Subgroups — C₃₁×#33198 #21489957

Order 7936.

898 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#477	C₂	1	2	yes	maximal
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1987	C₂	1	2	yes	maximal
5	C₃₁×#1947	C₂	1	4	yes	maximal
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1803	C₂	1	4	yes	maximal
8	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#477	—	2	8	no	—
13	C₂×C₃₁×#107	C₂²	1	4	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#103	C₂²	1	4	yes	—
17	C₃₁×#384	—	2	1	no	—
18	C₃₁×#384	—	2	1	no	—
19	C₃₁×#384	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
21	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₄³×C₃₁	—	2	1	no	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#340	C₂²	1	8	yes	—
27	C₂²×Q₈×C₃₁	—	2	2	no	—
28	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
30	C₃₁×C₄⋊Q₈	—	2	4	no	—
31	C₃₁×C₄⋊Q₈	—	2	4	no	—
32	C₃₁×C₄⋊Q₈	—	4	8	no	—
33	C₄×Q₈×C₃₁	—	2	2	no	—
34	C₄×Q₈×C₃₁	—	2	4	no	—
35	C₄×Q₈×C₃₁	D₄	1	4	yes	—
36	C₄×Q₈×C₃₁	Q₈	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	D₄	1	2	yes	—
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#107	—	2	8	no	—
55	C₃₁×C₄⋊C₈	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	8	no	—
57	C₃₁×#103	—	2	8	no	—
58	C₃₁×#95	—	2	1	no	—
59	C₃₁×#95	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	2	1	no	—
61	C₂×Q₈×C₃₁	—	2	2	no	—
62	C₂×Q₈×C₃₁	—	4	8	no	—
63	C₂×Q₈×C₃₁	—	4	4	no	—
64	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
65	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	8	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	—	2	8	no	—
79	C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	4	8	no	—
84	C₄⋊C₄×C₃₁	—	4	4	no	—
85	C₄⋊C₄×C₃₁	—	4	8	no	—
86	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
88	C₄²×C₃₁	—	2	4	no	—
89	C₄²×C₃₁	—	2	8	no	—
90	C₄²×C₃₁	—	2	4	no	—
91	C₄²×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	—	4	4	no	—
93	C₄²×C₃₁	—	4	2	no	—
94	C₄²×C₃₁	Q₁₆	1	4	yes	—
95	C₄²×C₃₁	C₂×D₄	1	2	yes	—
96	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
97	C₄²×C₃₁	#41	1	2	yes	—
98	C₄²×C₃₁	#41	1	4	yes	—
99	#33198	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₃₁	—	2	2	no	—
102	Q₈×C₃₁	—	2	4	no	—
103	Q₈×C₃₁	—	8	8	no	—
104	Q₈×C₃₁	C₄⋊Q₈	1	4	yes	complemented
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	8	no	—
117	C₂×C₄×C₃₁	#121	1	2	yes	—
118	C₂×C₄×C₃₁	#121	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
123	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
124	C₂×C₄×C₃₁	#137	1	2	yes	—
125	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
129	C₈×C₃₁	—	4	4	no	—
130	C₂×#477	C₂×C₃₁	1	2	yes	—
131	#2868	C₂×C₃₁	1	1	yes	characteristic
132	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
133	#1987	C₂×C₃₁	1	2	yes	—
134	#1947	C₂×C₃₁	1	4	yes	—
135	#1842	C₂×C₃₁	1	1	yes	characteristic
136	#1803	C₂×C₃₁	1	4	yes	—
137	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
139	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
142	C₄×C₃₁	—	2	4	no	—
143	C₄×C₃₁	—	2	4	no	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	4	4	no	—
146	C₄×C₃₁	—	4	2	no	—
147	C₄×C₃₁	—	8	4	no	—
148	C₄×C₃₁	#462	1	4	yes	—
149	C₄×C₃₁	#477	1	8	yes	complemented
150	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₄×C₃₁	#532	1	1	yes	characteristic
152	C₄×C₃₁	#533	1	2	yes	—
153	C₂×C₄⋊Q₈	—	2	2	no	—
154	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
157	#477	—	2	8	no	—
158	C₂×#107	C₂²×C₃₁	1	4	yes	—
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
