Subgroups — C₃₁×#33757 #21490516

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3050	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2217	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#525	—	2	2	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#467	C₂²	1	2	yes	—
19	C₃₁×#467	C₂²	1	2	yes	—
20	C₃₁×#466	C₂²	1	2	yes	—
21	C₃₁×#466	C₂²	1	2	yes	—
22	C₃₁×#465	—	2	2	no	—
23	C₃₁×#465	—	2	2	no	—
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₃₁×#416	—	2	2	no	—
26	C₃₁×#367	C₂²	1	2	yes	complemented
27	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#358	C₂²	1	2	yes	complemented
30	C₃₁×#358	C₂²	1	2	yes	complemented
31	C₃₁×#357	C₂²	1	2	yes	complemented
32	C₃₁×#357	C₂²	1	2	yes	complemented
33	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₂×#41×C₃₁	—	2	2	no	—
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×#135	—	4	2	no	—
38	C₃₁×#135	—	4	2	no	—
39	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₃₁	—	2	2	no	—
41	C₂×D₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	—	4	1	no	—
43	C₂²×C₈×C₃₁	—	4	1	no	—
44	C₂²×C₈×C₃₁	D₄	1	2	yes	—
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#123	—	4	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₃₁×#121	—	2	2	no	—
53	C₃₁×#120	—	4	2	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	—	4	2	no	—
56	C₃₁×#117	—	2	2	no	—
57	C₃₁×Q₃₂	—	4	1	no	—
58	C₃₁×SD₃₂	—	4	2	no	—
59	C₃₁×D₁₆	—	4	1	no	—
60	C₂×C₁₆×C₃₁	—	4	1	no	—
61	C₂×C₁₆×C₃₁	—	4	1	no	—
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	C₂³	1	2	yes	—
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#106	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
73	#41×C₃₁	—	4	4	no	—
74	#41×C₃₁	—	4	4	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	4	2	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	D₈	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₃₁	—	2	2	no	—
88	Q₁₆×C₃₁	—	2	2	no	—
89	SD₁₆×C₃₁	—	4	4	no	—
90	D₈×C₃₁	—	2	2	no	—
91	D₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	4	2	no	—
97	C₂×C₈×C₃₁	—	4	1	no	—
98	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₄²×C₃₁	—	4	2	no	—
111	C₁₆×C₃₁	—	2	1	no	—
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	—	2	1	no	—
114	#33757	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	2	2	no	—
116	C₂³×C₃₁	—	4	1	no	—
117	C₂³×C₃₁	—	4	1	no	—
118	C₂³×C₃₁	—	4	2	no	—
119	C₂³×C₃₁	C₂×D₈	1	2	yes	—
120	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₃₁	—	4	2	no	—
122	Q₈×C₃₁	—	4	2	no	—
123	D₄×C₃₁	—	4	2	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	8	4	no	—
126	D₄×C₃₁	—	8	4	no	—
127	D₄×C₃₁	—	8	4	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	8	4	no	—
138	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
145	#3050	C₂×C₃₁	1	2	yes	complemented
146	#2237	C₂×C₃₁	1	2	yes	complemented
147	#2234	C₂×C₃₁	1	2	yes	complemented
148	#2217	C₂×C₃₁	1	2	yes	complemented
149	#2216	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2215	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2189	C₂×C₃₁	1	2	yes	complemented
152	#2186	C₂×C₃₁	1	2	yes	complemented
153	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	4	4	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	#508	1	2	yes	—
164	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	#534	1	1	yes	characteristic
171	C₄×C₃₁	#575	1	1	yes	characteristic
172	C₂×#135	C₂²×C₃₁	1	2	yes	—
173	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
174	#525	—	2	2	no	—
175	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
176	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
177	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
178	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
179	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
180	#467	C₂²×C₃₁	1	2	yes	—
181	#467	C₂²×C₃₁	1	2	yes	—
182	#466	C₂²×C₃₁	1	2	yes	—
183	#466	C₂²×C₃₁	1	2	yes	—
184	#465	—	2	2	no	—
185	#465	—	2	2	no	—
186	#427	C₂²×C₃₁	1	2	yes	—
187	#416	—	2	2	no	—
188	#367	C₂²×C₃₁	1	2	yes	complemented
189	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
