Subgroups — C₃₁×#33168 #21489927

Order 7936.

1122 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#463	C₂	1	3	yes	maximal
2	C₂×C₃₁×#461	C₂	1	3	yes	maximal, complemented
3	C₃₁×#2658	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1983	C₂	1	3	yes	maximal, complemented
6	C₃₁×#1981	C₂	1	3	yes	maximal, complemented
7	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	C₂²	1	3	yes	—
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₂×C₃₁×#124	C₂²	1	3	yes	—
12	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
13	C₃₁×#463	—	2	12	no	—
14	C₃₁×#461	—	2	12	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₃₁×#103	C₂²	1	3	yes	—
18	C₂×C₃₁×#103	C₂²	1	6	yes	—
19	C₂×C₃₁×#102	C₂²	1	3	yes	—
20	C₃₁×#399	—	2	1	no	—
21	C₃₁×#399	—	2	3	no	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
23	C₄×C₂²⋊C₄×C₃₁	—	2	3	no	—
24	C₃₁×#376	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#95	—	2	1	no	—
26	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
27	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	—	2	3	no	—
30	C₂×SD₁₆×C₃₁	—	2	12	no	—
31	C₂×SD₁₆×C₃₁	—	2	6	no	—
32	C₂×D₈×C₃₁	—	2	6	no	—
33	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×C₄⋊Q₈	—	2	6	no	—
36	C₃₁×C₄⋊Q₈	—	2	6	no	—
37	C₃₁×#124	—	2	6	no	—
38	C₃₁×#124	—	2	6	no	—
39	C₄×D₄×C₃₁	—	2	6	no	—
40	C₄×D₄×C₃₁	D₄	1	12	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
46	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
47	C₂×C₄²×C₃₁	—	2	3	no	—
48	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
49	C₃₁×C₄⋊C₈	—	2	6	no	—
50	C₃₁×C₄⋊C₈	—	2	6	no	—
51	C₃₁×#103	—	2	12	no	—
52	C₃₁×#103	—	2	6	no	—
53	C₃₁×#102	—	2	6	no	—
54	C₃₁×#95	—	2	3	no	—
55	C₃₁×#95	D₄	1	1	yes	characteristic
56	C₃₁×#95	D₄	1	3	yes	—
57	C₂⁴×C₃₁	—	2	1	no	—
58	C₂⁴×C₃₁	—	2	1	no	—
59	C₂×Q₈×C₃₁	—	2	3	no	—
60	C₂×Q₈×C₃₁	—	2	6	no	—
61	C₂×Q₈×C₃₁	—	2	3	no	—
62	C₂×Q₈×C₃₁	—	2	6	no	—
63	C₂×D₄×C₃₁	—	2	3	no	—
64	C₂×D₄×C₃₁	—	2	3	no	—
65	C₂×D₄×C₃₁	—	2	3	no	—
66	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₃₁	—	2	3	no	—
68	C₂²×C₄×C₃₁	—	2	12	no	—
69	C₂²×C₄×C₃₁	—	2	3	no	—
70	C₂²×C₄×C₃₁	—	8	6	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	SD₁₆×C₃₁	—	4	12	no	—
75	D₈×C₃₁	—	4	4	no	—
76	C₂×C₈×C₃₁	—	2	3	no	—
77	C₂×C₈×C₃₁	—	2	3	no	—
78	C₂×C₈×C₃₁	—	2	6	no	—
79	C₄⋊C₄×C₃₁	—	2	3	no	—
80	C₄⋊C₄×C₃₁	—	4	6	no	—
81	C₄⋊C₄×C₃₁	—	4	6	no	—
82	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
83	C₂²⋊C₄×C₃₁	—	2	12	no	—
84	C₂²⋊C₄×C₃₁	—	4	6	no	—
85	C₂²⋊C₄×C₃₁	—	4	6	no	—
86	C₄²×C₃₁	—	8	3	no	—
87	C₄²×C₃₁	C₂×D₄	1	6	yes	—
88	C₄²×C₃₁	#41	1	6	yes	—
89	#33168	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	6	no	—
91	C₂³×C₃₁	—	4	3	no	—
92	C₂³×C₃₁	—	4	1	no	—
93	C₂³×C₃₁	—	4	3	no	—
94	C₂³×C₃₁	—	4	3	no	—
95	C₂³×C₃₁	—	4	1	no	—
96	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₃₁	—	4	3	no	—
98	Q₈×C₃₁	—	4	3	no	—
99	Q₈×C₃₁	—	4	6	no	—
