Subgroups — C₂×C₃₁×#3054 #21517200

Order 7936.

2918 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3054	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#470	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#469	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#468	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#416	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#135	—	2	8	no	—
13	C₂×C₃₁×#135	C₂²	1	8	yes	complemented
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
16	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×M₁₆×C₃₁	—	2	8	no	—
18	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₃₁	C₂²	1	8	yes	complemented
21	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#525	—	2	8	no	—
24	C₃₁×#525	C₂²	1	16	yes	complemented
25	C₂×C₃₁×#123	—	2	2	no	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₂×C₃₁×#121	C₂²	1	4	yes	—
28	C₂×C₃₁×#120	—	2	2	no	—
29	C₂×C₄×D₄×C₃₁	—	2	4	no	—
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₃₁×#470	C₂²	1	16	yes	complemented
32	C₃₁×#469	C₂²	1	16	yes	complemented
33	C₃₁×#468	C₂²	1	32	yes	complemented
34	C₃₁×#427	C₂²	1	8	yes	complemented
35	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
36	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
37	C₃₁×#416	C₂²	1	16	yes	complemented
38	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
39	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
40	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
41	C₂×#41×C₃₁	—	2	4	no	—
42	C₂×#41×C₃₁	—	2	8	no	—
43	C₂×#41×C₃₁	—	2	8	no	—
44	C₂×#41×C₃₁	C₂³	1	8	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₃₁	—	2	4	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₃₁	—	2	4	no	—
50	C₂³×C₄×C₃₁	—	2	4	no	—
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	D₄	1	4	yes	—
54	C₂³×C₄×C₃₁	D₄	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#135	—	2	32	no	—
59	C₂×Q₁₆×C₃₁	—	2	4	no	—
60	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	complemented
61	C₂×SD₁₆×C₃₁	—	2	8	no	—
62	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
63	C₂×D₈×C₃₁	—	2	4	no	—
64	C₂×D₈×C₃₁	C₂³	1	4	yes	complemented
65	C₂×M₁₆×C₃₁	—	2	8	no	—
66	C₂×M₁₆×C₃₁	—	2	8	no	—
67	C₂×M₁₆×C₃₁	—	2	32	no	—
68	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
70	C₂²×C₈×C₃₁	—	2	4	no	—
71	C₂²×C₈×C₃₁	D₄	1	8	yes	complemented
72	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
73	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
77	C₃₁×#123	—	2	16	no	—
78	C₃₁×C₂²⋊Q₈	—	2	8	no	—
79	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
80	C₃₁×#121	—	2	8	no	—
81	C₃₁×#121	C₂³	1	16	yes	—
82	C₃₁×#120	—	2	16	no	—
83	C₄×D₄×C₃₁	—	2	32	no	—
84	C₃₁×#117	—	2	4	no	—
85	C₃₁×#117	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
91	C₂×C₄²×C₃₁	—	2	2	no	—
92	C₃₁×#107	C₂³	1	8	yes	complemented
93	C₃₁×#106	C₂³	1	8	yes	complemented
94	C₃₁×#103	C₂³	1	16	yes	complemented
95	C₃₁×#102	C₂³	1	16	yes	complemented
96	C₂⁴×C₃₁	—	2	4	no	—
97	C₂⁴×C₃₁	—	2	2	no	—
98	C₂⁴×C₃₁	—	4	4	no	—
99	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
102	#41×C₃₁	—	2	16	no	—
103	#41×C₃₁	—	2	16	no	—
104	#41×C₃₁	—	4	16	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	4	no	—