190	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
191	#358	C₂²×C₃₁	1	2	yes	complemented
192	#358	C₂²×C₃₁	1	2	yes	complemented
193	#357	C₂²×C₃₁	1	2	yes	complemented
194	#357	C₂²×C₃₁	1	2	yes	complemented
195	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
196	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
197	C₂×C₃₁	—	2	2	no	—
198	C₂×C₃₁	—	4	2	no	—
199	C₂×C₃₁	—	16	2	no	—
200	C₂×C₃₁	#3186	1	1	yes	characteristic
201	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
202	C₂×C₃₁	#3414	1	1	yes	characteristic
203	C₂×#41	—	2	2	no	—
204	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
205	#135	—	4	2	no	—
206	#135	—	4	2	no	—
207	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×D₈	C₂³×C₃₁	1	2	yes	—
210	C₂²×C₈	—	4	1	no	—
211	C₂²×C₈	—	4	1	no	—
212	C₂²×C₈	D₄×C₃₁	1	2	yes	—
213	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
215	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
216	#123	—	4	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	2	2	no	—
221	#120	—	4	2	no	—
222	C₄×D₄	—	4	2	no	—
223	C₄×D₄	—	4	2	no	—
224	#117	—	2	2	no	—
225	Q₃₂	—	4	1	no	—
226	SD₃₂	—	4	2	no	—
227	D₁₆	—	4	1	no	—
228	C₂×C₁₆	—	4	1	no	—
229	C₂×C₁₆	—	4	1	no	—
230	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
232	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
233	#107	C₂³×C₃₁	1	2	yes	—
234	#107	C₂³×C₃₁	1	2	yes	—
235	#106	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	C₃₁	#33757	1	1	yes	characteristic, direct product factor, Sylow
241	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
242	#41	—	4	4	no	—
243	#41	—	4	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	D₈×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	SD₁₆	—	4	4	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
269	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	1	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×D₈×C₃₁	1	2	yes	—
288	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	2	no	—
311	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
312	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	C₃₁×#508	1	2	yes	—
323	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	C₃₁×#534	1	1	yes	characteristic
330	C₄	C₃₁×#575	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	—	16	2	no	—
334	C₂	C₃₁×#3186	1	1	yes	characteristic
335	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
336	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 ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3050	C₂	semidirect
2. C₃₁×#2237	C₂	semidirect
3. C₃₁×#2234	C₂	semidirect
4. C₃₁×#2217	C₂	semidirect
5. C₃₁×#2216	C₂	semidirect
6. C₃₁×#2215	C₂	semidirect
7. C₃₁×#2189	C₂	semidirect
8. C₃₁×#2186	C₂	semidirect
9. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
13. C₂×C₃₁×Q₃₂	C₂²	semidirect
14. C₂×C₃₁×SD₃₂	C₂²	semidirect
15. C₂×C₃₁×D₁₆	C₂²	semidirect
26. C₃₁×#367	C₂²	semidirect
27. C₃₁×#366	C₂²	semidirect
28. C₃₁×#366	C₂²	semidirect
29. C₃₁×#358	C₂²	semidirect
30. C₃₁×#358	C₂²	semidirect
31. C₃₁×#357	C₂²	semidirect
32. C₃₁×#357	C₂²	semidirect
114. #33757	C₃₁	direct (N × H)
145. #3050	C₂×C₃₁	semidirect
146. #2237	C₂×C₃₁	semidirect
147. #2234	C₂×C₃₁	semidirect
148. #2217	C₂×C₃₁	semidirect
149. #2216	C₂×C₃₁	semidirect
150. #2215	C₂×C₃₁	semidirect
151. #2189	C₂×C₃₁	semidirect
152. #2186	C₂×C₃₁	semidirect
153. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
175. C₂×Q₃₂	C₂²×C₃₁	semidirect
176. C₂×SD₃₂	C₂²×C₃₁	semidirect
177. C₂×D₁₆	C₂²×C₃₁	semidirect
188. #367	C₂²×C₃₁	semidirect
189. #366	C₂²×C₃₁	semidirect
190. #366	C₂²×C₃₁	semidirect
191. #358	C₂²×C₃₁	semidirect
192. #358	C₂²×C₃₁	semidirect
193. #357	C₂²×C₃₁	semidirect
194. #357	C₂²×C₃₁	semidirect
240. C₃₁	#33757	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×#3050	2
C₃₁×#2237	2
C₃₁×#2234	2
C₃₁×#2217	2
C₃₁×#2216	2
C₃₁×#2215	2
C₃₁×#2189	2
C₃₁×#2186	2
C₂×C₃₁×C₂²⋊C₁₆	2
#33757	31

Subgroups — C31×#33757 #21490516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33757 #21490516