Subgroups — C₂×C₃₁×#2349 #21490870

Order 7936.

4198 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₃₁	C₂	1	2	yes	maximal
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2349	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#389	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#385	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#380	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₃₁×#95	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₃₁	C₂²	1	4	yes	—
12	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	—	2	16	no	—
14	C₂×C₄×D₄×C₃₁	C₄	1	16	yes	complemented
15	C₂×C₄×D₄×C₃₁	C₂²	1	16	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#389	C₂²	1	16	yes	complemented
26	C₃₁×#385	C₂²	1	32	yes	complemented
27	C₃₁×#382	C₂²	1	8	yes	complemented
28	C₃₁×#380	C₂²	1	16	yes	complemented
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
30	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#95	C₂²	1	2	yes	—
32	C₂×C₃₁×#95	C₂²	1	4	yes	—
33	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#95	C₂²	1	2	yes	—
35	C₂×C₃₁×#95	C₂²	1	16	yes	complemented
36	C₂⁵×C₃₁	C₂³	1	2	yes	—
37	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
38	C₂²×D₄×C₃₁	D₄	1	8	yes	—
39	C₂²×D₄×C₃₁	Q₈	1	8	yes	—
40	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
41	C₂³×C₄×C₃₁	—	2	16	no	—
42	C₂³×C₄×C₃₁	—	2	16	no	—
43	C₂³×C₄×C₃₁	—	4	8	no	—
44	C₂³×C₄×C₃₁	—	4	8	no	—
45	C₂³×C₄×C₃₁	C₂×C₄	1	16	yes	complemented
46	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
48	C₂³×C₄×C₃₁	C₂³	1	16	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₄×D₄×C₃₁	—	2	64	no	—
56	C₄×D₄×C₃₁	C₂×C₄	1	64	yes	complemented
57	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
59	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
67	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
68	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
69	C₂×C₄²×C₃₁	—	2	8	no	—
70	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
71	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
73	C₃₁×#95	C₂³	1	4	yes	—
74	C₃₁×#95	C₂³	1	8	yes	—
75	C₃₁×#95	C₂³	1	16	yes	—
76	C₃₁×#95	C₂³	1	4	yes	—
77	C₃₁×#95	C₂³	1	8	yes	—
78	C₂⁴×C₃₁	—	2	8	no	—
79	C₂⁴×C₃₁	—	2	8	no	—
80	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
81	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
82	C₂⁴×C₃₁	C₂×Q₈	1	8	yes	—
83	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
84	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₂×D₄×C₃₁	C₄⋊C₄	1	64	yes	complemented
86	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
87	C₂×D₄×C₃₁	C₂×D₄	1	16	yes	—
88	C₂×D₄×C₃₁	C₂×Q₈	1	16	yes	—
89	C₂²×C₄×C₃₁	—	2	4	no	—
90	C₂²×C₄×C₃₁	—	2	4	no	—
91	C₂²×C₄×C₃₁	—	2	8	no	—
92	C₂²×C₄×C₃₁	—	2	4	no	—
93	C₂²×C₄×C₃₁	—	2	4	no	—
94	C₂²×C₄×C₃₁	—	2	4	no	—
95	C₂²×C₄×C₃₁	—	2	8	no	—
96	C₂²×C₄×C₃₁	—	2	32	no	—
97	C₂²×C₄×C₃₁	—	2	4	no	—
98	C₂²×C₄×C₃₁	—	2	8	no	—
99	C₂²×C₄×C₃₁	—	2	4	no	—
100	C₂²×C₄×C₃₁	—	2	8	no	—
101	C₂²×C₄×C₃₁	—	2	32	no	—
102	C₂²×C₄×C₃₁	—	4	32	no	—
103	C₂²×C₄×C₃₁	—	4	16	no	—
104	C₂²×C₄×C₃₁	—	4	16	no	—
105	C₂²×C₄×C₃₁	—	4	64	no	—
106	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
111	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
113	C₂²×C₄×C₃₁	#41	1	8	yes	—
114	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
115	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
116	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
121	C₄⋊C₄×C₃₁	—	2	16	no	—
122	C₄⋊C₄×C₃₁	—	2	16	no	—
123	C₄⋊C₄×C₃₁	—	4	16	no	—
124	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
125	C₂²⋊C₄×C₃₁	—	2	32	no	—
126	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
127	C₄²×C₃₁	—	2	16	no	—
128	C₄²×C₃₁	C₂²×C₄	1	16	yes	complemented
129	C₂×#2349	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	16	no	—
131	C₂³×C₃₁	—	2	16	no	—
132	C₂³×C₃₁	—	2	32	no	—
133	C₂³×C₃₁	—	2	32	no	—
134	C₂³×C₃₁	—	2	8	no	—
135	C₂³×C₃₁	—	2	8	no	—
136	C₂³×C₃₁	C₂×C₄⋊C₄	1	32	yes	—
137	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
138	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