100	D₄×C₃₁	—	2	6	no	—
101	D₄×C₃₁	—	4	1	no	—
102	D₄×C₃₁	—	4	3	no	—
103	D₄×C₃₁	#127	1	4	yes	—
104	C₂×C₄×C₃₁	—	2	3	no	—
105	C₂×C₄×C₃₁	—	2	6	no	—
106	C₂×C₄×C₃₁	—	2	3	no	—
107	C₂×C₄×C₃₁	—	2	6	no	—
108	C₂×C₄×C₃₁	—	4	6	no	—
109	C₂×C₄×C₃₁	—	4	3	no	—
110	C₂×C₄×C₃₁	—	4	6	no	—
111	C₂×C₄×C₃₁	—	8	12	no	—
112	C₂×C₄×C₃₁	#121	1	6	yes	—
113	C₂×C₄×C₃₁	#121	1	6	yes	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
116	C₈×C₃₁	—	4	1	no	—
117	C₈×C₃₁	—	4	3	no	—
118	C₂×#463	C₂×C₃₁	1	3	yes	—
119	C₂×#461	C₂×C₃₁	1	3	yes	complemented
120	#2658	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#2275	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#1983	C₂×C₃₁	1	3	yes	complemented
123	#1981	C₂×C₃₁	1	3	yes	complemented
124	#1843	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	4	6	no	—
127	C₂²×C₃₁	—	4	6	no	—
128	C₂²×C₃₁	—	8	1	no	—
129	C₂²×C₃₁	—	8	3	no	—
130	C₂²×C₃₁	—	8	3	no	—
131	C₂²×C₃₁	C₂×#121	1	3	yes	—
132	C₂²×C₃₁	#577	1	3	yes	—
133	C₂²×C₃₁	#578	1	1	yes	characteristic
134	C₄×C₃₁	—	2	6	no	—
135	C₄×C₃₁	—	2	6	no	—
136	C₄×C₃₁	—	4	6	no	—
137	C₄×C₃₁	—	8	3	no	—
138	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
139	C₄×C₃₁	#532	1	3	yes	—
140	C₂²×SD₁₆	C₂²×C₃₁	1	3	yes	—
141	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
143	C₂×#124	C₂²×C₃₁	1	3	yes	—
144	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
145	#463	—	2	12	no	—
146	#461	—	2	12	no	—
147	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
148	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
149	C₂×#103	C₂²×C₃₁	1	3	yes	—
150	C₂×#103	C₂²×C₃₁	1	6	yes	—
151	C₂×#102	C₂²×C₃₁	1	3	yes	—
152	#399	—	2	1	no	—
153	#399	—	2	3	no	—
154	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
155	C₄×C₂²⋊C₄	—	2	3	no	—
156	#376	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#95	—	2	1	no	—
158	C₂×C₃₁	—	4	4	no	—
159	C₂×C₃₁	—	16	1	no	—
160	C₂×C₃₁	#2856	1	1	yes	characteristic
161	C₂×C₃₁	#3042	1	3	yes	—
162	C₂×C₃₁	#3046	1	3	yes	—
163	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
164	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
165	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
166	C₂³×C₄	—	2	3	no	—
167	C₂×SD₁₆	—	2	12	no	—
168	C₂×SD₁₆	—	2	6	no	—
169	C₂×D₈	—	2	6	no	—
170	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
172	C₄⋊Q₈	—	2	6	no	—
173	C₄⋊Q₈	—	2	6	no	—
174	#124	—	2	6	no	—
175	#124	—	2	6	no	—
176	C₄×D₄	—	2	6	no	—
177	C₄×D₄	D₄×C₃₁	1	12	yes	—
178	C₂×C₄⋊C₄	—	2	3	no	—
179	C₂×C₄⋊C₄	—	2	3	no	—
180	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
181	C₂×C₂²⋊C₄	—	2	3	no	—
182	C₂×C₂²⋊C₄	—	2	3	no	—
183	C₂×C₂²⋊C₄	—	2	3	no	—
184	C₂×C₄²	—	2	3	no	—
185	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
186	C₄⋊C₈	—	2	6	no	—
187	C₄⋊C₈	—	2	6	no	—
188	#103	—	2	12	no	—
189	#103	—	2	6	no	—
190	#102	—	2	6	no	—
191	#95	—	2	3	no	—
192	#95	D₄×C₃₁	1	1	yes	characteristic
