Subgroups — C₃₁×#23080 #21479839

Order 7936.

674 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3076	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1937	C₂	1	2	yes	maximal
7	C₃₁×#1929	C₂	1	2	yes	maximal
8	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1772	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#479	C₄	1	2	yes	complemented
15	C₃₁×#478	C₄	1	2	yes	complemented
16	C₃₁×#477	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#476	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#475	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#474	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#446	—	2	2	no	—
21	C₃₁×#446	—	2	1	no	—
22	C₈×Q₈×C₃₁	—	2	1	no	—
23	C₃₁×#433	—	2	2	no	—
24	C₃₁×#432	—	2	1	no	—
25	C₃₁×#431	—	2	1	no	—
26	C₂×C₃₁×#108	C₂²	1	2	yes	—
27	C₃₁×#427	C₂²	1	2	yes	—
28	C₃₁×#424	C₂²	1	2	yes	—
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
34	C₃₁×#416	C₂²	1	2	yes	—
35	C₃₁×#408	C₂²	1	2	yes	—
36	C₃₁×#405	C₂²	1	2	yes	—
37	C₃₁×#405	C₂²	1	2	yes	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
40	C₄³×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#341	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	1	yes	characteristic
43	C₃₁×#339	C₂²	1	2	yes	—
44	C₃₁×#339	C₂²	1	2	yes	—
45	C₃₁×#339	C₂²	1	1	yes	characteristic
46	C₃₁×#336	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	C₃₁×#125	C₂×C₄	1	2	yes	—
56	C₃₁×C₂²⋊Q₈	—	2	1	no	—
57	C₃₁×C₂²⋊Q₈	—	2	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	2	1	no	—
60	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₃₁	—	2	1	no	—
63	C₄×D₄×C₃₁	—	2	1	no	—
64	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#117	—	2	1	no	—
67	C₃₁×#117	C₂³	1	1	yes	characteristic
68	C₃₁×#117	C₂³	1	2	yes	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#108	—	2	4	no	—
76	C₃₁×#107	C₂×C₄	1	2	yes	—
77	C₃₁×#107	C₂×C₄	1	2	yes	—
78	C₃₁×#106	C₂×C₄	1	2	yes	—
79	C₃₁×#106	C₂×C₄	1	2	yes	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
87	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
88	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
89	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
90	C₃₁×#104	—	2	2	no	—
91	C₃₁×#104	—	2	2	no	—
92	C₃₁×#103	C₂×C₄	1	2	yes	—
93	C₃₁×#103	C₂×C₄	1	2	yes	—
94	C₃₁×#102	C₂×C₄	1	2	yes	—
95	C₃₁×#102	C₂×C₄	1	2	yes	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×#97	—	2	2	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#97	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	C₄×C₈×C₃₁	—	2	1	no	—
104	C₄×C₈×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	2	1	no	—
106	#41×C₃₁	—	2	1	no	—
107	#41×C₃₁	—	2	1	no	—
108	#41×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	M₁₆×C₃₁	—	2	1	no	—
123	M₁₆×C₃₁	—	2	1	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	—	2	1	no	—
126	M₁₆×C₃₁	—	2	2	no	—
127	M₁₆×C₃₁	#41	1	2	yes	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
146	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₃₁	—	2	1	no	—
150	C₂²⋊C₄×C₃₁	—	2	2	no	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	4	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₃₁	#41	1	2	yes	—
167	#23080	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	1	no	—
171	Q₈×C₃₁	—	2	1	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	Q₈×C₃₁	—	8	1	no	—
174	D₄×C₃₁	—	2	1	no	—
175	D₄×C₃₁	—	2	1	no	—
176	D₄×C₃₁	—	4	1	no	—
177	D₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	8	1	no	—
195	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
204	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	2	2	no	—
207	C₈×C₃₁	—	2	2	no	—
208	C₈×C₃₁	—	2	2	no	—
209	C₈×C₃₁	—	2	2	no	—