139	C₂³×C₃₁	C₂²×D₄	1	8	yes	—
140	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
141	C₂³×C₃₁	C₂²×Q₈	1	8	yes	—
142	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
143	C₂³×C₃₁	C₂×#41	1	4	yes	—
144	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
145	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
146	D₄×C₃₁	C₂×C₄⋊C₄	1	64	yes	complemented
147	C₂×C₄×C₃₁	—	2	8	no	—
148	C₂×C₄×C₃₁	—	2	8	no	—
149	C₂×C₄×C₃₁	—	2	16	no	—
150	C₂×C₄×C₃₁	—	2	16	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	8	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	8	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	8	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	8	no	—
162	C₂×C₄×C₃₁	—	4	32	no	—
163	C₂×C₄×C₃₁	—	4	64	no	—
164	C₂×C₄×C₃₁	—	4	8	no	—
165	C₂×C₄×C₃₁	—	4	16	no	—
166	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	16	yes	—
167	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
168	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
169	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
170	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
171	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
172	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
173	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
174	C₂²×C₄×D₄	C₂×C₃₁	1	2	yes	—
175	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
176	#2349	C₂×C₃₁	1	16	yes	direct product factor
177	C₂×#389	C₂×C₃₁	1	2	yes	complemented
178	C₂×#385	C₂×C₃₁	1	4	yes	complemented
179	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₂×#380	C₂×C₃₁	1	2	yes	complemented
181	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
182	C₂²×#95	C₂×C₃₁	1	2	yes	complemented
183	C₂²×C₃₁	—	2	64	no	—
184	C₂²×C₃₁	—	2	8	no	—
185	C₂²×C₃₁	—	2	8	no	—
186	C₂²×C₃₁	—	2	16	no	—
187	C₂²×C₃₁	—	2	16	no	—
188	C₂²×C₃₁	C₂²×C₄⋊C₄	1	16	yes	—
189	C₂²×C₃₁	C₂²×C₄⋊C₄	1	4	yes	—
190	C₂²×C₃₁	C₄×#41	1	4	yes	—
191	C₂²×C₃₁	#518	1	4	yes	—
192	C₂²×C₃₁	#546	1	8	yes	—
193	C₂²×C₃₁	#554	1	8	yes	—
194	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
195	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
197	C₂²×C₃₁	C₂²×#41	1	2	yes	—
198	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
199	C₄×C₃₁	—	2	4	no	—
200	C₄×C₃₁	—	2	4	no	—
201	C₄×C₃₁	—	2	8	no	—
202	C₄×C₃₁	—	2	8	no	—
203	C₄×C₃₁	—	4	4	no	—
204	C₄×C₃₁	—	4	8	no	—
205	C₄×C₃₁	C₂²×C₄⋊C₄	1	8	yes	—
206	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
207	C₂⁴×C₄	C₂²×C₃₁	1	4	yes	—
208	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
209	C₂×C₄×D₄	—	2	16	no	—
210	C₂×C₄×D₄	C₄×C₃₁	1	16	yes	complemented
211	C₂×C₄×D₄	C₂²×C₃₁	1	16	yes	—
212	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
213	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
214	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
215	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
216	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
217	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
218	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
219	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
220	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
221	#389	C₂²×C₃₁	1	16	yes	complemented
222	#385	C₂²×C₃₁	1	32	yes	complemented
223	#382	C₂²×C₃₁	1	8	yes	complemented
224	#380	C₂²×C₃₁	1	16	yes	complemented
225	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
226	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#95	C₂²×C₃₁	1	2	yes	—
228	C₂×#95	C₂²×C₃₁	1	4	yes	—
229	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#95	C₂²×C₃₁	1	2	yes	—
231	C₂×#95	C₂²×C₃₁	1	16	yes	complemented
232	C₂×C₃₁	—	2	16	no	—
233	C₂×C₃₁	#2349	1	8	yes	direct product factor
234	C₂×C₃₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
236	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
237	C₂×C₃₁	C₂×#546	1	2	yes	—
238	C₂×C₃₁	C₂×#554	1	2	yes	—
239	C₂⁵	C₂³×C₃₁	1	2	yes	—
240	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
241	C₂²×D₄	D₄×C₃₁	1	8	yes	—
242	C₂²×D₄	Q₈×C₃₁	1	8	yes	—
243	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
