Subgroups — C₃₁×#33838 #21490597

Order 7936.

738 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#509	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×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2187	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#547	—	2	1	no	—
15	D₄²×C₃₁	—	2	1	no	—
16	C₃₁×#509	—	2	8	no	—
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
20	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
22	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
23	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#493	C₂²	1	1	yes	characteristic
25	C₃₁×#467	C₂²	1	2	yes	complemented
26	C₃₁×#466	C₂²	1	2	yes	—
27	C₃₁×#461	—	2	1	no	—
28	C₃₁×#459	—	2	1	no	—
29	C₃₁×#452	—	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, complemented
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#358	C₂²	1	2	yes	complemented
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#357	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	—	2	2	no	—
44	C₂²×D₄×C₃₁	—	4	1	no	—
45	C₃₁×#135	—	2	4	no	—
46	C₃₁×#135	—	4	2	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	C₂³	1	2	yes	—
55	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#123	—	4	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	4	1	no	—
64	C₃₁×#121	—	4	1	no	—
65	C₃₁×#120	—	4	2	no	—
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
70	C₃₁×SD₃₂	—	2	4	no	—
71	C₃₁×SD₃₂	—	2	4	no	—
72	C₃₁×SD₃₂	—	4	2	no	—
73	C₃₁×D₁₆	—	2	4	no	—
74	C₃₁×D₁₆	D₄	1	4	yes	complemented
75	C₃₁×M₃₂	—	2	4	no	—
76	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
77	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	C₂³	1	1	yes	characteristic
80	C₃₁×#107	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#102	—	2	1	no	—
84	C₃₁×#102	—	2	1	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
87	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂⁴×C₃₁	—	4	2	no	—
89	#41×C₃₁	—	4	4	no	—
90	#41×C₃₁	—	8	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	4	4	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	4	1	no	—
101	C₂×D₄×C₃₁	—	8	2	no	—
102	C₂×D₄×C₃₁	—	8	4	no	—
103	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	—	4	1	no	—
105	C₂²×C₄×C₃₁	—	4	2	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
108	Q₁₆×C₃₁	—	2	2	no	—
109	Q₁₆×C₃₁	—	2	2	no	—
110	Q₁₆×C₃₁	—	8	1	no	—
111	SD₁₆×C₃₁	—	4	4	no	—
112	D₈×C₃₁	—	2	2	no	—
113	D₈×C₃₁	—	2	1	no	—
114	D₈×C₃₁	—	2	1	no	—
115	D₈×C₃₁	—	2	2	no	—
116	D₈×C₃₁	—	4	4	no	—
117	D₈×C₃₁	—	8	1	no	—
118	D₈×C₃₁	C₂×D₄	1	2	yes	—
119	D₈×C₃₁	C₂×D₄	1	2	yes	complemented
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	4	1	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
131	C₂²⋊C₄×C₃₁	—	4	2	no	—
132	C₂²⋊C₄×C₃₁	—	4	1	no	—
133	C₂²⋊C₄×C₃₁	—	4	1	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
136	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₁₆×C₃₁	—	2	2	no	—
138	C₁₆×C₃₁	—	2	1	no	—
139	C₁₆×C₃₁	C₂×D₄	1	2	yes	complemented
140	#33838	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	4	1	no	—
142	C₂³×C₃₁	—	4	1	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	8	4	no	—
145	C₂³×C₃₁	—	8	4	no	—
146	C₂³×C₃₁	—	16	2	no	—
147	C₂³×C₃₁	C₂×D₈	1	2	yes	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	2	2	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	1	no	—
155	D₄×C₃₁	—	4	1	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	8	4	no	—
158	D₄×C₃₁	—	8	4	no	—
159	D₄×C₃₁	—	8	4	no	—
160	D₄×C₃₁	—	8	2	no	—
161	D₄×C₃₁	—	16	1	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	8	4	no	—
168	C₂×C₄×C₃₁	—	8	2	no	—
169	C₂×C₄×C₃₁	—	8	2	no	—
170	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₃₁	—	2	2	no	—
175	C₈×C₃₁	—	2	1	no	—
176	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×#509	C₂×C₃₁	1	2	yes	complemented
179	#3283	C₂×C₃₁	1	1	yes	characteristic, complemented
180	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#2248	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#2220	C₂×C₃₁	1	2	yes	complemented