108	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
109	C₂×D₄×C₃₁	—	2	16	no	—
110	C₂×D₄×C₃₁	—	2	8	no	—
111	C₂×D₄×C₃₁	—	2	4	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	4	no	—
114	C₂×D₄×C₃₁	—	4	16	no	—
115	C₂×D₄×C₃₁	—	4	8	no	—
116	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	8	no	—
119	C₂²×C₄×C₃₁	—	2	16	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	8	no	—
127	C₂²×C₄×C₃₁	—	4	8	no	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	#41	1	4	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
139	Q₁₆×C₃₁	—	2	8	no	—
140	SD₁₆×C₃₁	—	2	16	no	—
141	D₈×C₃₁	—	2	8	no	—
142	M₁₆×C₃₁	—	2	16	no	—
143	M₁₆×C₃₁	—	2	16	no	—
144	C₂×C₈×C₃₁	—	2	4	no	—
145	C₂×C₈×C₃₁	—	2	8	no	—
146	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
147	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
148	C₂×C₈×C₃₁	C₂×D₄	1	16	yes	complemented
149	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
150	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
151	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
152	C₄⋊C₄×C₃₁	—	2	16	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
155	C₂²⋊C₄×C₃₁	—	2	8	no	—
156	C₂²⋊C₄×C₃₁	—	2	16	no	—
157	C₂²⋊C₄×C₃₁	—	2	16	no	—
158	C₄²×C₃₁	—	2	8	no	—
159	C₂×#3054	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	4	no	—
161	C₂³×C₃₁	—	2	8	no	—
162	C₂³×C₃₁	—	2	4	no	—
163	C₂³×C₃₁	—	2	8	no	—
164	C₂³×C₃₁	—	2	4	no	—
165	C₂³×C₃₁	—	4	8	no	—
166	C₂³×C₃₁	—	4	16	no	—
167	C₂³×C₃₁	—	4	4	no	—
168	C₂³×C₃₁	—	4	8	no	—
169	C₂³×C₃₁	#121	1	4	yes	—
170	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
171	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₃₁	—	2	4	no	—
174	Q₈×C₃₁	—	2	4	no	—
175	Q₈×C₃₁	—	4	4	no	—
176	D₄×C₃₁	—	2	4	no	—
177	D₄×C₃₁	—	2	4	no	—
178	D₄×C₃₁	—	4	32	no	—
179	D₄×C₃₁	—	4	16	no	—
180	D₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	—	2	8	no	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	2	16	no	—
184	C₂×C₄×C₃₁	—	2	8	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	2	8	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	4	4	no	—
190	C₂×C₄×C₃₁	—	4	8	no	—
191	C₂×C₄×C₃₁	—	4	16	no	—
192	C₂×C₄×C₃₁	#121	1	4	yes	—
193	C₂×C₄×C₃₁	#121	1	8	yes	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
199	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
201	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	C₂²×D₄	1	4	yes	—
204	C₈×C₃₁	C₂²×D₄	1	4	yes	—
205	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂×#525	C₂×C₃₁	1	2	yes	complemented
208	#3054	C₂×C₃₁	1	16	yes	direct product factor
209	C₂×#470	C₂×C₃₁	1	2	yes	complemented
210	C₂×#469	C₂×C₃₁	1	2	yes	complemented
211	C₂×#468	C₂×C₃₁	1	4	yes	complemented
212	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂×#416	C₂×C₃₁	1	2	yes	complemented
214	C₂²×C₃₁	—	2	8	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	—	2	8	no	—
218	C₂²×C₃₁	—	4	16	no	—
219	C₂²×C₃₁	—	4	8	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	4	8	no	—
222	C₂²×C₃₁	—	8	8	no	—
223	C₂²×C₃₁	C₂×#121	1	4	yes	—
224	C₂²×C₃₁	C₂×#121	1	2	yes	—
225	C₂²×C₃₁	#575	1	4	yes	—
226	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
227	C₄×C₃₁	—	2	4	no	—
