Subgroups — C₃₁×#33934 #21490693

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2227	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2210	C₂	1	2	yes	maximal
8	C₃₁×#2208	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#556	—	2	1	no	—
13	D₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	2	yes	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#474	—	2	1	no	—
21	C₃₁×#474	—	2	2	no	—
22	C₃₁×#464	—	2	2	no	—
23	C₃₁×#461	—	2	1	no	—
24	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#366	C₂²	1	1	yes	characteristic
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃₁×#358	C₂²	1	2	yes	—
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#357	C₂²	1	1	yes	characteristic
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	—	4	1	no	—
42	C₂×Q₁₆×C₃₁	—	2	1	no	—
43	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
44	C₂×SD₁₆×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×#126	—	4	2	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	4	1	no	—
53	C₃₁×C₂²⋊Q₈	—	4	1	no	—
54	C₃₁×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₃₁×#117	—	4	1	no	—
61	C₃₁×SD₃₂	—	2	4	no	—
62	C₃₁×D₁₆	—	2	2	no	—
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	—	2	2	no	—
68	C₃₁×#107	C₂³	1	1	yes	characteristic
69	C₃₁×#107	C₂³	1	1	yes	characteristic
70	C₃₁×#107	C₂³	1	2	yes	—
71	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
73	C₃₁×#103	—	2	2	no	—
74	C₃₁×#102	—	2	1	no	—
75	C₃₁×#102	—	2	1	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	4	2	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	—	8	4	no	—
83	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	Q₁₆×C₃₁	—	2	2	no	—
90	Q₁₆×C₃₁	—	2	2	no	—
91	Q₁₆×C₃₁	—	8	1	no	—
92	SD₁₆×C₃₁	—	8	2	no	—
93	D₈×C₃₁	—	2	1	no	—
94	D₈×C₃₁	—	2	1	no	—
95	D₈×C₃₁	—	2	1	no	—
96	D₈×C₃₁	#41	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	2	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
109	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	1	no	—
113	C₂²⋊C₄×C₃₁	—	4	1	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	2	yes	—
118	C₁₆×C₃₁	—	2	1	no	—
119	C₁₆×C₃₁	—	2	2	no	—
120	C₁₆×C₃₁	—	2	1	no	—
121	#33934	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	4	1	no	—
123	C₂³×C₃₁	—	4	1	no	—
124	Q₈×C₃₁	—	2	2	no	—
125	Q₈×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	8	4	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	4	1	no	—
131	D₄×C₃₁	—	4	1	no	—
132	D₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	8	4	no	—
138	C₂×C₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	—	8	2	no	—
140	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
149	#3393	C₂×C₃₁	1	1	yes	characteristic, complemented
150	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2243	C₂×C₃₁	1	2	yes	complemented
152	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2227	C₂×C₃₁	1	2	yes	complemented
154	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2210	C₂×C₃₁	1	2	yes	—
156	#2208	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2175	C₂×C₃₁	1	2	yes	complemented
159	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic
160	C₂²×C₃₁	—	4	2	no	—
161	C₂²×C₃₁	—	8	1	no	—
162	C₂²×C₃₁	—	8	1	no	—
163	C₂²×C₃₁	—	8	1	no	—
164	C₂²×C₃₁	—	8	1	no	—
165	C₂²×C₃₁	—	8	1	no	—
166	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
167	C₄×C₃₁	—	2	2	no	—
168	C₄×C₃₁	—	2	2	no	—
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	—	8	1	no	—
172	C₄×C₃₁	—	8	2	no	—
173	C₄×C₃₁	—	8	1	no	—
174	C₄×C₃₁	#509	1	2	yes	—
175	C₄×C₃₁	#548	1	1	yes	characteristic
176	C₄×C₃₁	#577	1	1	yes	characteristic
177	#556	—	2	1	no	—
178	D₄×Q₈	—	2	1	no	—
179	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
180	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
181	#500	C₂²×C₃₁	1	1	yes	characteristic
182	#500	C₂²×C₃₁	1	2	yes	—
183	#495	C₂²×C₃₁	1	2	yes	—