161	C₂×#103	C₂²×C₃₁	1	4	yes	—
162	#384	—	2	1	no	—
163	#384	—	2	1	no	—
164	#384	C₂²×C₃₁	1	4	yes	—
165	C₄×C₄⋊C₄	—	2	2	no	—
166	C₄×C₄⋊C₄	—	2	1	no	—
167	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
169	C₄³	—	2	1	no	—
170	C₄³	C₂²×C₃₁	1	1	yes	characteristic
171	#340	C₂²×C₃₁	1	8	yes	—
172	C₂×C₃₁	#2857	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#462	1	1	yes	characteristic
174	C₂×C₃₁	#3038	1	1	yes	characteristic
175	C₂×C₃₁	C₂×#477	1	2	yes	—
176	C₂×C₃₁	#3079	1	2	yes	—
177	C₂²×Q₈	—	2	2	no	—
178	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
180	C₄⋊Q₈	—	2	4	no	—
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	4	8	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	D₄×C₃₁	1	4	yes	—
186	C₄×Q₈	Q₈×C₃₁	1	8	yes	complemented
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
194	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	D₄×C₃₁	1	2	yes	—
200	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
204	#107	—	2	8	no	—
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	8	no	—
207	#103	—	2	8	no	—
208	#95	—	2	1	no	—
209	#95	—	2	2	no	—
210	C₃₁	#33198	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	8	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
216	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	8	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	8	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	Q₁₆×C₃₁	1	4	yes	—
246	C₄²	C₂×D₄×C₃₁	1	2	yes	—
247	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
248	C₄²	#41×C₃₁	1	2	yes	—
249	C₄²	#41×C₃₁	1	4	yes	—
250	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	2	4	no	—
253	Q₈	—	8	8	no	—
254	Q₈	C₃₁×C₄⋊Q₈	1	4	yes	complemented
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	C₃₁×#121	1	2	yes	—
268	C₂×C₄	C₃₁×#121	1	2	yes	—
269	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
270	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
271	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
272	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
273	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
274	C₂×C₄	C₃₁×#137	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
277	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
278	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
279	C₈	—	4	4	no	—
280	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
281	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
282	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
283	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
284	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	2	no	—
290	C₄	—	8	4	no	—
291	C₄	C₃₁×#462	1	4	yes	—
292	C₄	C₃₁×#477	1	8	yes	complemented
293	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
294	C₄	C₃₁×#532	1	1	yes	characteristic
295	C₄	C₃₁×#533	1	2	yes	—
296	C₂	C₃₁×#2857	1	1	yes	characteristic
297	C₂	C₂×C₃₁×#462	1	1	yes	characteristic
298	C₂	C₃₁×#3038	1	1	yes	characteristic
299	C₂	C₂×C₃₁×#477	1	2	yes	—
300	C₂	C₃₁×#3079	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₄×Q₈×C₃₁	Q₈	semidirect
99. #33198	C₃₁	direct (N × H)
104. Q₈×C₃₁	C₄⋊Q₈	semidirect
149. C₄×C₃₁	#477	semidirect
186. C₄×Q₈	Q₈×C₃₁	semidirect
210. C₃₁	#33198	direct (N × H)
254. Q₈	C₃₁×C₄⋊Q₈	semidirect
292. C₄	C₃₁×#477	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#477	2
C₃₁×#2868	2
C₄²×Q₈×C₃₁	2
C₃₁×#1987	2
C₃₁×#1947	2
C₃₁×#1842	2
C₃₁×#1803	2
#33198	31

Subgroups — C31×#33198 #21489957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33198 #21489957