Subgroups — C₃₁×#23042 #21479801

Order 7936.

824 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3067	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2956	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1932	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1881	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1858	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₃₁×#568	C₂²	1	2	yes	—
15	C₃₁×#568	C₂²	1	2	yes	—
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₃₁×#473	C₄	1	2	yes	complemented
18	C₃₁×#472	C₄	1	2	yes	complemented
19	C₃₁×#471	C₄	1	4	yes	complemented
20	C₃₁×#444	—	2	2	no	—
21	C₃₁×#444	—	2	4	no	—
22	C₃₁×#443	—	2	2	no	—
23	C₃₁×#443	—	2	4	no	—
24	C₃₁×#430	—	2	2	no	—
25	C₃₁×#430	C₂²	1	2	yes	—
26	C₃₁×#430	C₂²	1	2	yes	—
27	C₂×C₃₁×#108	—	2	2	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
31	C₃₁×#421	C₂²	1	2	yes	—
32	C₂×C₃₁×#104	C₂²	1	2	yes	—
33	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
34	C₃₁×#416	C₂²	1	1	yes	characteristic
35	C₃₁×#413	C₂²	1	2	yes	—
36	C₃₁×#408	C₂²	1	2	yes	—
37	C₃₁×#405	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#344	C₂²	1	2	yes	—
41	C₃₁×#341	C₂²	1	1	yes	characteristic
42	C₃₁×#341	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₂×#41×C₃₁	C₂³	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#137	—	2	2	no	—
47	C₃₁×#137	C₂×C₄	1	4	yes	—
48	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
49	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
50	C₃₁×#135	—	2	4	no	—
51	C₃₁×#135	—	2	1	no	—
52	C₃₁×#135	—	2	1	no	—
53	C₃₁×#135	—	2	1	no	—
54	C₃₁×#135	—	2	1	no	—
55	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₃₁×#131	—	2	1	no	—
59	C₃₁×#131	—	2	1	no	—
60	C₃₁×#131	—	2	4	no	—
61	C₃₁×#131	—	2	4	no	—
62	C₃₁×#131	—	2	4	no	—
63	C₃₁×#131	—	4	2	no	—
64	C₃₁×#131	C₂×C₄	1	2	yes	—
65	C₃₁×#131	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₃₁	—	2	1	no	—
67	C₂×M₁₆×C₃₁	—	2	4	no	—
68	C₂×M₁₆×C₃₁	—	4	2	no	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#117	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#108	—	2	4	no	—
80	C₃₁×#108	—	2	4	no	—
81	C₃₁×#108	—	2	1	no	—
82	C₃₁×#108	—	2	1	no	—
83	C₃₁×#108	C₂×C₄	1	2	yes	—
84	C₃₁×#108	C₂×C₄	1	2	yes	—
85	C₃₁×#108	D₄	1	2	yes	—
86	C₃₁×#108	D₄	1	2	yes	—
87	C₃₁×#104	—	2	4	no	—
88	C₃₁×#104	—	2	2	no	—
89	C₃₁×#104	—	2	2	no	—
90	C₃₁×#104	D₄	1	2	yes	—
91	C₃₁×#104	D₄	1	2	yes	—
92	C₃₁×#103	—	2	1	no	—
93	C₃₁×#102	—	2	1	no	—
94	C₃₁×#101	C₂×C₄	1	4	yes	—
95	C₃₁×#100	C₂×C₄	1	4	yes	—
96	C₃₁×C₂²⋊C₈	—	4	2	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₃₁×#97	—	2	2	no	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	#41×C₃₁	—	2	1	no	—
104	#41×C₃₁	—	2	1	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	C₂²×C₄	1	2	yes	—
110	#41×C₃₁	C₂×D₄	1	2	yes	—
111	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₃₁	—	2	4	no	—
122	Q₁₆×C₃₁	—	2	1	no	—
123	Q₁₆×C₃₁	—	2	1	no	—
124	SD₁₆×C₃₁	—	2	4	no	—
125	SD₁₆×C₃₁	—	2	4	no	—
126	SD₁₆×C₃₁	—	2	2	no	—
127	SD₁₆×C₃₁	—	2	2	no	—
128	D₈×C₃₁	—	2	4	no	—
129	D₈×C₃₁	—	2	1	no	—
130	D₈×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	2	1	no	—
132	M₁₆×C₃₁	—	2	2	no	—
133	M₁₆×C₃₁	—	2	1	no	—
134	M₁₆×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	2	2	no	—
139	M₁₆×C₃₁	—	4	4	no	—
140	M₁₆×C₃₁	—	4	4	no	—
141	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
142	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
143	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
144	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	4	4	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
162	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₂²⋊C₄×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	—	2	1	no	—
167	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
169	#23042	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
173	Q₈×C₃₁	—	2	1	no	—
174	Q₈×C₃₁	—	2	2	no	—
175	Q₈×C₃₁	—	2	2	no	—
176	Q₈×C₃₁	C₄×D₄	1	2	yes	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	2	no	—
