Subgroups — C₃₁×#17764 #21474523

Order 7936.

964 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1560	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1559	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1551	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1550	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#556	C₄	1	2	yes	—
14	C₃₁×#555	—	2	1	no	—
15	C₃₁×#554	—	2	1	no	—
16	C₃₁×#553	—	2	2	no	—
17	C₃₁×#548	C₄	1	4	yes	—
18	C₃₁×#547	C₄	1	2	yes	—
19	C₃₁×#542	—	2	2	no	—
20	C₃₁×#537	—	2	1	no	—
21	C₃₁×#533	C₂²	1	2	yes	—
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#520	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#436	—	2	1	no	—
29	C₃₁×#435	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#424	—	2	1	no	—
33	C₃₁×#424	—	2	2	no	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	2	yes	—
36	C₃₁×#423	C₂²	1	2	yes	—
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
40	C₃₁×#333	C₂²	1	2	yes	—
41	C₃₁×#333	C₂²	1	2	yes	—
42	C₃₁×#333	C₂²	1	2	yes	—
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#332	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	1	yes	characteristic
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	1	yes	characteristic
48	C₃₁×#331	C₂²	1	1	yes	characteristic
49	C₃₁×#331	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	—	2	1	no	—
51	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
54	C₃₁×#131	—	4	4	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×#127	C₂³	1	1	yes	characteristic
67	C₃₁×#126	—	2	4	no	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#124	—	2	4	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	C₂×C₄	1	4	yes	—
72	C₃₁×#123	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	—	2	2	no	—
74	C₃₁×C₂²⋊Q₈	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
77	C₃₁×#121	—	2	2	no	—
78	C₃₁×#121	—	2	4	no	—
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
85	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
87	C₄×D₄×C₃₁	—	2	4	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	1	no	—
91	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
94	C₄×D₄×C₃₁	D₄	1	2	yes	—
95	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
96	C₃₁×#117	—	2	2	no	—
97	C₃₁×#117	—	2	2	no	—
98	C₃₁×#117	D₄	1	1	yes	characteristic
99	C₃₁×#117	D₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	—	2	1	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	1	no	—
110	C₃₁×C₄⋊C₈	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×C₂²⋊C₈	—	2	1	no	—
118	C₃₁×C₂²⋊C₈	—	2	2	no	—
119	C₃₁×#97	—	2	2	no	—
120	C₄×C₈×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	4	2	no	—
124	#41×C₃₁	—	4	2	no	—
125	#41×C₃₁	—	8	4	no	—
126	C₂×Q₈×C₃₁	—	2	2	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	8	2	no	—
129	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
130	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
136	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	M₁₆×C₃₁	—	4	4	no	—
148	M₁₆×C₃₁	—	4	2	no	—
149	M₁₆×C₃₁	—	8	2	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₂×C₈×C₃₁	—	4	2	no	—
158	C₂×C₈×C₃₁	—	8	1	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
169	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
170	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	—	2	4	no	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
177	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
178	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
179	C₄²×C₃₁	—	2	4	no	—
180	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
182	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂×D₄	1	2	yes	—
184	C₄²×C₃₁	C₂×D₄	1	2	yes	—
185	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
187	#17764	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	Q₈×C₃₁	—	2	2	no	—
192	Q₈×C₃₁	—	4	4	no	—
193	Q₈×C₃₁	—	4	1	no	—
194	Q₈×C₃₁	—	4	1	no	—
195	Q₈×C₃₁	—	8	2	no	—
196	D₄×C₃₁	—	2	4	no	—
197	D₄×C₃₁	—	2	2	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	4	4	no	—
200	D₄×C₃₁	—	4	1	no	—
201	D₄×C₃₁	—	4	1	no	—
202	D₄×C₃₁	—	4	1	no	—
203	D₄×C₃₁	—	4	1	no	—
204	D₄×C₃₁	—	8	4	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	8	4	no	—