210	C₈×C₃₁	—	4	1	no	—
211	C₈×C₃₁	—	4	2	no	—
212	#3076	C₂×C₃₁	1	1	yes	characteristic
213	#2962	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#2925	C₂×C₃₁	1	1	yes	characteristic
215	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1937	C₂×C₃₁	1	2	yes	—
218	#1929	C₂×C₃₁	1	2	yes	—
219	#1922	C₂×C₃₁	1	2	yes	complemented
220	#1772	C₂×C₃₁	1	2	yes	complemented
221	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₄×#104	C₂×C₃₁	1	1	yes	characteristic
223	C₂²×C₃₁	—	2	1	no	—
224	C₂²×C₃₁	—	2	1	no	—
225	C₂²×C₃₁	—	4	1	no	—
226	C₂²×C₃₁	—	4	1	no	—
227	C₂²×C₃₁	—	4	1	no	—
228	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
229	C₂²×C₃₁	#552	1	1	yes	characteristic
230	C₂²×C₃₁	#556	1	1	yes	characteristic
231	C₄×C₃₁	—	2	4	no	—
232	C₄×C₃₁	—	2	1	no	—
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	4	2	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	1	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	1	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	#443	1	2	yes	—
242	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
243	C₄×C₃₁	#519	1	1	yes	characteristic
244	C₄×C₃₁	#547	1	1	yes	characteristic
245	C₄×C₃₁	#548	1	1	yes	characteristic
246	#533	C₂²×C₃₁	1	1	yes	characteristic
247	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
248	#479	C₄×C₃₁	1	2	yes	complemented
249	#478	C₄×C₃₁	1	2	yes	complemented
250	#477	C₄×C₃₁	1	1	yes	characteristic, complemented
251	#476	C₄×C₃₁	1	1	yes	characteristic, complemented
252	#475	C₄×C₃₁	1	1	yes	characteristic, complemented
253	#474	C₄×C₃₁	1	1	yes	characteristic, complemented
254	#446	—	2	2	no	—
255	#446	—	2	1	no	—
256	C₈×Q₈	—	2	1	no	—
257	#433	—	2	2	no	—
258	#432	—	2	1	no	—
259	#431	—	2	1	no	—
260	C₂×#108	C₂²×C₃₁	1	2	yes	—
261	#427	C₂²×C₃₁	1	2	yes	—
262	#424	C₂²×C₃₁	1	2	yes	—
263	#423	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
268	#416	C₂²×C₃₁	1	2	yes	—
269	#408	C₂²×C₃₁	1	2	yes	—
270	#405	C₂²×C₃₁	1	2	yes	—
271	#405	C₂²×C₃₁	1	2	yes	—
272	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
274	C₄³	C₂²×C₃₁	1	1	yes	characteristic
275	#341	C₂²×C₃₁	1	2	yes	—
276	#339	C₂²×C₃₁	1	1	yes	characteristic
277	#339	C₂²×C₃₁	1	2	yes	—
278	#339	C₂²×C₃₁	1	2	yes	—
279	#339	C₂²×C₃₁	1	1	yes	characteristic
280	#336	C₂²×C₃₁	1	2	yes	—
281	C₂×C₃₁	—	2	1	no	—
282	C₂×C₃₁	—	2	1	no	—
283	C₂×C₃₁	—	8	1	no	—
284	C₂×C₃₁	#2362	1	1	yes	characteristic
285	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
286	C₂×C₃₁	#2956	1	1	yes	characteristic
287	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
289	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
292	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
293	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
294	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
295	#125	C₂×C₄×C₃₁	1	2	yes	—
296	C₂²⋊Q₈	—	2	1	no	—
297	C₂²⋊Q₈	—	2	1	no	—
298	C₄×Q₈	—	2	1	no	—
299	C₄×Q₈	—	2	1	no	—
300	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
301	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₄×D₄	—	2	1	no	—
303	C₄×D₄	—	2	1	no	—
304	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
305	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
306	#117	—	2	1	no	—
307	#117	C₂³×C₃₁	1	1	yes	characteristic
308	#117	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
315	#108	—	2	4	no	—
316	#107	C₂×C₄×C₃₁	1	2	yes	—
317	#107	C₂×C₄×C₃₁	1	2	yes	—
318	#106	C₂×C₄×C₃₁	1	2	yes	—
319	#106	C₂×C₄×C₃₁	1	2	yes	—
320	C₄⋊C₈	—	2	2	no	—
321	C₄⋊C₈	—	2	2	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	—	2	1	no	—
324	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
326	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