228	C₄×C₃₁	—	4	4	no	—
229	C₄×C₃₁	—	4	8	no	—
230	C₄×C₃₁	C₂×#121	1	4	yes	—
231	C₄×C₃₁	C₂×#121	1	2	yes	—
232	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
233	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
234	C₂²×#41	C₂²×C₃₁	1	2	yes	—
235	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#135	—	2	8	no	—
237	C₂×#135	C₂²×C₃₁	1	8	yes	complemented
238	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
240	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
241	C₂²×M₁₆	—	2	8	no	—
242	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
243	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
244	C₂²×M₁₆	C₂²×C₃₁	1	8	yes	complemented
245	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
247	#525	—	2	8	no	—
248	#525	C₂²×C₃₁	1	16	yes	complemented
249	C₂×#123	—	2	2	no	—
250	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
251	C₂×#121	C₂²×C₃₁	1	4	yes	—
252	C₂×#120	—	2	2	no	—
253	C₂×C₄×D₄	—	2	4	no	—
254	C₂×#117	C₂²×C₃₁	1	2	yes	—
255	#470	C₂²×C₃₁	1	16	yes	complemented
256	#469	C₂²×C₃₁	1	16	yes	complemented
257	#468	C₂²×C₃₁	1	32	yes	complemented
258	#427	C₂²×C₃₁	1	8	yes	complemented
259	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
260	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
261	#416	C₂²×C₃₁	1	16	yes	complemented
262	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
263	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
264	C₂×C₃₁	—	2	4	no	—
265	C₂×C₃₁	—	4	4	no	—
266	C₂×C₃₁	—	8	4	no	—
267	C₂×C₃₁	#3054	1	4	yes	direct product factor
268	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
269	C₂×C₃₁	C₂×#575	1	2	yes	—
270	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
271	C₂×#41	—	2	4	no	—
272	C₂×#41	—	2	8	no	—
273	C₂×#41	—	2	8	no	—
274	C₂×#41	C₂³×C₃₁	1	8	yes	—
275	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
276	C₂²×D₄	—	2	4	no	—
277	C₂²×D₄	—	2	2	no	—
278	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
279	C₂³×C₄	—	2	4	no	—
280	C₂³×C₄	—	2	4	no	—
281	C₂³×C₄	—	2	2	no	—
282	C₂³×C₄	D₄×C₃₁	1	2	yes	—
283	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
284	C₂³×C₄	D₄×C₃₁	1	4	yes	—
285	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
286	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	#135	—	2	32	no	—
289	C₂×Q₁₆	—	2	4	no	—
290	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	complemented
291	C₂×SD₁₆	—	2	8	no	—
292	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
293	C₂×D₈	—	2	4	no	—
294	C₂×D₈	C₂³×C₃₁	1	4	yes	complemented
295	C₂×M₁₆	—	2	8	no	—
296	C₂×M₁₆	—	2	8	no	—
297	C₂×M₁₆	—	2	32	no	—
298	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
299	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
300	C₂²×C₈	—	2	4	no	—
301	C₂²×C₈	D₄×C₃₁	1	8	yes	complemented
302	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
303	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
304	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
307	#123	—	2	16	no	—
308	C₂²⋊Q₈	—	2	8	no	—
309	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
310	#121	—	2	8	no	—
311	#121	C₂³×C₃₁	1	16	yes	—
312	#120	—	2	16	no	—
313	C₄×D₄	—	2	32	no	—
314	#117	—	2	4	no	—
315	#117	C₂³×C₃₁	1	8	yes	—
