Subgroups — C₃₁×#19968 #21476727

Order 7936.

780 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3294	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1639	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1625	C₂	1	2	yes	maximal, complemented
10	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂²×D₈×C₃₁	—	2	1	no	—
14	C₃₁×#547	C₂²	1	2	yes	complemented
15	D₄²×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#486	C₂²	1	2	yes	complemented
19	C₃₁×#486	C₂²	1	2	yes	complemented
20	C₃₁×#480	—	2	2	no	—
21	C₃₁×#474	C₂²	1	2	yes	—
22	C₃₁×#467	—	2	1	no	—
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#466	—	2	2	no	—
25	C₃₁×#461	C₂²	1	2	yes	complemented
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	complemented
28	C₃₁×#452	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₃₁×#447	—	2	2	no	—
31	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
36	C₂×C₃₁×#102	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#332	C₂²	1	2	yes	complemented
40	C₃₁×#331	C₂²	1	2	yes	complemented
41	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₃₁×#135	—	4	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	4	4	no	—
57	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
58	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₃₁×#123	—	4	2	no	—
65	C₃₁×C₂²⋊Q₈	—	2	2	no	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#120	—	4	2	no	—
70	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₄×D₄×C₃₁	C₂³	1	2	yes	—
72	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×#107	D₄	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	4	2	no	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×#102	D₄	1	2	yes	—
89	C₃₁×#102	D₄	1	1	yes	characteristic
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂⁴×C₃₁	—	4	2	no	—
97	#41×C₃₁	—	4	4	no	—
98	#41×C₃₁	—	8	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	—	4	2	no	—
105	C₂×D₄×C₃₁	—	4	4	no	—
106	C₂×D₄×C₃₁	—	4	4	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	—	8	2	no	—
109	C₂×D₄×C₃₁	—	8	2	no	—
110	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	Q₁₆×C₃₁	—	4	2	no	—
118	Q₁₆×C₃₁	—	8	1	no	—
119	SD₁₆×C₃₁	—	4	4	no	—
120	D₈×C₃₁	—	2	2	no	—
121	D₈×C₃₁	—	4	4	no	—
122	D₈×C₃₁	—	4	2	no	—
123	D₈×C₃₁	—	8	1	no	—
124	D₈×C₃₁	—	8	2	no	—
125	D₈×C₃₁	D₈	1	4	yes	complemented
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	—	8	2	no	—
133	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
149	#19968	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	8	4	no	—
155	C₂³×C₃₁	—	8	4	no	—
156	C₂³×C₃₁	—	16	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	2	1	no	—
160	D₄×C₃₁	—	2	1	no	—
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	2	1	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	8	2	no	—
168	D₄×C₃₁	—	8	4	no	—
169	D₄×C₃₁	—	8	4	no	—
170	D₄×C₃₁	—	8	4	no	—
171	D₄×C₃₁	C₂×D₈	1	4	yes	—
172	D₄×C₃₁	#135	1	2	yes	complemented
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	8	4	no	—
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	C₈×C₃₁	—	2	1	no	—
188	C₈×C₃₁	—	2	1	no	—
189	C₈×C₃₁	—	2	1	no	—
190	C₈×C₃₁	—	4	1	no	—
191	C₈×C₃₁	—	4	1	no	—
192	C₈×C₃₁	—	4	1	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	C₂×D₈	1	2	yes	—
195	#3294	C₂×C₃₁	1	2	yes	complemented
196	#3283	C₂×C₃₁	1	1	yes	characteristic, complemented
197	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1681	C₂×C₃₁	1	2	yes	complemented
200	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1639	C₂×C₃₁	1	2	yes	complemented
203	#1625	C₂×C₃₁	1	2	yes	complemented
