Subgroups — C₃₁×#33778 #21490537

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2249	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2242	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂×C₃₁×#135	C₂²	1	2	yes	—
14	C₃₁×#532	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
19	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
20	C₃₁×#499	C₂²	1	1	yes	characteristic
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#495	C₂²	1	2	yes	—
23	C₃₁×#495	C₂²	1	1	yes	characteristic
24	C₃₁×#494	C₂²	1	1	yes	characteristic
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#492	—	2	1	no	—
27	C₃₁×#488	—	2	1	no	—
28	C₃₁×#467	C₂²	1	1	yes	characteristic
29	C₃₁×#466	C₂²	1	1	yes	characteristic
30	C₃₁×#465	—	2	1	no	—
31	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×SD₁₆×C₃₁	—	2	1	no	—
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#358	C₂²	1	2	yes	—
39	C₃₁×#357	C₂²	1	2	yes	—
40	C₃₁×#357	C₂²	1	2	yes	—
41	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
42	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₃₁×#135	—	4	2	no	—
45	C₃₁×#135	—	4	2	no	—
46	C₃₁×#135	—	4	4	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×SD₁₆×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×D₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#126	—	4	1	no	—
61	C₃₁×#125	—	2	1	no	—
62	C₃₁×#124	—	2	1	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#123	—	4	1	no	—
65	C₃₁×C₂²⋊Q₈	—	2	1	no	—
66	C₃₁×#121	—	2	1	no	—
67	C₃₁×#121	—	4	1	no	—
68	C₃₁×#121	—	4	1	no	—
69	C₄×Q₈×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	4	1	no	—
72	C₄×D₄×C₃₁	—	4	1	no	—
73	C₄×D₄×C₃₁	—	4	1	no	—
74	C₃₁×#117	—	4	1	no	—
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₃₁×Q₃₂	—	4	2	no	—
77	C₃₁×SD₃₂	—	4	2	no	—
78	C₃₁×SD₃₂	—	4	2	no	—
79	C₃₁×D₁₆	—	4	2	no	—
80	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
81	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
82	C₃₁×#107	C₂³	1	1	yes	characteristic
83	C₃₁×#107	C₂³	1	1	yes	characteristic
84	C₃₁×#106	—	2	1	no	—
85	C₃₁×#103	—	2	1	no	—
86	C₃₁×#103	—	2	1	no	—
87	C₃₁×#102	—	2	1	no	—
88	C₃₁×#102	—	2	1	no	—
89	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	#41×C₃₁	—	4	4	no	—
94	#41×C₃₁	—	4	4	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₃₁	—	2	1	no	—
109	Q₁₆×C₃₁	—	2	2	no	—
110	Q₁₆×C₃₁	—	2	1	no	—
111	Q₁₆×C₃₁	—	2	2	no	—
112	Q₁₆×C₃₁	—	4	2	no	—
113	SD₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	4	2	no	—
115	SD₁₆×C₃₁	—	4	4	no	—
116	D₈×C₃₁	—	2	1	no	—
117	D₈×C₃₁	—	2	2	no	—
118	D₈×C₃₁	—	2	1	no	—
119	D₈×C₃₁	—	2	2	no	—
120	D₈×C₃₁	—	4	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	4	1	no	—
131	C₄⋊C₄×C₃₁	—	4	1	no	—
132	C₄⋊C₄×C₃₁	—	4	1	no	—
133	C₄⋊C₄×C₃₁	—	4	1	no	—
134	C₄⋊C₄×C₃₁	—	4	1	no	—
135	C₄⋊C₄×C₃₁	—	4	1	no	—
136	C₂²⋊C₄×C₃₁	—	4	1	no	—
137	C₂²⋊C₄×C₃₁	—	4	1	no	—
138	C₂²⋊C₄×C₃₁	—	4	1	no	—
139	C₂²⋊C₄×C₃₁	—	4	1	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₂²⋊C₄×C₃₁	—	4	1	no	—
143	C₄²×C₃₁	—	4	1	no	—
144	C₄²×C₃₁	D₈	1	1	yes	characteristic
145	C₄²×C₃₁	D₈	1	1	yes	characteristic
146	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₁₆×C₃₁	—	2	2	no	—
149	C₁₆×C₃₁	—	2	2	no	—
150	#33778	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	4	1	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	1	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	1	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	1	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	1	no	—
162	D₄×C₃₁	—	8	2	no	—
163	D₄×C₃₁	—	8	2	no	—
164	D₄×C₃₁	—	8	4	no	—
165	D₄×C₃₁	—	8	2	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	8	4	no	—
