Subgroups — C₃₁×#11374 #21468133

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1733	C₂	1	2	yes	maximal
3	C₂×C₃₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1729	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	2	yes	—
11	C₃₁×#430	—	2	2	no	—
12	C₃₁×#430	C₄	1	4	yes	complemented
13	C₂×C₃₁×#108	—	2	2	no	—
14	C₃₁×#339	—	2	4	no	—
15	C₃₁×H(3,ℤ/4)	—	2	2	no	—
16	C₃₁×#336	—	2	2	no	—
17	C₃₁×#336	—	2	4	no	—
18	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
29	C₂²×C₈×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
36	C₃₁×#117	—	2	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₃₁×#117	—	2	2	no	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	D₄	1	4	yes	—
48	C₃₁×#108	—	2	8	no	—
49	C₂⁴×C₃₁	#41	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	#41	1	2	yes	—
59	C₂²×C₄×C₃₁	#41	1	2	yes	—
60	C₂²×C₄×C₃₁	#41	1	2	yes	—
61	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
62	M₁₆×C₃₁	—	2	4	no	—
63	M₁₆×C₃₁	—	2	4	no	—
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	2	4	no	—
66	M₁₆×C₃₁	—	4	2	no	—
67	M₁₆×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₂×C₈×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	2	no	—
83	C₂²⋊C₄×C₃₁	—	4	2	no	—
84	C₂²⋊C₄×C₃₁	—	4	2	no	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	2	4	no	—
87	C₄²×C₃₁	—	4	1	no	—
88	C₄²×C₃₁	—	4	1	no	—
89	#11374	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂³×C₃₁	#123	1	2	yes	—
94	C₂³×C₃₁	#124	1	1	yes	characteristic
95	C₂³×C₃₁	#126	1	2	yes	—
96	C₂×C₄×C₃₁	—	4	8	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#117	1	2	yes	—
102	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#121	1	2	yes	—
105	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#123	1	2	yes	—
109	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#123	1	2	yes	—
112	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#125	1	2	yes	—
114	C₂×C₄×C₃₁	#125	1	2	yes	—
115	C₂×C₄×C₃₁	#126	1	2	yes	—
116	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
118	C₈×C₃₁	—	2	4	no	—
119	C₈×C₃₁	—	4	4	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₂×#430	C₂×C₃₁	1	1	yes	characteristic
122	#1733	C₂×C₃₁	1	2	yes	—
123	C₂×H(3,ℤ/4)	C₂×C₃₁	1	1	yes	characteristic
124	#1729	C₂×C₃₁	1	1	yes	characteristic
125	#1729	C₂×C₃₁	1	2	yes	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	4	no	—
128	C₂²×C₃₁	—	4	1	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	2	no	—
131	C₂²×C₃₁	#387	1	2	yes	—
132	C₂²×C₃₁	#391	1	1	yes	characteristic
133	C₂²×C₃₁	#397	1	2	yes	—
134	C₂²×C₃₁	#399	1	1	yes	characteristic
135	C₂²×C₃₁	#400	1	1	yes	characteristic
136	C₄×C₃₁	—	8	4	no	—
137	C₄×C₃₁	—	8	2	no	—
138	C₄×C₃₁	#381	1	1	yes	characteristic
139	C₄×C₃₁	#387	1	2	yes	—
140	C₄×C₃₁	#390	1	1	yes	characteristic
141	C₄×C₃₁	#398	1	2	yes	—
142	C₄×C₃₁	#399	1	1	yes	characteristic
143	C₄×C₃₁	#400	1	1	yes	characteristic
144	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
145	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#117	C₂²×C₃₁	1	2	yes	—
149	#430	—	2	2	no	—
150	#430	C₄×C₃₁	1	4	yes	complemented
151	C₂×#108	—	2	2	no	—
152	#339	—	2	4	no	—
153	H(3,ℤ/4)	—	2	2	no	—
154	#336	—	2	2	no	—
155	#336	—	2	4	no	—
156	C₂×C₃₁	—	2	2	no	—
157	C₂×C₃₁	—	2	2	no	—
158	C₂×C₃₁	—	2	1	no	—
159	C₂×C₃₁	—	2	1	no	—
160	C₂×C₃₁	#1445	1	1	yes	characteristic
161	C₂×C₃₁	#2098	1	2	yes	—
162	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
171	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
172	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	D₄×C₃₁	1	2	yes	—
179	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
188	C₂×C₂²⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	D₄×C₃₁	1	4	yes	—
192	#108	—	2	8	no	—
193	C₃₁	#11374	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	#41×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	#41×C₃₁	1	2	yes	—
204	C₂²×C₄	#41×C₃₁	1	2	yes	—
205	C₂²×C₄	#41×C₃₁	1	2	yes	—
206	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₂³	—	4	2	no	—
235	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
236	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂³	C₃₁×#123	1	2	yes	—
238	C₂³	C₃₁×#124	1	1	yes	characteristic
239	C₂³	C₃₁×#126	1	2	yes	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₃₁×#117	1	2	yes	—
246	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
247	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₃₁×#121	1	2	yes	—
249	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
250	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
251	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₄	C₃₁×#123	1	2	yes	—
253	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
254	C₂×C₄	C₃₁×#123	1	2	yes	—
255	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
256	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
257	C₂×C₄	C₃₁×#125	1	2	yes	—
258	C₂×C₄	C₃₁×#125	1	2	yes	—
259	C₂×C₄	C₃₁×#126	1	2	yes	—
260	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
261	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	C₃₁×#387	1	2	yes	—
271	C₂²	C₃₁×#391	1	1	yes	characteristic
272	C₂²	C₃₁×#397	1	2	yes	—
273	C₂²	C₃₁×#399	1	1	yes	characteristic
274	C₂²	C₃₁×#400	1	1	yes	characteristic
275	C₄	—	8	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₃₁×#381	1	1	yes	characteristic
278	C₄	C₃₁×#387	1	2	yes	—
279	C₄	C₃₁×#390	1	1	yes	characteristic
280	C₄	C₃₁×#398	1	2	yes	—
281	C₄	C₃₁×#399	1	1	yes	characteristic
282	C₄	C₃₁×#400	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	1	no	—
287	C₂	C₃₁×#1445	1	1	yes	characteristic
288	C₂	C₃₁×#2098	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#399
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#430	C₄	semidirect
89. #11374	C₃₁	direct (N × H)
150. #430	C₄×C₃₁	semidirect
193. C₃₁	#11374	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×#430	2
C₃₁×#1733	2
C₂×C₃₁×H(3,ℤ/4)	2
C₃₁×#1729	2
#11374	31

Subgroups — C31×#11374 #21468133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11374 #21468133