327	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
328	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
330	#104	—	2	2	no	—
331	#104	—	2	2	no	—
332	#103	C₂×C₄×C₃₁	1	2	yes	—
333	#103	C₂×C₄×C₃₁	1	2	yes	—
334	#102	C₂×C₄×C₃₁	1	2	yes	—
335	#102	C₂×C₄×C₃₁	1	2	yes	—
336	C₂²⋊C₈	—	2	2	no	—
337	C₂²⋊C₈	—	2	2	no	—
338	#97	—	2	2	no	—
339	#97	—	2	2	no	—
340	#97	—	2	1	no	—
341	C₄×C₈	—	2	2	no	—
342	C₄×C₈	—	2	2	no	—
343	C₄×C₈	—	2	1	no	—
344	C₄×C₈	—	2	4	no	—
345	C₃₁	#23080	1	1	yes	characteristic, direct product factor, Sylow
346	#41	—	2	1	no	—
347	#41	—	2	1	no	—
348	#41	—	2	1	no	—
349	#41	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
363	M₁₆	—	2	1	no	—
364	M₁₆	—	2	1	no	—
365	M₁₆	—	2	2	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	#41×C₃₁	1	2	yes	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
379	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
387	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
388	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
389	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
390	C₂²⋊C₄	—	2	1	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	—	4	2	no	—
403	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
404	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₄²	#41×C₃₁	1	2	yes	—
408	C₂³	—	2	1	no	—
409	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
410	Q₈	—	2	1	no	—
411	Q₈	—	2	1	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	8	1	no	—
414	D₄	—	2	1	no	—
415	D₄	—	2	1	no	—
416	D₄	—	4	1	no	—
417	D₄	—	4	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	8	1	no	—
435	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
444	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	2	2	no	—
448	C₈	—	2	2	no	—
449	C₈	—	2	2	no	—
450	C₈	—	4	1	no	—
451	C₈	—	4	2	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
458	C₂²	C₃₁×#552	1	1	yes	characteristic
459	C₂²	C₃₁×#556	1	1	yes	characteristic
460	C₄	—	2	4	no	—
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	2	1	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	1	no	—
469	C₄	—	4	2	no	—
470	C₄	C₃₁×#443	1	2	yes	—
471	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
472	C₄	C₃₁×#519	1	1	yes	characteristic
473	C₄	C₃₁×#547	1	1	yes	characteristic
474	C₄	C₃₁×#548	1	1	yes	characteristic
475	C₂	—	2	1	no	—
476	C₂	—	2	1	no	—
477	C₂	—	8	1	no	—
478	C₂	C₃₁×#2362	1	1	yes	characteristic
479	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
480	C₂	C₃₁×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2962	C₂	semidirect
4. C₃₁×#1994	C₂	semidirect
5. C₃₁×#1950	C₂	semidirect
8. C₃₁×#1922	C₂	semidirect
9. C₃₁×#1772	C₂	semidirect
10. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
14. C₃₁×#479	C₄	semidirect
15. C₃₁×#478	C₄	semidirect
16. C₃₁×#477	C₄	semidirect
17. C₃₁×#476	C₄	semidirect
18. C₃₁×#475	C₄	semidirect
19. C₃₁×#474	C₄	semidirect
167. #23080	C₃₁	direct (N × H)
213. #2962	C₂×C₃₁	semidirect
215. #1994	C₂×C₃₁	semidirect
216. #1950	C₂×C₃₁	semidirect
219. #1922	C₂×C₃₁	semidirect
220. #1772	C₂×C₃₁	semidirect
221. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
248. #479	C₄×C₃₁	semidirect
249. #478	C₄×C₃₁	semidirect
250. #477	C₄×C₃₁	semidirect
251. #476	C₄×C₃₁	semidirect
252. #475	C₄×C₃₁	semidirect
253. #474	C₄×C₃₁	semidirect
345. C₃₁	#23080	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×#3076	2
C₃₁×#2962	2
C₃₁×#2925	2
C₃₁×#1994	2
C₃₁×#1950	2
C₃₁×#1937	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1772	2
C₄×C₃₁×C₄⋊C₈	2
C₄×C₃₁×#104	2
#23080	31

Subgroups — C31×#23080 #21479839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23080 #21479839