175	C₂×C₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	2	1	no	—
184	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
185	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
186	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	#3168	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#2249	C₂×C₃₁	1	2	yes	complemented
190	#2242	C₂×C₃₁	1	2	yes	complemented
191	#2209	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#2208	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#2207	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#2189	C₂×C₃₁	1	2	yes	complemented
196	#2186	C₂×C₃₁	1	2	yes	complemented
197	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₃₁	—	2	1	no	—
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	8	1	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	—	8	1	no	—
205	C₂²×C₃₁	—	8	2	no	—
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	8	1	no	—
213	C₄×C₃₁	—	8	1	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	—	8	1	no	—
216	C₄×C₃₁	#508	1	2	yes	—
217	C₄×C₃₁	#535	1	1	yes	characteristic
218	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂×#135	—	2	1	no	—
220	C₂×#135	C₂²×C₃₁	1	2	yes	—
221	#532	—	2	1	no	—
222	#529	—	2	1	no	—
223	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
224	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
225	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
226	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
227	#499	C₂²×C₃₁	1	1	yes	characteristic
228	#495	—	2	1	no	—
229	#495	C₂²×C₃₁	1	2	yes	—
230	#495	C₂²×C₃₁	1	1	yes	characteristic
231	#494	C₂²×C₃₁	1	1	yes	characteristic
232	#493	C₂²×C₃₁	1	1	yes	characteristic
233	#492	—	2	1	no	—
234	#488	—	2	1	no	—
235	#467	C₂²×C₃₁	1	1	yes	characteristic
236	#466	C₂²×C₃₁	1	1	yes	characteristic
237	#465	—	2	1	no	—
238	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
239	C₄×SD₁₆	—	2	1	no	—
240	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
243	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	#358	C₂²×C₃₁	1	2	yes	—
245	#358	C₂²×C₃₁	1	2	yes	—
246	#357	C₂²×C₃₁	1	2	yes	—
247	#357	C₂²×C₃₁	1	2	yes	—
248	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
249	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
250	C₂×C₃₁	—	4	1	no	—
251	C₂×C₃₁	—	16	1	no	—
252	C₂×C₃₁	—	16	2	no	—
253	C₂×C₃₁	#3205	1	1	yes	characteristic
254	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
255	C₂×C₃₁	#3414	1	1	yes	characteristic
256	C₂×#41	—	2	2	no	—
257	#135	—	4	2	no	—
258	#135	—	4	2	no	—
259	#135	—	4	4	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
262	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×SD₁₆	—	2	2	no	—
265	C₂×SD₁₆	—	2	1	no	—
266	C₂×D₈	—	2	1	no	—
267	C₂×D₈	C₂³×C₃₁	1	2	yes	—
268	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₄⋊Q₈	—	2	1	no	—
272	#127	—	2	1	no	—
273	#126	—	4	1	no	—
274	#125	—	2	1	no	—
275	#124	—	2	1	no	—
276	#124	—	2	2	no	—
277	#123	—	4	1	no	—
278	C₂²⋊Q₈	—	2	1	no	—
279	#121	—	2	1	no	—
280	#121	—	4	1	no	—
281	#121	—	4	1	no	—
282	C₄×Q₈	—	2	1	no	—
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	4	1	no	—
285	C₄×D₄	—	4	1	no	—
286	C₄×D₄	—	4	1	no	—
287	#117	—	4	1	no	—
288	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
289	Q₃₂	—	4	2	no	—
290	SD₃₂	—	4	2	no	—
291	SD₃₂	—	4	2	no	—
292	D₁₆	—	4	2	no	—
293	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
294	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
295	#107	C₂³×C₃₁	1	1	yes	characteristic
296	#107	C₂³×C₃₁	1	1	yes	characteristic
297	#106	—	2	1	no	—
298	#103	—	2	1	no	—
299	#103	—	2	1	no	—
300	#102	—	2	1	no	—
301	#102	—	2	1	no	—