181	D₄×C₃₁	—	2	1	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	—	2	2	no	—
204	C₈×C₃₁	—	2	4	no	—
205	C₈×C₃₁	—	4	2	no	—
206	C₈×C₃₁	—	4	1	no	—
207	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
208	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
209	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
210	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
211	#3067	C₂×C₃₁	1	1	yes	characteristic
212	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#2956	C₂×C₃₁	1	2	yes	complemented
214	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1974	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1932	C₂×C₃₁	1	2	yes	complemented
218	#1881	C₂×C₃₁	1	2	yes	complemented
219	#1858	C₂×C₃₁	1	2	yes	complemented
220	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	8	1	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	#520	1	1	yes	characteristic
228	C₂²×C₃₁	#547	1	1	yes	characteristic
229	C₂²×C₃₁	#548	1	1	yes	characteristic
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	8	1	no	—
235	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
236	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
237	C₄×C₃₁	D₄²	1	1	yes	characteristic
238	C₄×C₃₁	#553	1	1	yes	characteristic
239	#575	C₂²×C₃₁	1	2	yes	—
240	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
241	#568	C₂²×C₃₁	1	2	yes	—
242	#568	C₂²×C₃₁	1	2	yes	—
243	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
244	#473	C₄×C₃₁	1	2	yes	complemented
245	#472	C₄×C₃₁	1	2	yes	complemented
246	#471	C₄×C₃₁	1	4	yes	complemented
247	#444	—	2	2	no	—
248	#444	—	2	4	no	—
249	#443	—	2	2	no	—
250	#443	—	2	4	no	—
251	#430	—	2	2	no	—
252	#430	C₂²×C₃₁	1	2	yes	—
253	#430	C₂²×C₃₁	1	2	yes	—
254	C₂×#108	—	2	2	no	—
255	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
258	#421	C₂²×C₃₁	1	2	yes	—
259	C₂×#104	C₂²×C₃₁	1	2	yes	—
260	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
261	#416	C₂²×C₃₁	1	1	yes	characteristic
262	#413	C₂²×C₃₁	1	2	yes	—
263	#408	C₂²×C₃₁	1	2	yes	—
264	#405	C₂²×C₃₁	1	1	yes	characteristic
265	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
267	#344	C₂²×C₃₁	1	2	yes	—
268	#341	C₂²×C₃₁	1	1	yes	characteristic
269	#341	C₂²×C₃₁	1	2	yes	—
270	#339	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₃₁	—	2	1	no	—
272	C₂×C₃₁	—	2	1	no	—
273	C₂×C₃₁	—	2	1	no	—
274	C₂×C₃₁	—	4	2	no	—
275	C₂×C₃₁	—	8	2	no	—
276	C₂×C₃₁	#2361	1	1	yes	characteristic
277	C₂×#41	C₂³×C₃₁	1	2	yes	—
278	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
279	#137	—	2	2	no	—
280	#137	C₂×C₄×C₃₁	1	4	yes	—
281	AGL(1,ℤ/8)	—	2	2	no	—
282	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	4	yes	—
283	#135	—	2	4	no	—
284	#135	—	2	1	no	—
285	#135	—	2	1	no	—
286	#135	—	2	1	no	—
287	#135	—	2	1	no	—
288	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
289	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
290	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
291	#131	—	2	1	no	—
292	#131	—	2	1	no	—
293	#131	—	2	4	no	—
294	#131	—	2	4	no	—
295	#131	—	2	4	no	—
296	#131	—	4	2	no	—
297	#131	C₂×C₄×C₃₁	1	2	yes	—
298	#131	C₂×C₄×C₃₁	1	2	yes	—
299	C₂×M₁₆	—	2	1	no	—
300	C₂×M₁₆	—	2	4	no	—
301	C₂×M₁₆	—	4	2	no	—
302	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
303	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
305	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
306	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	#117	C₂³×C₃₁	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	#108	—	2	4	no	—
313	#108	—	2	4	no	—
314	#108	—	2	1	no	—
315	#108	—	2	1	no	—
316	#108	C₂×C₄×C₃₁	1	2	yes	—
317	#108	C₂×C₄×C₃₁	1	2	yes	—
318	#108	D₄×C₃₁	1	2	yes	—
319	#108	D₄×C₃₁	1	2	yes	—
320	#104	—	2	4	no	—
321	#104	—	2	2	no	—
322	#104	—	2	2	no	—
323	#104	D₄×C₃₁	1	2	yes	—
324	#104	D₄×C₃₁	1	2	yes	—
325	#103	—	2	1	no	—
326	#102	—	2	1	no	—
327	#101	C₂×C₄×C₃₁	1	4	yes	—
328	#100	C₂×C₄×C₃₁	1	4	yes	—
329	C₂²⋊C₈	—	4	2	no	—
330	#97	—	2	1	no	—