217	C₂×C₄×C₃₁	—	8	2	no	—
218	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
219	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
225	C₂×C₄×C₃₁	#137	1	2	yes	—
226	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	2	no	—
231	C₈×C₃₁	—	4	1	no	—
232	C₈×C₃₁	—	4	2	no	—
233	C₈×C₃₁	—	4	1	no	—
234	C₈×C₃₁	—	4	1	no	—
235	#3511	C₂×C₃₁	1	1	yes	characteristic
236	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1560	C₂×C₃₁	1	2	yes	complemented
239	#1559	C₂×C₃₁	1	2	yes	complemented
240	#1551	C₂×C₃₁	1	2	yes	complemented
241	#1550	C₂×C₃₁	1	2	yes	complemented
242	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	2	no	—
249	C₂²×C₃₁	—	2	2	no	—
250	C₂²×C₃₁	—	2	2	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	#417	1	2	yes	—
255	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
256	C₄×C₃₁	—	2	1	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	4	4	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	#417	1	2	yes	—
265	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
267	C₄×C₃₁	#577	1	1	yes	characteristic
268	C₄×C₃₁	#578	1	1	yes	characteristic
269	C₂×#131	—	2	1	no	—
270	#556	C₄×C₃₁	1	2	yes	—
271	#555	—	2	1	no	—
272	#554	—	2	1	no	—
273	#553	—	2	2	no	—
274	#548	C₄×C₃₁	1	4	yes	—
275	#547	C₄×C₃₁	1	2	yes	—
276	#542	—	2	2	no	—
277	#537	—	2	1	no	—
278	#533	C₂²×C₃₁	1	2	yes	—
279	#532	C₂²×C₃₁	1	2	yes	—
280	#532	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
282	#520	—	2	1	no	—
283	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
284	#446	—	2	1	no	—
285	#436	—	2	1	no	—
286	#435	—	2	2	no	—
287	#432	C₂²×C₃₁	1	2	yes	—
288	#431	C₂²×C₃₁	1	1	yes	characteristic
289	#424	—	2	1	no	—
290	#424	—	2	2	no	—
291	#423	C₂²×C₃₁	1	1	yes	characteristic
292	#423	C₂²×C₃₁	1	2	yes	—
293	#423	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
297	#333	C₂²×C₃₁	1	2	yes	—
298	#333	C₂²×C₃₁	1	2	yes	—
299	#333	C₂²×C₃₁	1	2	yes	—
300	#333	C₂²×C₃₁	1	2	yes	—
301	#332	C₂²×C₃₁	1	1	yes	characteristic
302	#332	C₂²×C₃₁	1	1	yes	characteristic
303	#332	C₂²×C₃₁	1	2	yes	—
304	#331	C₂²×C₃₁	1	1	yes	characteristic
305	#331	C₂²×C₃₁	1	1	yes	characteristic
306	#331	C₂²×C₃₁	1	2	yes	—
307	C₂×C₃₁	—	2	2	no	—
308	C₂×C₃₁	—	4	2	no	—
309	C₂×C₃₁	—	8	2	no	—
310	C₂×C₃₁	#2888	1	1	yes	characteristic
311	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
312	C₂×C₃₁	#2898	1	1	yes	characteristic
313	C₂×#41	—	2	1	no	—
314	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
315	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
316	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
317	#131	—	4	4	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	1	no	—
320	C₂×M₁₆	—	2	2	no	—
321	C₂²×C₈	—	2	2	no	—
322	C₂²×C₈	—	2	1	no	—
323	C₂²×C₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	1	no	—
325	C₄⋊Q₈	—	2	1	no	—
326	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
327	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
328	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
329	#127	C₂³×C₃₁	1	1	yes	characteristic
330	#126	—	2	4	no	—
331	#125	—	2	2	no	—
332	#124	—	2	4	no	—
333	#124	—	2	1	no	—
334	#124	C₂×C₄×C₃₁	1	4	yes	—
335	#123	—	2	4	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₂²⋊Q₈	—	2	4	no	—
339	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
340	#121	—	2	2	no	—
341	#121	—	2	4	no	—
342	#121	—	2	2	no	—
343	#121	—	2	1	no	—
344	#121	—	2	2	no	—
345	#121	C₂×C₄×C₃₁	1	2	yes	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
348	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
349	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
350	C₄×D₄	—	2	4	no	—
351	C₄×D₄	—	2	2	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	—	2	1	no	—
354	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
355	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
357	C₄×D₄	D₄×C₃₁	1	2	yes	—
358	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
359	#117	—	2	2	no	—
360	#117	—	2	2	no	—
361	#117	D₄×C₃₁	1	1	yes	characteristic
362	#117	D₄×C₃₁	1	2	yes	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
368	C₄⋊C₈	—	2	2	no	—
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	—	2	2	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	2	no	—
