Subgroups — C₃₁×#11419 #21468178

Order 7936.

532 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1314	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×M₃₂	—	2	1	no	—
9	C₂²×C₁₆×C₃₁	—	2	1	no	—
10	C₃₁×#408	—	2	1	no	—
11	C₃₁×#408	—	2	1	no	—
12	C₃₁×#408	—	2	1	no	—
13	C₃₁×#408	—	2	1	no	—
14	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
18	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
19	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₂×#41×C₃₁	—	2	1	no	—
21	C₂×#41×C₃₁	—	2	1	no	—
22	C₂×#41×C₃₁	—	2	1	no	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₂×#41×C₃₁	—	4	2	no	—
25	C₂×#41×C₃₁	—	4	2	no	—
26	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	C₈	1	1	yes	characteristic
28	C₂²×D₄×C₃₁	C₈	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂×M₁₆×C₃₁	—	4	2	no	—
34	C₂²×C₈×C₃₁	—	4	2	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×M₃₂	—	4	2	no	—
43	C₂×C₁₆×C₃₁	—	2	2	no	—
44	C₂×C₁₆×C₃₁	—	2	2	no	—
45	C₂×C₁₆×C₃₁	—	4	2	no	—
46	C₃₁×C₂²⋊C₈	—	2	2	no	—
47	C₃₁×C₂²⋊C₈	—	2	2	no	—
48	C₃₁×C₂²⋊C₈	—	2	2	no	—
49	C₃₁×C₂²⋊C₈	—	2	2	no	—
50	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
51	C₂⁴×C₃₁	C₂×C₈	1	2	yes	—
52	#41×C₃₁	—	4	4	no	—
53	#41×C₃₁	—	4	4	no	—
54	#41×C₃₁	—	4	2	no	—
55	#41×C₃₁	—	4	2	no	—
56	#41×C₃₁	—	4	2	no	—
57	#41×C₃₁	—	4	2	no	—
58	C₂×Q₈×C₃₁	—	2	1	no	—
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	—	4	2	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	—	2	1	no	—
65	C₂×D₄×C₃₁	—	4	2	no	—
66	C₂×D₄×C₃₁	—	4	4	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	M₁₆×C₃₁	—	4	2	no	—
81	M₁₆×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	4	1	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	—	4	1	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₁₆×C₃₁	—	4	2	no	—
95	C₁₆×C₃₁	—	4	2	no	—
96	#11419	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	—	2	4	no	—
101	C₂³×C₃₁	—	4	1	no	—
102	C₂³×C₃₁	—	4	2	no	—
103	C₂³×C₃₁	—	4	2	no	—
104	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
105	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
106	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
107	Q₈×C₃₁	—	4	2	no	—
108	Q₈×C₃₁	—	4	2	no	—
109	D₄×C₃₁	—	4	2	no	—
110	D₄×C₃₁	—	4	4	no	—
111	D₄×C₃₁	—	4	4	no	—
112	D₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#99	1	2	yes	—
128	C₂×C₄×C₃₁	#100	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	—	2	1	no	—
134	C₈×C₃₁	—	2	1	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
137	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#1314	C₂×C₃₁	1	4	yes	complemented
140	C₂²×C₃₁	—	2	1	no	—
141	C₂²×C₃₁	—	2	1	no	—
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	2	1	no	—
147	C₂²×C₃₁	—	4	4	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	—	4	2	no	—
150	C₂²×C₃₁	—	4	2	no	—
151	C₂²×C₃₁	—	8	4	no	—
152	C₂²×C₃₁	#323	1	2	yes	—
153	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₂²×C₃₁	#504	1	2	yes	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	#323	1	2	yes	—
159	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
160	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
161	C₂²×#41	C₄×C₃₁	1	1	yes	characteristic
162	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×M₃₂	—	2	1	no	—
165	C₂²×C₁₆	—	2	1	no	—
166	#408	—	2	1	no	—
167	#408	—	2	1	no	—
168	#408	—	2	1	no	—
169	#408	—	2	1	no	—
170	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
171	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₂²⋊C₁₆	—	2	1	no	—
173	C₂²⋊C₁₆	—	2	1	no	—
174	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
175	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
176	C₂×C₃₁	—	2	2	no	—
177	C₂×C₃₁	—	2	2	no	—
178	C₂×C₃₁	—	4	1	no	—
179	C₂×C₃₁	—	8	2	no	—
180	C₂×C₃₁	C₂×#323	1	1	yes	characteristic
181	C₂×C₃₁	#2113	1	1	yes	characteristic
182	C₂×C₃₁	#2115	1	1	yes	characteristic
183	C₂×#41	—	2	1	no	—
184	C₂×#41	—	2	1	no	—
185	C₂×#41	—	2	1	no	—
186	C₂×#41	—	2	1	no	—
187	C₂×#41	—	4	2	no	—
188	C₂×#41	—	4	2	no	—
189	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
190	C₂²×D₄	C₈×C₃₁	1	1	yes	characteristic
191	C₂²×D₄	C₈×C₃₁	1	2	yes	—
192	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
193	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	4	2	no	—
197	C₂²×C₈	—	4	2	no	—
198	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
199	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
205	M₃₂	—	4	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	4	2	no	—
209	C₂²⋊C₈	—	2	2	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₃₁	#11419	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
215	C₂⁴	C₂×C₈×C₃₁	1	2	yes	—
216	#41	—	4	4	no	—
217	#41	—	4	2	no	—
218	#41	—	4	2	no	—
219	#41	—	4	4	no	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
241	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₁₆	—	4	2	no	—
259	C₁₆	—	4	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
268	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
269	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₃₁×#99	1	2	yes	—
291	C₂×C₄	C₃₁×#100	1	2	yes	—
292	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	8	4	no	—
311	C₂²	C₃₁×#323	1	2	yes	—
312	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
313	C₂²	C₃₁×#504	1	2	yes	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	C₃₁×#323	1	2	yes	—
318	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
319	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×C₃₁×#323	1	1	yes	characteristic
325	C₂	C₃₁×#2113	1	1	yes	characteristic
326	C₂	C₃₁×#2115	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2112	C₂	semidirect
3. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
4. C₃₁×#1314	C₂	semidirect
96. #11419	C₃₁	direct (N × H)
137. #2112	C₂×C₃₁	semidirect
138. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
139. #1314	C₂×C₃₁	semidirect
213. C₃₁	#11419	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#408	2
C₃₁×#2112	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1314	2
#11419	31

Subgroups — C31×#11419 #21468178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11419 #21468178