×C₃₁	—	2	8	no	—
311	C₄×C₃₁	—	2	8	no	—
312	C₄×C₃₁	—	2	8	no	—
313	C₄×C₃₁	—	4	4	no	—
314	C₄×C₃₁	—	4	4	no	—
315	C₄×C₃₁	—	4	4	no	—
316	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
317	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
318	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₂²⋊Q₈	C₄×C₃₁	1	16	yes	complemented
321	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
322	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
323	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
324	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
325	#519	—	2	8	no	—
326	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
327	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
328	C₂×#117	—	2	8	no	—
329	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#117	C₂²×C₃₁	1	2	yes	—
333	C₂×#117	C₂²×C₃₁	1	4	yes	—
334	C₂×#117	C₂²×C₃₁	1	2	yes	—
335	C₂×#117	C₂²×C₃₁	1	2	yes	—
336	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
337	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
338	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
339	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
340	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
341	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
342	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
343	#391	C₂²×C₃₁	1	4	yes	complemented
344	#391	C₂²×C₃₁	1	4	yes	complemented
345	#389	C₂²×C₃₁	1	4	yes	complemented
346	#389	C₂²×C₃₁	1	4	yes	complemented
347	#389	C₂²×C₃₁	1	8	yes	complemented
348	#388	C₂²×C₃₁	1	8	yes	—
349	#388	C₂²×C₃₁	1	8	yes	—
350	#387	C₂²×C₃₁	1	8	yes	complemented
351	#387	C₂²×C₃₁	1	8	yes	complemented
352	#385	C₂²×C₃₁	1	8	yes	—
353	#381	C₂²×C₃₁	1	4	yes	—
354	#381	C₂²×C₃₁	1	2	yes	—
355	#380	C₂²×C₃₁	1	4	yes	—
356	#380	C₂²×C₃₁	1	2	yes	—
357	#380	C₂²×C₃₁	1	2	yes	—
358	#379	C₂²×C₃₁	1	2	yes	—
359	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
360	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
361	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
362	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
365	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
366	C₂×C₃₁	—	2	4	no	—
367	C₂×C₃₁	—	2	4	no	—
368	C₂×C₃₁	—	4	2	no	—
369	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
370	C₂×C₃₁	C₂×#519	1	1	yes	characteristic
371	C₂×C₃₁	#3428	1	1	yes	characteristic
372	C₂×C₃₁	#3489	1	2	yes	—
373	C₂×C₃₁	#3495	1	2	yes	—
374	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
375	C₂²×Q₈	—	2	4	no	—
376	C₂²×Q₈	—	2	4	no	—
377	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
378	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
379	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
380	C₂³×C₄	—	2	4	no	—
381	C₂³×C₄	—	2	4	no	—
382	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	—
383	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
384	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
391	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
392	C₂²⋊Q₈	C₂×C₄×C₃₁	1	16	yes	—
393	C₂²⋊Q₈	C₂×C₄×C₃₁	1	16	yes	—
394	C₂²⋊Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
395	C₂²⋊Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
396	C₄×Q₈	—	2	16	no	—
397	C₄×Q₈	—	2	16	no	—
398	#117	—	2	4	no	—
399	#117	—	2	4	no	—
400	#117	—	2	4	no	—
401	#117	—	2	8	no	—
402	#117	—	2	8	no	—
403	#117	—	2	16	no	—
404	#117	—	2	8	no	—
405	#117	D₄×C₃₁	1	8	yes	—
406	C₂×C₄⋊C₄	—	2	8	no	—
407	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
408	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
413	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₂²⋊C₄	—	2	8	no	—
421	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
431	C₂×C₄²	—	2	8	no	—
432	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
433	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
435	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
436	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
437	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
438	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
439	#95	C₂³×C₃₁	1	4	yes	—
440	#95	C₂³×C₃₁	1	8	yes	—
441	#95	C₂³×C₃₁	1	4	yes	—
442	#95	C₂³×C₃₁	1	8	yes	—
443	#95	C₂³×C₃₁	1	8	yes	—
444	#95	C₂³×C₃₁	1	8	yes	—
445	C₃₁	#34203	1	1	yes	characteristic, direct product factor, Sylow
446	C₂⁴	—	2	2	no	—
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	2	4	no	—
449	C₂⁴	—	4	2	no	—
450	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
451	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
452	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
453	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
454	C₂×Q₈	—	2	32	no	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
460	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	32	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	8	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	—	4	4	no	—
478	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
479	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
480	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
481	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
482	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
483	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
484	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
485	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
486	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
487	C₂²×C₄	#41×C₃₁	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	16	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	4	8	no	—
504	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