331	#97	—	2	2	no	—
332	C₄×C₈	—	2	1	no	—
333	C₄×C₈	—	2	1	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	—	2	2	no	—
336	C₃₁	#23042	1	1	yes	characteristic, direct product factor, Sylow
337	#41	—	2	1	no	—
338	#41	—	2	1	no	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	2	2	no	—
343	#41	C₂²×C₄×C₃₁	1	2	yes	—
344	#41	C₂×D₄×C₃₁	1	2	yes	—
345	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
346	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
355	Q₁₆	—	2	4	no	—
356	Q₁₆	—	2	1	no	—
357	Q₁₆	—	2	1	no	—
358	SD₁₆	—	2	4	no	—
359	SD₁₆	—	2	4	no	—
360	SD₁₆	—	2	2	no	—
361	SD₁₆	—	2	2	no	—
362	D₈	—	2	4	no	—
363	D₈	—	2	1	no	—
364	D₈	—	2	1	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	2	2	no	—
369	M₁₆	—	2	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	2	no	—
373	M₁₆	—	4	4	no	—
374	M₁₆	—	4	4	no	—
375	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
376	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
377	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
378	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	4	4	no	—
390	C₂×C₈	—	4	4	no	—
391	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
392	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
393	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
396	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
397	C₄⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂³	—	2	1	no	—
404	C₂³	—	2	2	no	—
405	C₂³	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
406	Q₈	—	2	1	no	—
407	Q₈	—	2	2	no	—
408	Q₈	—	2	2	no	—
409	Q₈	C₄×D₄×C₃₁	1	2	yes	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	2	2	no	—
413	D₄	—	2	2	no	—
414	D₄	—	2	1	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	C₄×D₄×C₃₁	1	2	yes	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	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	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	2	4	no	—
436	C₈	—	2	2	no	—
437	C₈	—	2	4	no	—
438	C₈	—	4	2	no	—
439	C₈	—	4	1	no	—
440	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
441	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
442	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
443	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
444	C₂²	—	2	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	2	no	—
449	C₂²	C₃₁×#520	1	1	yes	characteristic
450	C₂²	C₃₁×#547	1	1	yes	characteristic
451	C₂²	C₃₁×#548	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
458	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
459	C₄	D₄²×C₃₁	1	1	yes	characteristic
460	C₄	C₃₁×#553	1	1	yes	characteristic
461	C₂	—	2	1	no	—
462	C₂	—	2	1	no	—
463	C₂	—	2	1	no	—
464	C₂	—	4	2	no	—
465	C₂	—	8	2	no	—
466	C₂	C₃₁×#2361	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2361
prime radical	C₃₁×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2957	C₂	semidirect
3. C₃₁×#2956	C₂	semidirect
4. C₃₁×#2916	C₂	semidirect
5. C₃₁×#1974	C₂	semidirect
6. C₃₁×#1953	C₂	semidirect
7. C₃₁×#1932	C₂	semidirect
8. C₃₁×#1881	C₂	semidirect
9. C₃₁×#1858	C₂	semidirect
10. C₄×C₃₁×#108	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
17. C₃₁×#473	C₄	semidirect
18. C₃₁×#472	C₄	semidirect
19. C₃₁×#471	C₄	semidirect
169. #23042	C₃₁	direct (N × H)
212. #2957	C₂×C₃₁	semidirect
213. #2956	C₂×C₃₁	semidirect
214. #2916	C₂×C₃₁	semidirect
215. #1974	C₂×C₃₁	semidirect
216. #1953	C₂×C₃₁	semidirect
217. #1932	C₂×C₃₁	semidirect
218. #1881	C₂×C₃₁	semidirect
219. #1858	C₂×C₃₁	semidirect
220. C₄×#108	C₂×C₃₁	semidirect
221. #1764	C₂×C₃₁	semidirect
244. #473	C₄×C₃₁	semidirect
245. #472	C₄×C₃₁	semidirect
246. #471	C₄×C₃₁	semidirect
336. C₃₁	#23042	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×#3067	2
C₃₁×#2957	2
C₃₁×#2956	2
C₃₁×#2916	2
C₃₁×#1974	2
C₃₁×#1953	2
C₃₁×#1932	2
C₃₁×#1881	2
C₃₁×#1858	2
C₄×C₃₁×#108	2
C₃₁×#1764	2
#23042	31

Subgroups — C31×#23042 #21479801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23042 #21479801