204	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂²×C₃₁	—	2	4	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	—	4	2	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	16	8	no	—
219	C₂²×C₃₁	D₄²	1	1	yes	characteristic
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	8	2	no	—
225	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
226	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
227	C₄×C₃₁	#576	1	1	yes	characteristic
228	C₄×C₃₁	#576	1	1	yes	characteristic
229	C₂×#135	—	2	1	no	—
230	C₂²×D₈	—	2	1	no	—
231	#547	C₂²×C₃₁	1	2	yes	complemented
232	D₄²	C₂²×C₃₁	1	2	yes	—
233	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
234	#493	C₂²×C₃₁	1	1	yes	characteristic
235	#486	C₂²×C₃₁	1	2	yes	complemented
236	#486	C₂²×C₃₁	1	2	yes	complemented
237	#480	—	2	2	no	—
238	#474	C₂²×C₃₁	1	2	yes	—
239	#467	—	2	1	no	—
240	#466	—	2	1	no	—
241	#466	—	2	2	no	—
242	#461	C₂²×C₃₁	1	2	yes	complemented
243	#459	C₂²×C₃₁	1	2	yes	—
244	#459	C₂²×C₃₁	1	2	yes	complemented
245	#452	—	2	2	no	—
246	#447	—	2	2	no	—
247	#447	—	2	2	no	—
248	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
251	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
253	C₂×#102	—	2	2	no	—
254	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
255	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
256	#332	C₂²×C₃₁	1	2	yes	complemented
257	#331	C₂²×C₃₁	1	2	yes	complemented
258	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
259	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
260	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
261	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
262	C₂×C₃₁	—	4	4	no	—
263	C₂×C₃₁	—	4	2	no	—
264	C₂×C₃₁	—	8	1	no	—
265	C₂×C₃₁	—	16	2	no	—
266	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
267	C₂×C₃₁	#3280	1	1	yes	characteristic
268	C₂×C₃₁	#3290	1	1	yes	characteristic
269	C₂×#41	—	2	2	no	—
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	—	2	2	no	—
272	#135	—	4	4	no	—
273	C₂×Q₁₆	—	2	1	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂×D₈	—	2	2	no	—
277	C₂×D₈	—	2	2	no	—
278	C₂×D₈	—	2	1	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	—	4	4	no	—
281	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
282	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
283	C₂²×C₈	—	2	1	no	—
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	—	2	2	no	—
286	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
287	#127	C₂³×C₃₁	1	2	yes	—
288	#123	—	4	2	no	—
289	C₂²⋊Q₈	—	2	2	no	—
290	#121	—	2	2	no	—
291	#121	—	2	2	no	—
292	#121	—	2	2	no	—
293	#120	—	4	2	no	—
294	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₄×D₄	C₂³×C₃₁	1	2	yes	—
296	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	—	2	2	no	—
298	#107	—	2	2	no	—
299	#107	D₄×C₃₁	1	1	yes	characteristic
300	#107	D₄×C₃₁	1	1	yes	characteristic
301	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
303	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
304	#103	—	2	2	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	4	2	no	—
311	#102	D₄×C₃₁	1	1	yes	characteristic
312	#102	D₄×C₃₁	1	2	yes	—
313	#102	D₄×C₃₁	1	1	yes	characteristic
314	C₂²⋊C₈	—	2	1	no	—
315	C₂²⋊C₈	—	2	2	no	—
316	C₂²⋊C₈	—	2	1	no	—
317	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₃₁	#19968	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	4	2	no	—
322	#41	—	4	4	no	—
323	#41	—	8	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	—	4	4	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	4	2	no	—
342	Q₁₆	—	4	2	no	—
343	Q₁₆	—	8	1	no	—
344	SD₁₆	—	4	4	no	—
345	D₈	—	2	2	no	—
346	D₈	—	4	4	no	—
347	D₈	—	4	2	no	—
348	D₈	—	8	1	no	—
349	D₈	—	8	2	no	—
350	D₈	D₈×C₃₁	1	4	yes	complemented