244	C₂³×C₄	—	2	16	no	—
245	C₂³×C₄	—	2	16	no	—
246	C₂³×C₄	—	4	8	no	—
247	C₂³×C₄	—	4	8	no	—
248	C₂³×C₄	C₂×C₄×C₃₁	1	16	yes	complemented
249	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
250	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
251	C₂³×C₄	C₂³×C₃₁	1	16	yes	—
252	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
253	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
254	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
255	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
256	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
257	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
258	C₄×D₄	—	2	64	no	—
259	C₄×D₄	C₂×C₄×C₃₁	1	64	yes	complemented
260	C₂×C₄⋊C₄	—	2	8	no	—
261	C₂×C₄⋊C₄	—	2	8	no	—
262	C₂×C₄⋊C₄	—	4	8	no	—
263	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
264	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
265	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
266	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
267	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
268	C₂×C₂²⋊C₄	—	2	16	no	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
270	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
271	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
272	C₂×C₄²	—	2	8	no	—
273	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
274	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
275	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
276	#95	C₂³×C₃₁	1	4	yes	—
277	#95	C₂³×C₃₁	1	8	yes	—
278	#95	C₂³×C₃₁	1	16	yes	—
279	#95	C₂³×C₃₁	1	4	yes	—
280	#95	C₂³×C₃₁	1	8	yes	—
281	C₃₁	C₂×#2349	1	1	yes	characteristic, direct product factor, Sylow
282	C₂⁴	—	2	8	no	—
283	C₂⁴	—	2	8	no	—
284	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
285	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
286	C₂⁴	C₂×Q₈×C₃₁	1	8	yes	—
287	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
288	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
289	C₂×D₄	C₄⋊C₄×C₃₁	1	64	yes	complemented
290	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
291	C₂×D₄	C₂×D₄×C₃₁	1	16	yes	—
292	C₂×D₄	C₂×Q₈×C₃₁	1	16	yes	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	8	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	2	32	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	8	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	8	no	—
305	C₂²×C₄	—	2	32	no	—
306	C₂²×C₄	—	4	32	no	—
307	C₂²×C₄	—	4	16	no	—
308	C₂²×C₄	—	4	16	no	—
309	C₂²×C₄	—	4	64	no	—
310	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
311	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
312	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
313	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
314	C₂²×C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
315	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
316	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
317	C₂²×C₄	#41×C₃₁	1	8	yes	—
318	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
320	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
323	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
324	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
325	C₄⋊C₄	—	2	16	no	—
326	C₄⋊C₄	—	2	16	no	—
327	C₄⋊C₄	—	4	16	no	—
328	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	complemented
329	C₂²⋊C₄	—	2	32	no	—
330	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
331	C₄²	—	2	16	no	—
332	C₄²	C₂²×C₄×C₃₁	1	16	yes	complemented
333	C₂³	—	2	16	no	—
334	C₂³	—	2	16	no	—
335	C₂³	—	2	32	no	—
336	C₂³	—	2	32	no	—
337	C₂³	—	2	8	no	—
338	C₂³	—	2	8	no	—
339	C₂³	C₂×C₄⋊C₄×C₃₁	1	32	yes	—
340	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
341	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
342	C₂³	C₂²×D₄×C₃₁	1	8	yes	—
343	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
344	C₂³	C₂²×Q₈×C₃₁	1	8	yes	—
345	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
346	C₂³	C₂×#41×C₃₁	1	4	yes	—
347	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