505	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
506	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
507	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
508	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	—	2	8	no	—
514	C₂²⋊C₄	—	4	4	no	—
515	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
516	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
517	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
518	C₄²	—	2	8	no	—
519	C₄²	—	2	4	no	—
520	C₄²	—	2	16	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	4	no	—
523	C₄²	—	4	8	no	—
524	C₄²	C₂×D₄×C₃₁	1	8	yes	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	8	no	—
531	C₂³	—	2	8	no	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	1	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	4	8	no	—
538	C₂³	—	4	8	no	—
539	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
540	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
541	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
542	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
543	C₂³	C₂×#41×C₃₁	1	2	yes	—
544	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
545	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
546	Q₈	—	2	16	no	—
547	Q₈	—	2	16	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	16	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	16	no	—
574	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	—
575	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
576	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
577	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
578	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
580	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
582	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
584	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
585	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	4	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	2	4	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	4	16	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
601	C₂²	C₃₁×#519	1	4	yes	complemented
602	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
603	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
604	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
605	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
606	C₂²	C₃₁×#585	1	2	yes	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	8	no	—
610	C₄	—	2	8	no	—
611	C₄	—	2	8	no	—
612	C₄	—	4	4	no	—
613	C₄	—	4	4	no	—
614	C₄	—	4	4	no	—
615	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
616	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
617	C₂	—	2	4	no	—
618	C₂	—	2	4	no	—
619	C₂	—	4	2	no	—
620	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
621	C₂	C₂×C₃₁×#519	1	1	yes	characteristic
622	C₂	C₃₁×#3428	1	1	yes	characteristic
623	C₂	C₃₁×#3489	1	2	yes	—
624	C₂	C₃₁×#3495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₃₁×#519	C₂	semidirect
4. C₃₁×#2362	C₂	semidirect
5. C₃₁×#2360	C₂	semidirect
6. C₃₁×#2344	C₂	semidirect
8. C₃₁×#2323	C₂	semidirect
9. C₃₁×#2316	C₂	semidirect
10. C₃₁×#2310	C₂	semidirect
11. C₄×C₃₁×C₂²⋊Q₈	C₂	semidirect
13. C₃₁×#2274	C₂	semidirect
16. C₂×C₃₁×C₂²⋊Q₈	C₄	semidirect
22. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
23. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
39. C₃₁×#391	C₂²	semidirect
40. C₃₁×#391	C₂²	semidirect
41. C₃₁×#389	C₂²	semidirect
42. C₃₁×#389	C₂²	semidirect
43. C₃₁×#389	C₂²	semidirect
46. C₃₁×#387	C₂²	semidirect
47. C₃₁×#387	C₂²	semidirect
55. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
56. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
57. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
65. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
82. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
83. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
95. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
96. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
212. #34203	C₃₁	direct (N × H)
275. C₂×#519	C₂×C₃₁	semidirect
277. #2362	C₂×C₃₁	semidirect
278. #2360	C₂×C₃₁	semidirect
279. #2344	C₂×C₃₁	semidirect
281. #2323	C₂×C₃₁	semidirect
282. #2316	C₂×C₃₁	semidirect
283. #2310	C₂×C₃₁	semidirect
284. C₄×C₂²⋊Q₈	C₂×C₃₁	semidirect
286. #2274	C₂×C₃₁	semidirect
302. C₂²×C₃₁	#519	semidirect
320. C₂×C₂²⋊Q₈	C₄×C₃₁	semidirect
326. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
327. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
343. #391	C₂²×C₃₁	semidirect
344. #391	C₂²×C₃₁	semidirect
345. #389	C₂²×C₃₁	semidirect
346. #389	C₂²×C₃₁	semidirect
347. #389	C₂²×C₃₁	semidirect
350. #387	C₂²×C₃₁	semidirect
351. #387	C₂²×C₃₁	semidirect
359. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
360. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
361. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
377. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
394. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
395. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
407. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
408. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
445. C₃₁	#34203	direct (N × H)
601. C₂²	C₃₁×#519	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×C₂²⋊Q₈	2
C₂×C₃₁×#519	2
C₂²×C₃₁×#117	2
C₃₁×#2362	2
C₃₁×#2360	2
C₃₁×#2344	2
C₃₁×#2342	2
C₃₁×#2323	2
C₃₁×#2316	2
C₃₁×#2310	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2280	2
C₃₁×#2274	2
#34203	31

Subgroups — C31×#34203 #21490962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34203 #21490962