Subgroups — C₂×C₃₁×#2337 #21490839

Order 7936.

3046 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2337	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#391	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#386	C₂	1	2	yes	maximal
7	C₂×C₃₁×#385	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#383	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
11	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#124	C₄	1	16	yes	complemented
15	C₂×C₃₁×#124	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#391	C₂²	1	16	yes	complemented
28	C₃₁×#388	C₂²	1	16	yes	complemented
29	C₃₁×#388	C₂²	1	16	yes	complemented
30	C₃₁×#386	C₂²	1	16	yes	—
31	C₃₁×#385	C₂²	1	16	yes	complemented
32	C₃₁×#383	C₂²	1	8	yes	complemented
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
34	C₄×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	complemented
35	C₂×C₃₁×#95	C₂²	1	2	yes	—
36	C₂×C₃₁×#95	C₂²	1	2	yes	—
37	C₂×C₃₁×#95	C₂²	1	2	yes	—
38	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#95	C₂²	1	2	yes	—
40	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#95	C₂²	1	2	yes	—
42	C₂⁵×C₃₁	C₂³	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	4	4	no	—
44	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
46	C₂²×D₄×C₃₁	—	4	4	no	—
47	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂³×C₄×C₃₁	—	2	8	no	—
50	C₂³×C₄×C₃₁	—	4	4	no	—
51	C₂³×C₄×C₃₁	—	4	4	no	—
52	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
61	C₃₁×#124	C₂×C₄	1	64	yes	complemented
62	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
63	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
71	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₄²×C₃₁	—	4	2	no	—
77	C₂×C₄²×C₃₁	—	4	2	no	—
78	C₂×C₄²×C₃₁	—	4	4	no	—
79	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
80	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
84	C₃₁×#95	C₂³	1	8	yes	—
85	C₃₁×#95	C₂³	1	8	yes	—
86	C₃₁×#95	C₂³	1	8	yes	—
87	C₃₁×#95	C₂³	1	4	yes	—
88	C₃₁×#95	C₂³	1	8	yes	—
89	C₃₁×#95	C₂³	1	4	yes	—
90	C₃₁×#95	C₂³	1	8	yes	—
91	C₂⁴×C₃₁	—	2	4	no	—
92	C₂⁴×C₃₁	—	2	4	no	—
93	C₂⁴×C₃₁	—	2	4	no	—
94	C₂⁴×C₃₁	—	2	4	no	—
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	2	4	no	—
97	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
98	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
99	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₂×Q₈×C₃₁	—	4	16	no	—
101	C₂×Q₈×C₃₁	—	4	8	no	—
102	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
103	C₂×D₄×C₃₁	—	4	16	no	—
104	C₂×D₄×C₃₁	—	4	8	no	—
105	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	16	no	—
124	C₂²×C₄×C₃₁	—	4	8	no	—
125	C₂²×C₄×C₃₁	—	4	8	no	—
126	C₂²×C₄×C₃₁	—	4	32	no	—
127	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
130	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
131	C₂²×C₄×C₃₁	#41	1	8	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
140	C₄⋊C₄×C₃₁	—	2	16	no	—
141	C₄⋊C₄×C₃₁	—	4	8	no	—
142	C₂²⋊C₄×C₃₁	—	2	16	no	—
143	C₂²⋊C₄×C₃₁	—	4	8	no	—
144	C₂²⋊C₄×C₃₁	—	4	8	no	—
145	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
146	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
147	C₄²×C₃₁	—	4	4	no	—
148	C₄²×C₃₁	—	4	4	no	—
149	C₄²×C₃₁	—	4	8	no	—
150	C₄²×C₃₁	C₂²×C₄	1	16	yes	complemented
151	C₂×#2337	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	16	no	—
153	C₂³×C₃₁	—	2	16	no	—
154	C₂³×C₃₁	—	2	16	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	4	no	—
157	C₂³×C₃₁	—	2	4	no	—
158	C₂³×C₃₁	—	2	4	no	—
159	C₂³×C₃₁	—	2	4	no	—
160	C₂³×C₃₁	—	2	4	no	—
161	C₂³×C₃₁	—	4	8	no	—
162	C₂³×C₃₁	—	4	8	no	—
163	C₂³×C₃₁	—	4	8	no	—
164	C₂³×C₃₁	—	4	8	no	—
165	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
166	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
167	C₂³×C₃₁	C₂×#41	1	4	yes	—
168	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
169	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
170	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
171	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
172	Q₈×C₃₁	—	4	16	no	—
173	D₄×C₃₁	—	4	16	no	—
174	C₂×C₄×C₃₁	—	2	8	no	—
175	C₂×C₄×C₃₁	—	2	8	no	—
176	C₂×C₄×C₃₁	—	2	16	no	—
177	C₂×C₄×C₃₁	—	2	8	no	—
178	C₂×C₄×C₃₁	—	2	16	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	4	32	no	—
197	C₂×C₄×C₃₁	—	4	8	no	—
198	C₂×C₄×C₃₁	—	4	8	no	—
199	C₂×C₄×C₃₁	—	4	8	no	—
200	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
201	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
202	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
203	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
204	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
205	C₂²×#124	C₂×C₃₁	1	1	yes	characteristic
206	#2337	C₂×C₃₁	1	16	yes	direct product factor
207	C₂×#391	C₂×C₃₁	1	2	yes	complemented
208	C₂×#388	C₂×C₃₁	1	2	yes	complemented
209	C₂×#388	C₂×C₃₁	1	2	yes	complemented
210	C₂×#386	C₂×C₃₁	1	2	yes	—
211	C₂×#385	C₂×C₃₁	1	2	yes	complemented
212	C₂×#383	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
214	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
215	C₂²×C₃₁	—	2	8	no	—