348	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
349	D₄	C₂×C₄⋊C₄×C₃₁	1	64	yes	complemented
350	C₂×C₄	—	2	8	no	—
351	C₂×C₄	—	2	8	no	—
352	C₂×C₄	—	2	16	no	—
353	C₂×C₄	—	2	16	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	4	32	no	—
366	C₂×C₄	—	4	64	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	—	4	16	no	—
369	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	16	yes	—
370	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
371	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
372	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
373	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
374	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
375	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
376	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
377	C₂²	—	2	64	no	—
378	C₂²	—	2	8	no	—
379	C₂²	—	2	8	no	—
380	C₂²	—	2	16	no	—
381	C₂²	—	2	16	no	—
382	C₂²	C₂²×C₄⋊C₄×C₃₁	1	16	yes	—
383	C₂²	C₂²×C₄⋊C₄×C₃₁	1	4	yes	—
384	C₂²	C₄×#41×C₃₁	1	4	yes	—
385	C₂²	C₃₁×#518	1	4	yes	—
386	C₂²	C₃₁×#546	1	8	yes	—
387	C₂²	C₃₁×#554	1	8	yes	—
388	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
389	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
390	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
391	C₂²	C₂²×#41×C₃₁	1	2	yes	—
392	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
393	C₄	—	2	4	no	—
394	C₄	—	2	4	no	—
395	C₄	—	2	8	no	—
396	C₄	—	2	8	no	—
397	C₄	—	4	8	no	—
398	C₄	—	4	4	no	—
399	C₄	C₂²×C₄⋊C₄×C₃₁	1	8	yes	—
400	C₂	—	2	16	no	—
401	C₂	C₃₁×#2349	1	8	yes	direct product factor
402	C₂	C₂³×C₄⋊C₄×C₃₁	1	1	yes	characteristic
403	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
404	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
405	C₂	C₂×C₃₁×#546	1	2	yes	—
406	C₂	C₂×C₃₁×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2349	C₂	direct (N × H)
4. C₂×C₃₁×#389	C₂	semidirect
5. C₂×C₃₁×#385	C₂	semidirect
6. C₂×C₃₁×#382	C₂	semidirect
7. C₂×C₃₁×#380	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
9. C₂²×C₃₁×#95	C₂	semidirect
14. C₂×C₄×D₄×C₃₁	C₄	semidirect
25. C₃₁×#389	C₂²	semidirect
26. C₃₁×#385	C₂²	semidirect
27. C₃₁×#382	C₂²	semidirect
28. C₃₁×#380	C₂²	semidirect
29. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
35. C₂×C₃₁×#95	C₂²	semidirect
45. C₂³×C₄×C₃₁	C₂×C₄	semidirect
56. C₄×D₄×C₃₁	C₂×C₄	semidirect
60. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
66. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
70. C₂×C₄²×C₃₁	C₂×C₄	semidirect
85. C₂×D₄×C₃₁	C₄⋊C₄	semidirect
110. C₂²×C₄×C₃₁	C₂²×C₄	semidirect
124. C₄⋊C₄×C₃₁	C₂²×C₄	semidirect
126. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
128. C₄²×C₃₁	C₂²×C₄	semidirect
129. C₂×#2349	C₃₁	direct (N × H)
146. D₄×C₃₁	C₂×C₄⋊C₄	semidirect
176. #2349	C₂×C₃₁	direct (N × H)
177. C₂×#389	C₂×C₃₁	semidirect
178. C₂×#385	C₂×C₃₁	semidirect
179. C₂×#382	C₂×C₃₁	semidirect
180. C₂×#380	C₂×C₃₁	semidirect
181. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
182. C₂²×#95	C₂×C₃₁	semidirect
210. C₂×C₄×D₄	C₄×C₃₁	semidirect
221. #389	C₂²×C₃₁	semidirect
222. #385	C₂²×C₃₁	semidirect
223. #382	C₂²×C₃₁	semidirect
224. #380	C₂²×C₃₁	semidirect
225. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
231. C₂×#95	C₂²×C₃₁	semidirect
233. C₂×C₃₁	#2349	direct (N × H)
248. C₂³×C₄	C₂×C₄×C₃₁	semidirect
259. C₄×D₄	C₂×C₄×C₃₁	semidirect
263. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
269. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
273. C₂×C₄²	C₂×C₄×C₃₁	semidirect
281. C₃₁	C₂×#2349	direct (N × H)
289. C₂×D₄	C₄⋊C₄×C₃₁	semidirect
314. C₂²×C₄	C₂²×C₄×C₃₁	semidirect
328. C₄⋊C₄	C₂²×C₄×C₃₁	semidirect
330. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
332. C₄²	C₂²×C₄×C₃₁	semidirect
349. D₄	C₂×C₄⋊C₄×C₃₁	semidirect
401. C₂	C₃₁×#2349	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2349	2
C₂×C₃₁×#389	2
C₂×C₃₁×#385	2
C₂×C₃₁×#382	2
C₂×C₃₁×#380	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂²×C₃₁×#95	2
C₂×#2349	31

Subgroups — C2×C31×#2349 #21490870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2349 #21490870