316	C₂×C₄⋊C₄	—	2	4	no	—
317	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	4	no	—
320	C₂×C₂²⋊C₄	—	2	4	no	—
321	C₂×C₄²	—	2	2	no	—
322	#107	C₂³×C₃₁	1	8	yes	complemented
323	#106	C₂³×C₃₁	1	8	yes	complemented
324	#103	C₂³×C₃₁	1	16	yes	complemented
325	#102	C₂³×C₃₁	1	16	yes	complemented
326	C₃₁	C₂×#3054	1	1	yes	characteristic, direct product factor, Sylow
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	2	2	no	—
329	C₂⁴	—	4	4	no	—
330	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
331	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
333	#41	—	2	16	no	—
334	#41	—	2	16	no	—
335	#41	—	4	16	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	4	no	—
338	C₂×Q₈	—	2	4	no	—
339	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
340	C₂×D₄	—	2	16	no	—
341	C₂×D₄	—	2	8	no	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	4	16	no	—
346	C₂×D₄	—	4	8	no	—
347	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	2	16	no	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	8	no	—
358	C₂²×C₄	—	4	8	no	—
359	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
366	C₂²×C₄	#41×C₃₁	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
368	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
370	Q₁₆	—	2	8	no	—
371	SD₁₆	—	2	16	no	—
372	D₈	—	2	8	no	—
373	M₁₆	—	2	16	no	—
374	M₁₆	—	2	16	no	—
375	C₂×C₈	—	2	4	no	—
376	C₂×C₈	—	2	8	no	—
377	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
378	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
379	C₂×C₈	C₂×D₄×C₃₁	1	16	yes	complemented
380	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
381	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
382	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
383	C₄⋊C₄	—	2	16	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
386	C₂²⋊C₄	—	2	8	no	—
387	C₂²⋊C₄	—	2	16	no	—
388	C₂²⋊C₄	—	2	16	no	—
389	C₄²	—	2	8	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	8	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	2	8	no	—
394	C₂³	—	2	4	no	—
395	C₂³	—	4	8	no	—
396	C₂³	—	4	16	no	—
397	C₂³	—	4	4	no	—
398	C₂³	—	4	8	no	—
399	C₂³	C₃₁×#121	1	4	yes	—
400	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
401	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
402	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
403	Q₈	—	2	4	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	4	no	—
406	D₄	—	2	4	no	—
407	D₄	—	2	4	no	—
408	D₄	—	4	32	no	—
409	D₄	—	4	16	no	—
410	D₄	—	4	4	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	16	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	4	no	—
420	C₂×C₄	—	4	8	no	—
421	C₂×C₄	—	4	16	no	—
422	C₂×C₄	C₃₁×#121	1	4	yes	—
423	C₂×C₄	C₃₁×#121	1	8	yes	—
424	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
428	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
429	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
430	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
431	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
432	C₈	—	2	4	no	—
433	C₈	C₂²×D₄×C₃₁	1	4	yes	—
434	C₈	C₂²×D₄×C₃₁	1	4	yes	—
435	C₂²	—	2	8	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	2	8	no	—
439	C₂²	—	4	16	no	—