216	C₂²×C₃₁	—	2	8	no	—
217	C₂²×C₃₁	—	2	8	no	—
218	C₂²×C₃₁	—	4	32	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	4	4	no	—
222	C₂²×C₃₁	—	4	4	no	—
223	C₂²×C₃₁	C₄×#41	1	4	yes	—
224	C₂²×C₃₁	#518	1	4	yes	—
225	C₂²×C₃₁	#520	1	4	yes	—
226	C₂²×C₃₁	#538	1	8	yes	—
227	C₂²×C₃₁	#542	1	8	yes	—
228	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
229	C₂²×C₃₁	C₂²×#41	1	2	yes	—
230	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
231	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
233	C₄×C₃₁	—	2	4	no	—
234	C₄×C₃₁	—	2	4	no	—
235	C₄×C₃₁	—	2	8	no	—
236	C₄×C₃₁	—	2	4	no	—
237	C₄×C₃₁	—	2	8	no	—
238	C₄×C₃₁	—	4	4	no	—
239	C₄×C₃₁	—	4	4	no	—
240	C₄×C₃₁	—	4	4	no	—
241	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
244	C₂×#124	C₄×C₃₁	1	16	yes	complemented
245	C₂×#124	C₂²×C₃₁	1	8	yes	—
246	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
248	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
250	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
251	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
252	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
254	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
255	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
256	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
257	#391	C₂²×C₃₁	1	16	yes	complemented
258	#388	C₂²×C₃₁	1	16	yes	complemented
259	#388	C₂²×C₃₁	1	16	yes	complemented
260	#386	C₂²×C₃₁	1	16	yes	—
261	#385	C₂²×C₃₁	1	16	yes	complemented
262	#383	C₂²×C₃₁	1	8	yes	complemented
263	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
264	C₄×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	complemented
265	C₂×#95	C₂²×C₃₁	1	2	yes	—
266	C₂×#95	C₂²×C₃₁	1	2	yes	—
267	C₂×#95	C₂²×C₃₁	1	2	yes	—
268	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#95	C₂²×C₃₁	1	2	yes	—
270	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#95	C₂²×C₃₁	1	2	yes	—
272	C₂×C₃₁	—	4	8	no	—
273	C₂×C₃₁	#2337	1	8	yes	direct product factor
274	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
275	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
276	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
277	C₂×C₃₁	C₂×#538	1	2	yes	—
278	C₂×C₃₁	C₂×#542	1	2	yes	—
279	C₂⁵	C₂³×C₃₁	1	2	yes	—
280	C₂²×Q₈	—	4	4	no	—
281	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
282	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
283	C₂²×D₄	—	4	4	no	—
284	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
285	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
286	C₂³×C₄	—	2	8	no	—
287	C₂³×C₄	—	4	4	no	—
288	C₂³×C₄	—	4	4	no	—
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
291	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
292	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
294	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
297	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
298	#124	C₂×C₄×C₃₁	1	64	yes	complemented
299	C₂×C₄⋊C₄	—	2	8	no	—
300	C₂×C₄⋊C₄	—	4	4	no	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
303	C₂×C₂²⋊C₄	—	2	8	no	—
304	C₂×C₂²⋊C₄	—	4	4	no	—
305	C₂×C₂²⋊C₄	—	4	4	no	—
306	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
307	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
308	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
313	C₂×C₄²	—	4	2	no	—
314	C₂×C₄²	—	4	2	no	—
315	C₂×C₄²	—	4	4	no	—
316	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
317	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
318	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
319	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
320	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
321	#95	C₂³×C₃₁	1	8	yes	—
322	#95	C₂³×C₃₁	1	8	yes	—
323	#95	C₂³×C₃₁	1	8	yes	—
324	#95	C₂³×C₃₁	1	4	yes	—
325	#95	C₂³×C₃₁	1	8	yes	—
326	#95	C₂³×C₃₁	1	4	yes	—
327	#95	C₂³×C₃₁	1	8	yes	—
328	C₃₁	C₂×#2337	1	1	yes	characteristic, direct product factor, Sylow
329	C₂⁴	—	2	4	no	—
330	C₂⁴	—	2	4	no	—
331	C₂⁴	—	2	4	no	—
332	C₂⁴	—	2	4	no	—
333	C₂⁴	—	2	4	no	—
334	C₂⁴	—	2	4	no	—
335	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
336	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
337	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
338	C₂×Q₈	—	4	16	no	—
339	C₂×Q₈	—	4	8	no	—
340	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	complemented
341	C₂×D₄	—	4	16	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	16	no	—
362	C₂²×C₄	—	4	8	no	—
363	C₂²×C₄	—	4	8	no	—
364	C₂²×C₄	—	4	32	no	—
365	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
366	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
367	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
368	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
369	C₂²×C₄	#41×C₃₁	1	8	yes	—
370	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
372	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
374	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
375	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
376	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