193	#95	D₄×C₃₁	1	3	yes	—
194	C₃₁	#33168	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	—	2	1	no	—
196	C₂⁴	—	2	1	no	—
197	C₂×Q₈	—	2	3	no	—
198	C₂×Q₈	—	2	6	no	—
199	C₂×Q₈	—	2	3	no	—
200	C₂×Q₈	—	2	6	no	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	—	2	3	no	—
203	C₂×D₄	—	2	3	no	—
204	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
205	C₂²×C₄	—	2	3	no	—
206	C₂²×C₄	—	2	12	no	—
207	C₂²×C₄	—	2	3	no	—
208	C₂²×C₄	—	8	6	no	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
211	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
212	SD₁₆	—	4	12	no	—
213	D₈	—	4	4	no	—
214	C₂×C₈	—	2	3	no	—
215	C₂×C₈	—	2	3	no	—
216	C₂×C₈	—	2	6	no	—
217	C₄⋊C₄	—	2	3	no	—
218	C₄⋊C₄	—	4	6	no	—
219	C₄⋊C₄	—	4	6	no	—
220	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
221	C₂²⋊C₄	—	2	12	no	—
222	C₂²⋊C₄	—	4	6	no	—
223	C₂²⋊C₄	—	4	6	no	—
224	C₄²	—	8	3	no	—
225	C₄²	C₂×D₄×C₃₁	1	6	yes	—
226	C₄²	#41×C₃₁	1	6	yes	—
227	C₂³	—	2	6	no	—
228	C₂³	—	4	3	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	3	no	—
231	C₂³	—	4	3	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
234	Q₈	—	4	3	no	—
235	Q₈	—	4	3	no	—
236	Q₈	—	4	6	no	—
237	D₄	—	2	6	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	3	no	—
240	D₄	C₃₁×#127	1	4	yes	—
241	C₂×C₄	—	2	3	no	—
242	C₂×C₄	—	2	6	no	—
243	C₂×C₄	—	2	3	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	—	4	3	no	—
247	C₂×C₄	—	4	6	no	—
248	C₂×C₄	—	8	12	no	—
249	C₂×C₄	C₃₁×#121	1	6	yes	—
250	C₂×C₄	C₃₁×#121	1	6	yes	—
251	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
252	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
253	C₈	—	4	1	no	—
254	C₈	—	4	3	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	6	no	—
257	C₂²	—	4	6	no	—
258	C₂²	—	8	1	no	—
259	C₂²	—	8	3	no	—
260	C₂²	—	8	3	no	—
261	C₂²	C₂×C₃₁×#121	1	3	yes	—
262	C₂²	C₃₁×#577	1	3	yes	—
263	C₂²	C₃₁×#578	1	1	yes	characteristic
264	C₄	—	2	6	no	—
265	C₄	—	2	6	no	—
266	C₄	—	4	6	no	—
267	C₄	—	8	3	no	—
268	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
269	C₄	C₃₁×#532	1	3	yes	—
270	C₂	—	4	4	no	—
271	C₂	—	16	1	no	—
272	C₂	C₃₁×#2856	1	1	yes	characteristic
273	C₂	C₃₁×#3042	1	3	yes	—
274	C₂	C₃₁×#3046	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#461	C₂	semidirect
3. C₃₁×#2658	C₂	semidirect
4. C₃₁×#2275	C₂	semidirect
5. C₃₁×#1983	C₂	semidirect
6. C₃₁×#1981	C₂	semidirect
7. C₃₁×#1843	C₂	semidirect
89. #33168	C₃₁	direct (N × H)
119. C₂×#461	C₂×C₃₁	semidirect
120. #2658	C₂×C₃₁	semidirect
121. #2275	C₂×C₃₁	semidirect
122. #1983	C₂×C₃₁	semidirect
123. #1981	C₂×C₃₁	semidirect
124. #1843	C₂×C₃₁	semidirect
194. C₃₁	#33168	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×#463	2
C₂×C₃₁×#461	2
C₃₁×#2658	2
C₃₁×#2275	2
C₃₁×#1983	2
C₃₁×#1981	2
C₃₁×#1843	2
#33168	31

Subgroups — C31×#33168 #21489927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33168 #21489927