440	C₂²	—	4	8	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	8	no	—
443	C₂²	—	8	8	no	—
444	C₂²	C₂×C₃₁×#121	1	4	yes	—
445	C₂²	C₂×C₃₁×#121	1	2	yes	—
446	C₂²	C₃₁×#575	1	4	yes	—
447	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
448	C₄	—	2	4	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	8	no	—
451	C₄	C₂×C₃₁×#121	1	4	yes	—
452	C₄	C₂×C₃₁×#121	1	2	yes	—
453	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
454	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
455	C₂	—	2	4	no	—
456	C₂	—	4	4	no	—
457	C₂	—	8	4	no	—
458	C₂	C₃₁×#3054	1	4	yes	direct product factor
459	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
460	C₂	C₂×C₃₁×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#135	C₂	semidirect
2. C₂³×M₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#525	C₂	semidirect
4. C₃₁×#3054	C₂	direct (N × H)
5. C₂×C₃₁×#470	C₂	semidirect
6. C₂×C₃₁×#469	C₂	semidirect
7. C₂×C₃₁×#468	C₂	semidirect
8. C₂×C₃₁×#427	C₂	semidirect
9. C₂×C₃₁×#416	C₂	semidirect
13. C₂×C₃₁×#135	C₂²	semidirect
14. C₂²×Q₁₆×C₃₁	C₂²	semidirect
15. C₂²×SD₁₆×C₃₁	C₂²	semidirect
16. C₂²×D₈×C₃₁	C₂²	semidirect
20. C₂²×M₁₆×C₃₁	C₂²	semidirect
24. C₃₁×#525	C₂²	semidirect
31. C₃₁×#470	C₂²	semidirect
32. C₃₁×#469	C₂²	semidirect
33. C₃₁×#468	C₂²	semidirect
34. C₃₁×#427	C₂²	semidirect
35. C₂×C₃₁×#107	C₂²	semidirect
36. C₂×C₃₁×#106	C₂²	semidirect
37. C₃₁×#416	C₂²	semidirect
38. C₂×C₃₁×#103	C₂²	semidirect
39. C₂×C₃₁×#102	C₂²	semidirect
60. C₂×Q₁₆×C₃₁	C₂³	semidirect
62. C₂×SD₁₆×C₃₁	C₂³	semidirect
64. C₂×D₈×C₃₁	C₂³	semidirect
71. C₂²×C₈×C₃₁	D₄	semidirect
92. C₃₁×#107	C₂³	semidirect
93. C₃₁×#106	C₂³	semidirect
94. C₃₁×#103	C₂³	semidirect
95. C₃₁×#102	C₂³	semidirect
148. C₂×C₈×C₃₁	C₂×D₄	semidirect
159. C₂×#3054	C₃₁	direct (N × H)
205. C₂²×#135	C₂×C₃₁	semidirect
206. C₂³×M₁₆	C₂×C₃₁	semidirect
207. C₂×#525	C₂×C₃₁	semidirect
208. #3054	C₂×C₃₁	direct (N × H)
209. C₂×#470	C₂×C₃₁	semidirect
210. C₂×#469	C₂×C₃₁	semidirect
211. C₂×#468	C₂×C₃₁	semidirect
212. C₂×#427	C₂×C₃₁	semidirect
213. C₂×#416	C₂×C₃₁	semidirect
237. C₂×#135	C₂²×C₃₁	semidirect
238. C₂²×Q₁₆	C₂²×C₃₁	semidirect
239. C₂²×SD₁₆	C₂²×C₃₁	semidirect
240. C₂²×D₈	C₂²×C₃₁	semidirect
244. C₂²×M₁₆	C₂²×C₃₁	semidirect
248. #525	C₂²×C₃₁	semidirect
255. #470	C₂²×C₃₁	semidirect
256. #469	C₂²×C₃₁	semidirect
257. #468	C₂²×C₃₁	semidirect
258. #427	C₂²×C₃₁	semidirect
259. C₂×#107	C₂²×C₃₁	semidirect
260. C₂×#106	C₂²×C₃₁	semidirect
261. #416	C₂²×C₃₁	semidirect
262. C₂×#103	C₂²×C₃₁	semidirect
263. C₂×#102	C₂²×C₃₁	semidirect
267. C₂×C₃₁	#3054	direct (N × H)
290. C₂×Q₁₆	C₂³×C₃₁	semidirect
292. C₂×SD₁₆	C₂³×C₃₁	semidirect
294. C₂×D₈	C₂³×C₃₁	semidirect
301. C₂²×C₈	D₄×C₃₁	semidirect
322. #107	C₂³×C₃₁	semidirect
323. #106	C₂³×C₃₁	semidirect
324. #103	C₂³×C₃₁	semidirect
325. #102	C₂³×C₃₁	semidirect
326. C₃₁	C₂×#3054	direct (N × H)
379. C₂×C₈	C₂×D₄×C₃₁	semidirect
458. C₂	C₃₁×#3054	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×#135	2
C₂³×M₁₆×C₃₁	2
C₂×C₃₁×#525	2
C₃₁×#3054	2
C₂×C₃₁×#470	2
C₂×C₃₁×#469	2
C₂×C₃₁×#468	2
C₂×C₃₁×#427	2
C₂×C₃₁×#416	2
C₂×#3054	31

Subgroups — C2×C31×#3054 #21517200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3054 #21517200