183	#2208	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2187	C₂×C₃₁	1	2	yes	complemented
186	#2186	C₂×C₃₁	1	2	yes	complemented
187	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	8	1	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	—	8	1	no	—
197	C₂²×C₃₁	—	8	1	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	—	16	8	no	—
202	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
203	C₄×C₃₁	—	2	1	no	—
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	8	1	no	—
206	C₄×C₃₁	—	8	1	no	—
207	C₄×C₃₁	—	8	2	no	—
208	C₄×C₃₁	#509	1	2	yes	complemented
209	C₄×C₃₁	D₄²	1	1	yes	characteristic
210	C₄×C₃₁	#576	1	1	yes	characteristic
211	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
212	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
213	#547	—	2	1	no	—
214	D₄²	—	2	1	no	—
215	#509	—	2	8	no	—
216	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
217	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
218	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
219	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
221	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
222	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
223	#493	C₂²×C₃₁	1	1	yes	characteristic
224	#467	C₂²×C₃₁	1	2	yes	complemented
225	#466	C₂²×C₃₁	1	2	yes	—
226	#461	—	2	1	no	—
227	#459	—	2	1	no	—
228	#452	—	2	2	no	—
229	#447	—	2	2	no	—
230	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
232	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
233	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
234	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
235	#358	C₂²×C₃₁	1	2	yes	complemented
236	#357	C₂²×C₃₁	1	1	yes	characteristic
237	#357	C₂²×C₃₁	1	1	yes	characteristic, complemented
238	#357	C₂²×C₃₁	1	2	yes	complemented
239	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
240	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
241	C₂×C₃₁	—	4	2	no	—
242	C₂×C₃₁	—	8	2	no	—
243	C₂×C₃₁	—	8	2	no	—
244	C₂×C₃₁	—	16	2	no	—
245	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
246	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
247	C₂×C₃₁	#3416	1	1	yes	characteristic
248	C₂×#41	—	2	2	no	—
249	C₂²×D₄	—	2	2	no	—
250	C₂²×D₄	—	4	1	no	—
251	#135	—	2	4	no	—
252	#135	—	4	2	no	—
253	C₂×Q₁₆	—	2	1	no	—
254	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
255	C₂×SD₁₆	—	2	2	no	—
256	C₂×D₈	—	2	4	no	—
257	C₂×D₈	—	2	2	no	—
258	C₂×D₈	—	2	1	no	—
259	C₂×D₈	—	4	2	no	—
260	C₂×D₈	C₂³×C₃₁	1	2	yes	—
261	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
264	C₄⋊Q₈	—	2	1	no	—
265	#127	—	2	1	no	—
266	#123	—	4	2	no	—
267	C₂²⋊Q₈	—	2	2	no	—
268	#121	—	2	2	no	—
269	#121	—	4	1	no	—
270	#121	—	4	1	no	—
271	#120	—	4	2	no	—
272	C₄×D₄	—	2	1	no	—
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	—	4	1	no	—
276	SD₃₂	—	2	4	no	—
277	SD₃₂	—	2	4	no	—
278	SD₃₂	—	4	2	no	—
279	D₁₆	—	2	4	no	—
280	D₁₆	D₄×C₃₁	1	4	yes	complemented
281	M₃₂	—	2	4	no	—
282	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
284	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
285	#107	C₂³×C₃₁	1	1	yes	characteristic
286	#107	C₂³×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
288	#103	—	2	2	no	—
289	#102	—	2	1	no	—
290	#102	—	2	1	no	—
291	#102	—	2	2	no	—
292	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
293	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₃₁	#33838	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	—	4	2	no	—
296	#41	—	4	4	no	—
297	#41	—	8	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×D₄	—	2	1	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	4	4	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	8	2	no	—
309	C₂×D₄	—	8	4	no	—
310	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
315	Q₁₆	—	2	2	no	—
316	Q₁₆	—	2	2	no	—
317	Q₁₆	—	8	1	no	—
318	SD₁₆	—	4	4	no	—
319	D₈	—	2	2	no	—
320	D₈	—	2	1	no	—
321	D₈	—	2	1	no	—
322	D₈	—	2	2	no	—
323	D₈	—	4	4	no	—
324	D₈	—	8	1	no	—
325	D₈	C₂×D₄×C₃₁	1	2	yes	—
326	D₈	C₂×D₄×C₃₁	1	2	yes	complemented