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	4	2	no	—
356	C₂×C₈	—	4	2	no	—
357	C₂×C₈	—	8	2	no	—
358	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	4	2	no	—
365	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
367	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
374	C₂³	—	2	2	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	8	4	no	—
379	C₂³	—	8	4	no	—
380	C₂³	—	16	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	4	2	no	—
383	D₄	—	2	1	no	—
384	D₄	—	2	1	no	—
385	D₄	—	2	2	no	—
386	D₄	—	2	1	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	4	no	—
393	D₄	—	8	4	no	—
394	D₄	—	8	4	no	—
395	D₄	C₂×D₈×C₃₁	1	4	yes	—
396	D₄	C₃₁×#135	1	2	yes	complemented
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	4	no	—
408	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	—	2	1	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	1	no	—
416	C₈	—	4	1	no	—
417	C₈	—	4	2	no	—
418	C₈	C₂×D₈×C₃₁	1	2	yes	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	2	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	16	8	no	—
432	C₂²	D₄²×C₃₁	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	4	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	2	no	—
437	C₄	—	8	2	no	—
438	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
439	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
440	C₄	C₃₁×#576	1	1	yes	characteristic
441	C₄	C₃₁×#576	1	1	yes	characteristic
442	C₂	—	4	2	no	—
443	C₂	—	4	4	no	—
444	C₂	—	8	1	no	—
445	C₂	—	16	2	no	—
446	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
447	C₂	C₃₁×#3280	1	1	yes	characteristic
448	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3294	C₂	semidirect
2. C₃₁×#3283	C₂	semidirect
3. D₄×D₈×C₃₁	C₂	semidirect
4. C₃₁×#1709	C₂	semidirect
5. C₃₁×#1681	C₂	semidirect
6. C₃₁×#1669	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1639	C₂	semidirect
9. C₃₁×#1625	C₂	semidirect
10. C₈×D₈×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
14. C₃₁×#547	C₂²	semidirect
16. C₃₁×#494	C₂²	semidirect
18. C₃₁×#486	C₂²	semidirect
19. C₃₁×#486	C₂²	semidirect
25. C₃₁×#461	C₂²	semidirect
27. C₃₁×#459	C₂²	semidirect
34. C₈×D₄×C₃₁	C₂²	semidirect
35. C₈×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#335	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
42. C₃₁×D₄⋊C₈	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₃₁	C₂²	semidirect
125. D₈×C₃₁	D₈	semidirect
149. #19968	C₃₁	direct (N × H)
172. D₄×C₃₁	#135	semidirect
195. #3294	C₂×C₃₁	semidirect
196. #3283	C₂×C₃₁	semidirect
197. D₄×D₈	C₂×C₃₁	semidirect
198. #1709	C₂×C₃₁	semidirect
199. #1681	C₂×C₃₁	semidirect
200. #1669	C₂×C₃₁	semidirect
201. #1667	C₂×C₃₁	semidirect
202. #1639	C₂×C₃₁	semidirect
203. #1625	C₂×C₃₁	semidirect
204. C₈×D₈	C₂×C₃₁	semidirect
205. C₈×D₈	C₂×C₃₁	semidirect
231. #547	C₂²×C₃₁	semidirect
233. #494	C₂²×C₃₁	semidirect
235. #486	C₂²×C₃₁	semidirect
236. #486	C₂²×C₃₁	semidirect
242. #461	C₂²×C₃₁	semidirect
244. #459	C₂²×C₃₁	semidirect
251. C₈×D₄	C₂²×C₃₁	semidirect
252. C₈×D₄	C₂²×C₃₁	semidirect
254. #335	C₂²×C₃₁	semidirect
255. #335	C₂²×C₃₁	semidirect
256. #332	C₂²×C₃₁	semidirect
257. #331	C₂²×C₃₁	semidirect
259. D₄⋊C₈	C₂²×C₃₁	semidirect
260. D₄⋊C₈	C₂²×C₃₁	semidirect
261. C₈²	C₂²×C₃₁	semidirect
320. C₃₁	#19968	direct (N × H)
350. D₈	D₈×C₃₁	semidirect
396. D₄	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3294	2
C₃₁×#3283	2
D₄×D₈×C₃₁	2
C₃₁×#1709	2
C₃₁×#1681	2
C₃₁×#1669	2
C₃₁×#1667	2
C₃₁×#1639	2
C₃₁×#1625	2
C₈×D₈×C₃₁	2
#19968	31

Subgroups — C31×#19968 #21476727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19968 #21476727