378	C₄⋊C₄	—	2	16	no	—
379	C₄⋊C₄	—	4	8	no	—
380	C₂²⋊C₄	—	2	16	no	—
381	C₂²⋊C₄	—	4	8	no	—
382	C₂²⋊C₄	—	4	8	no	—
383	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
384	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
385	C₄²	—	4	4	no	—
386	C₄²	—	4	4	no	—
387	C₄²	—	4	8	no	—
388	C₄²	C₂²×C₄×C₃₁	1	16	yes	complemented
389	C₂³	—	2	16	no	—
390	C₂³	—	2	16	no	—
391	C₂³	—	2	16	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	2	4	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	2	4	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	4	8	no	—
399	C₂³	—	4	8	no	—
400	C₂³	—	4	8	no	—
401	C₂³	—	4	8	no	—
402	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
403	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
404	C₂³	C₂×#41×C₃₁	1	4	yes	—
405	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
406	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
407	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
408	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
409	Q₈	—	4	16	no	—
410	D₄	—	4	16	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	16	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	16	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	4	32	no	—
434	C₂×C₄	—	4	8	no	—
435	C₂×C₄	—	4	8	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
438	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
439	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
440	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
441	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
442	C₂²	—	2	8	no	—
443	C₂²	—	2	8	no	—
444	C₂²	—	2	8	no	—
445	C₂²	—	4	32	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	C₄×#41×C₃₁	1	4	yes	—
451	C₂²	C₃₁×#518	1	4	yes	—
452	C₂²	C₃₁×#520	1	4	yes	—
453	C₂²	C₃₁×#538	1	8	yes	—
454	C₂²	C₃₁×#542	1	8	yes	—
455	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
456	C₂²	C₂²×#41×C₃₁	1	2	yes	—
457	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
458	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
459	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
460	C₄	—	2	4	no	—
461	C₄	—	2	4	no	—
462	C₄	—	2	8	no	—
463	C₄	—	2	4	no	—
464	C₄	—	2	8	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	4	no	—
468	C₂	—	4	8	no	—
469	C₂	C₃₁×#2337	1	8	yes	direct product factor
470	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
471	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
472	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
473	C₂	C₂×C₃₁×#538	1	2	yes	—
474	C₂	C₂×C₃₁×#542	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
2. C₃₁×#2337	C₂	direct (N × H)
3. C₂×C₃₁×#391	C₂	semidirect
4. C₂×C₃₁×#388	C₂	semidirect
5. C₂×C₃₁×#388	C₂	semidirect
7. C₂×C₃₁×#385	C₂	semidirect
8. C₂×C₃₁×#383	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
14. C₂×C₃₁×#124	C₄	semidirect
27. C₃₁×#391	C₂²	semidirect
28. C₃₁×#388	C₂²	semidirect
29. C₃₁×#388	C₂²	semidirect
31. C₃₁×#385	C₂²	semidirect
32. C₃₁×#383	C₂²	semidirect
33. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
34. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
44. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
61. C₃₁×#124	C₂×C₄	semidirect
69. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
70. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
79. C₂×C₄²×C₃₁	C₂×C₄	semidirect
102. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
145. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
146. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
150. C₄²×C₃₁	C₂²×C₄	semidirect
151. C₂×#2337	C₃₁	direct (N × H)
206. #2337	C₂×C₃₁	direct (N × H)
207. C₂×#391	C₂×C₃₁	semidirect
208. C₂×#388	C₂×C₃₁	semidirect
209. C₂×#388	C₂×C₃₁	semidirect
211. C₂×#385	C₂×C₃₁	semidirect
212. C₂×#383	C₂×C₃₁	semidirect
213. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
214. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
244. C₂×#124	C₄×C₃₁	semidirect
257. #391	C₂²×C₃₁	semidirect
258. #388	C₂²×C₃₁	semidirect
259. #388	C₂²×C₃₁	semidirect
261. #385	C₂²×C₃₁	semidirect
262. #383	C₂²×C₃₁	semidirect
263. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
264. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
273. C₂×C₃₁	#2337	direct (N × H)
281. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
298. #124	C₂×C₄×C₃₁	semidirect
306. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
307. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
316. C₂×C₄²	C₂×C₄×C₃₁	semidirect
328. C₃₁	C₂×#2337	direct (N × H)
340. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
383. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
384. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
388. C₄²	C₂²×C₄×C₃₁	semidirect
469. C₂	C₃₁×#2337	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×C₃₁×#124	2
C₃₁×#2337	2
C₂×C₃₁×#391	2
C₂×C₃₁×#388	2
C₂×C₃₁×#386	2
C₂×C₃₁×#385	2
C₂×C₃₁×#383	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×#2337	31

Subgroups — C2×C31×#2337 #21490839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2337 #21490839