302	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
303	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
304	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₃₁	#33778	1	1	yes	characteristic, direct product factor, Sylow
307	#41	—	4	4	no	—
308	#41	—	4	4	no	—
309	C₂×Q₈	—	2	1	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	1	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	1	no	—
316	C₂×D₄	—	4	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	Q₁₆	—	2	1	no	—
323	Q₁₆	—	2	2	no	—
324	Q₁₆	—	2	1	no	—
325	Q₁₆	—	2	2	no	—
326	Q₁₆	—	4	2	no	—
327	SD₁₆	—	4	4	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	4	4	no	—
330	D₈	—	2	1	no	—
331	D₈	—	2	2	no	—
332	D₈	—	2	1	no	—
333	D₈	—	2	2	no	—
334	D₈	—	4	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	4	1	no	—
345	C₄⋊C₄	—	4	1	no	—
346	C₄⋊C₄	—	4	1	no	—
347	C₄⋊C₄	—	4	1	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₂²⋊C₄	—	4	1	no	—
351	C₂²⋊C₄	—	4	1	no	—
352	C₂²⋊C₄	—	4	1	no	—
353	C₂²⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₄²	—	4	1	no	—
358	C₄²	D₈×C₃₁	1	1	yes	characteristic
359	C₄²	D₈×C₃₁	1	1	yes	characteristic
360	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₁₆	—	2	2	no	—
363	C₁₆	—	2	2	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	2	no	—
366	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
367	Q₈	—	4	2	no	—
368	Q₈	—	4	1	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	4	1	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	1	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	1	no	—
375	D₄	—	8	2	no	—
376	D₄	—	8	2	no	—
377	D₄	—	8	4	no	—
378	D₄	—	8	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	8	4	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₈	—	2	1	no	—
396	C₈	—	2	1	no	—
397	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
398	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
399	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
400	C₂²	—	2	1	no	—
401	C₂²	—	2	1	no	—
402	C₂²	—	2	1	no	—
403	C₂²	—	8	1	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	1	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	1	no	—
409	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
410	C₄	—	2	1	no	—
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	8	1	no	—
415	C₄	—	8	1	no	—
416	C₄	—	8	2	no	—
417	C₄	—	8	1	no	—
418	C₄	C₃₁×#508	1	2	yes	—
419	C₄	C₃₁×#535	1	1	yes	characteristic
420	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
421	C₂	—	4	1	no	—
422	C₂	—	16	1	no	—
423	C₂	—	16	2	no	—
424	C₂	C₃₁×#3205	1	1	yes	characteristic
425	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
426	C₂	C₃₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3168	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2242	C₂	semidirect
5. C₃₁×#2209	C₂	semidirect
6. C₃₁×#2208	C₂	semidirect
7. C₃₁×#2207	C₂	semidirect
8. C₃₁×#2206	C₂	semidirect
9. C₃₁×#2189	C₂	semidirect
10. C₃₁×#2186	C₂	semidirect
11. C₃₁×#2162	C₂	semidirect
150. #33778	C₃₁	direct (N × H)
187. #3168	C₂×C₃₁	semidirect
188. #3147	C₂×C₃₁	semidirect
189. #2249	C₂×C₃₁	semidirect
190. #2242	C₂×C₃₁	semidirect
191. #2209	C₂×C₃₁	semidirect
192. #2208	C₂×C₃₁	semidirect
193. #2207	C₂×C₃₁	semidirect
194. #2206	C₂×C₃₁	semidirect
195. #2189	C₂×C₃₁	semidirect
196. #2186	C₂×C₃₁	semidirect
197. #2162	C₂×C₃₁	semidirect
306. C₃₁	#33778	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×#3168	2
C₃₁×#3147	2
C₃₁×#2249	2
C₃₁×#2242	2
C₃₁×#2209	2
C₃₁×#2208	2
C₃₁×#2207	2
C₃₁×#2206	2
C₃₁×#2189	2
C₃₁×#2186	2
C₃₁×#2162	2
#33778	31

Subgroups — C31×#33778 #21490537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33778 #21490537