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
331	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
338	C₂²⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	4	1	no	—
340	C₂²⋊C₄	—	4	1	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
343	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₁₆	—	2	2	no	—
345	C₁₆	—	2	1	no	—
346	C₁₆	C₂×D₄×C₃₁	1	2	yes	complemented
347	C₂³	—	4	1	no	—
348	C₂³	—	4	1	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	8	4	no	—
351	C₂³	—	8	4	no	—
352	C₂³	—	16	2	no	—
353	C₂³	C₂×D₈×C₃₁	1	2	yes	—
354	Q₈	—	4	2	no	—
355	Q₈	—	4	2	no	—
356	D₄	—	2	2	no	—
357	D₄	—	2	2	no	—
358	D₄	—	2	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	1	no	—
361	D₄	—	4	1	no	—
362	D₄	—	4	2	no	—
363	D₄	—	8	4	no	—
364	D₄	—	8	4	no	—
365	D₄	—	8	4	no	—
366	D₄	—	8	2	no	—
367	D₄	—	16	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	8	4	no	—
374	C₂×C₄	—	8	2	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
378	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
380	C₈	—	2	2	no	—
381	C₈	—	2	1	no	—
382	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
383	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	—	8	1	no	—
392	C₂²	—	8	1	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	—	16	8	no	—
397	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	2	no	—
400	C₄	—	8	1	no	—
401	C₄	—	8	1	no	—
402	C₄	—	8	2	no	—
403	C₄	C₃₁×#509	1	2	yes	complemented
404	C₄	D₄²×C₃₁	1	1	yes	characteristic
405	C₄	C₃₁×#576	1	1	yes	characteristic
406	C₂	—	4	2	no	—
407	C₂	—	8	2	no	—
408	C₂	—	8	2	no	—
409	C₂	—	16	2	no	—
410	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
411	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
412	C₂	C₃₁×#3416	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 ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#509	C₂	semidirect
2. C₃₁×#3283	C₂	semidirect
3. D₄×D₈×C₃₁	C₂	semidirect
4. C₃₁×#2248	C₂	semidirect
5. C₃₁×#2220	C₂	semidirect
6. C₃₁×#2208	C₂	semidirect
7. C₃₁×#2206	C₂	semidirect
8. C₃₁×#2187	C₂	semidirect
9. C₃₁×#2186	C₂	semidirect
10. C₄×C₃₁×D₁₆	C₂	semidirect
11. C₃₁×#2169	C₂	semidirect
12. C₂×C₃₁×#135	C₂²	semidirect
18. C₂×C₃₁×SD₃₂	C₂²	semidirect
19. C₂×C₃₁×SD₃₂	C₂²	semidirect
21. C₂×C₃₁×D₁₆	C₂²	semidirect
22. C₂×C₃₁×M₃₂	C₂²	semidirect
23. C₃₁×#494	C₂²	semidirect
25. C₃₁×#467	C₂²	semidirect
32. C₄×D₈×C₃₁	C₂²	semidirect
34. C₃₁×#366	C₂²	semidirect
35. C₃₁×C₄⋊C₁₆	C₂²	semidirect
36. C₃₁×#358	C₂²	semidirect
38. C₃₁×#357	C₂²	semidirect
39. C₃₁×#357	C₂²	semidirect
40. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
41. C₄×C₁₆×C₃₁	C₂²	semidirect
74. C₃₁×D₁₆	D₄	semidirect
119. D₈×C₃₁	C₂×D₄	semidirect
139. C₁₆×C₃₁	C₂×D₄	semidirect
140. #33838	C₃₁	direct (N × H)
178. C₂×#509	C₂×C₃₁	semidirect
179. #3283	C₂×C₃₁	semidirect
180. D₄×D₈	C₂×C₃₁	semidirect
181. #2248	C₂×C₃₁	semidirect
182. #2220	C₂×C₃₁	semidirect
183. #2208	C₂×C₃₁	semidirect
184. #2206	C₂×C₃₁	semidirect
185. #2187	C₂×C₃₁	semidirect
186. #2186	C₂×C₃₁	semidirect
187. C₄×D₁₆	C₂×C₃₁	semidirect
188. #2169	C₂×C₃₁	semidirect
208. C₄×C₃₁	#509	semidirect
211. C₂×#135	C₂²×C₃₁	semidirect
217. C₂×SD₃₂	C₂²×C₃₁	semidirect
218. C₂×SD₃₂	C₂²×C₃₁	semidirect
220. C₂×D₁₆	C₂²×C₃₁	semidirect
221. C₂×M₃₂	C₂²×C₃₁	semidirect
222. #494	C₂²×C₃₁	semidirect
224. #467	C₂²×C₃₁	semidirect
231. C₄×D₈	C₂²×C₃₁	semidirect
233. #366	C₂²×C₃₁	semidirect
234. C₄⋊C₁₆	C₂²×C₃₁	semidirect
235. #358	C₂²×C₃₁	semidirect
237. #357	C₂²×C₃₁	semidirect
238. #357	C₂²×C₃₁	semidirect
239. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
240. C₄×C₁₆	C₂²×C₃₁	semidirect
280. D₁₆	D₄×C₃₁	semidirect
294. C₃₁	#33838	direct (N × H)
326. D₈	C₂×D₄×C₃₁	semidirect
346. C₁₆	C₂×D₄×C₃₁	semidirect
403. C₄	C₃₁×#509	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#509	2
C₃₁×#3283	2
D₄×D₈×C₃₁	2
C₃₁×#2248	2
C₃₁×#2220	2
C₃₁×#2208	2
C₃₁×#2206	2
C₃₁×#2187	2
C₃₁×#2186	2
C₄×C₃₁×D₁₆	2
C₃₁×#2169	2
#33838	31

Subgroups — C31×#33838 #21490597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33838 #21490597