374	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
377	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
378	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²⋊C₈	—	2	1	no	—
381	C₂²⋊C₈	—	2	2	no	—
382	#97	—	2	2	no	—
383	C₄×C₈	—	2	1	no	—
384	C₃₁	#17764	1	1	yes	characteristic, direct product factor, Sylow
385	#41	—	2	2	no	—
386	#41	—	2	2	no	—
387	#41	—	4	2	no	—
388	#41	—	4	2	no	—
389	#41	—	8	4	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	8	2	no	—
393	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
394	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
400	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
401	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
407	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
411	M₁₆	—	4	4	no	—
412	M₁₆	—	4	2	no	—
413	M₁₆	—	8	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	4	4	no	—
421	C₂×C₈	—	4	2	no	—
422	C₂×C₈	—	8	1	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
431	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
432	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
433	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
434	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
441	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
442	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
443	C₄²	—	2	4	no	—
444	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
446	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₃₁	1	2	yes	—
448	C₄²	C₂×D₄×C₃₁	1	2	yes	—
449	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂³	—	2	2	no	—
452	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
453	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	Q₈	—	2	2	no	—
455	Q₈	—	4	4	no	—
456	Q₈	—	4	1	no	—
457	Q₈	—	4	1	no	—
458	Q₈	—	8	2	no	—
459	D₄	—	2	4	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	4	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	1	no	—
466	D₄	—	4	1	no	—
467	D₄	—	8	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	2	no	—
481	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
482	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
484	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
486	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
488	C₂×C₄	C₃₁×#137	1	2	yes	—
489	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₈	—	4	1	no	—
493	C₈	—	4	2	no	—
494	C₈	—	4	1	no	—
495	C₈	—	4	2	no	—
496	C₈	—	4	1	no	—
497	C₈	—	4	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	C₃₁×#417	1	2	yes	—
507	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
508	C₄	—	2	1	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	1	no	—
511	C₄	—	2	2	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	2	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	2	no	—
516	C₄	C₃₁×#417	1	2	yes	—
517	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
518	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
519	C₄	C₃₁×#577	1	1	yes	characteristic
520	C₄	C₃₁×#578	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	4	2	no	—
523	C₂	—	8	2	no	—
524	C₂	C₃₁×#2888	1	1	yes	characteristic
525	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
526	C₂	C₃₁×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2989	C₂	semidirect
3. C₃₁×#2906	C₂	semidirect
4. C₃₁×#1560	C₂	semidirect
5. C₃₁×#1559	C₂	semidirect
6. C₃₁×#1551	C₂	semidirect
7. C₃₁×#1550	C₂	semidirect
8. C₃₁×#1548	C₂	semidirect
9. C₃₁×#1547	C₂	semidirect
10. C₂×C₃₁×#333	C₂	semidirect
11. C₃₁×#1540	C₂	semidirect
187. #17764	C₃₁	direct (N × H)
236. #2989	C₂×C₃₁	semidirect
237. #2906	C₂×C₃₁	semidirect
238. #1560	C₂×C₃₁	semidirect
239. #1559	C₂×C₃₁	semidirect
240. #1551	C₂×C₃₁	semidirect
241. #1550	C₂×C₃₁	semidirect
242. #1548	C₂×C₃₁	semidirect
243. #1547	C₂×C₃₁	semidirect
244. C₂×#333	C₂×C₃₁	semidirect
245. #1540	C₂×C₃₁	semidirect
384. C₃₁	#17764	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3511	2
C₃₁×#2989	2
C₃₁×#2906	2
C₃₁×#1560	2
C₃₁×#1559	2
C₃₁×#1551	2
C₃₁×#1550	2
C₃₁×#1548	2
C₃₁×#1547	2
C₂×C₃₁×#333	2
C₃₁×#1540	2
#17764	31

Subgroups — C31×#17764 #21474523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17764 #21474523