184	#494	C₂²×C₃₁	1	1	yes	characteristic
185	#474	—	2	1	no	—
186	#474	—	2	2	no	—
187	#464	—	2	2	no	—
188	#461	—	2	1	no	—
189	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
191	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×D₈	C₂²×C₃₁	1	2	yes	—
193	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
194	#366	C₂²×C₃₁	1	1	yes	characteristic
195	#366	C₂²×C₃₁	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
198	#358	C₂²×C₃₁	1	2	yes	—
199	#358	C₂²×C₃₁	1	2	yes	—
200	#358	C₂²×C₃₁	1	2	yes	—
201	#357	C₂²×C₃₁	1	1	yes	characteristic
202	#357	C₂²×C₃₁	1	1	yes	characteristic
203	#357	C₂²×C₃₁	1	2	yes	—
204	#346	C₂²×C₃₁	1	2	yes	—
205	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×C₃₁	—	8	2	no	—
207	C₂×C₃₁	—	16	1	no	—
208	C₂×C₃₁	#3358	1	1	yes	characteristic
209	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
210	C₂×C₃₁	#3417	1	1	yes	characteristic
211	C₂²×Q₈	—	4	1	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
214	C₂×SD₁₆	—	2	2	no	—
215	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
216	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#126	—	4	2	no	—
221	#124	—	2	2	no	—
222	C₂²⋊Q₈	—	4	1	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	4	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	1	no	—
230	#117	—	4	1	no	—
231	SD₃₂	—	2	4	no	—
232	D₁₆	—	2	2	no	—
233	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
234	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
236	#107	—	2	1	no	—
237	#107	—	2	2	no	—
238	#107	C₂³×C₃₁	1	1	yes	characteristic
239	#107	C₂³×C₃₁	1	2	yes	—
240	#107	C₂³×C₃₁	1	1	yes	characteristic
241	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
242	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
243	#103	—	2	2	no	—
244	#102	—	2	1	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₃₁	1	2	yes	—
249	C₃₁	#33934	1	1	yes	characteristic, direct product factor, Sylow
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	8	4	no	—
254	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	8	1	no	—
263	SD₁₆	—	8	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	#41×C₃₁	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
270	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
280	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₃₁	1	2	yes	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	8	4	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	4	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
312	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
313	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
318	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	C₃₁×#509	1	2	yes	—
334	C₄	C₃₁×#548	1	1	yes	characteristic
335	C₄	C₃₁×#577	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	16	1	no	—
338	C₂	C₃₁×#3358	1	1	yes	characteristic
339	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
340	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3393	C₂	semidirect
2. Q₈×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2243	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
5. C₃₁×#2227	C₂	semidirect
6. C₃₁×#2224	C₂	semidirect
8. C₃₁×#2208	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
10. C₃₁×#2175	C₂	semidirect
121. #33934	C₃₁	direct (N × H)
149. #3393	C₂×C₃₁	semidirect
150. Q₈×D₈	C₂×C₃₁	semidirect
151. #2243	C₂×C₃₁	semidirect
152. #2241	C₂×C₃₁	semidirect
153. #2227	C₂×C₃₁	semidirect
154. #2224	C₂×C₃₁	semidirect
156. #2208	C₂×C₃₁	semidirect
157. #2183	C₂×C₃₁	semidirect
158. #2175	C₂×C₃₁	semidirect
249. C₃₁	#33934	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×#3393	2
Q₈×D₈×C₃₁	2
C₃₁×#2243	2
C₃₁×#2241	2
C₃₁×#2227	2
C₃₁×#2224	2
C₃₁×#2210	2
C₃₁×#2208	2
C₃₁×#2183	2
C₃₁×#2175	2
C₄×C₃₁×D₁₆	2
#33934	31

Subgroups — C31×#33934 #21490693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33934 #21490693