Subgroups — Q₈×C₃₁×#100 #21479273

Order 7936.

1198 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1997	C₂	1	6	yes	maximal, complemented
4	C₃₁×#1800	C₂	1	3	yes	maximal
5	C₄×C₃₁×#100	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₃₁	C₄	1	2	yes	—
7	D₄×Q₈×C₃₁	—	2	2	no	—
8	D₄×Q₈×C₃₁	—	2	6	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
10	C₂×C₃₁×C₂²⋊Q₈	C₄	1	6	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
13	C₃₁×#446	—	2	6	no	—
14	C₈×Q₈×C₃₁	—	2	2	no	—
15	C₃₁×#423	C₂²	1	6	yes	—
16	C₂×C₃₁×#100	C₂²	1	3	yes	—
17	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₃₁	C₂²	1	6	yes	—
19	C₃₁×#343	C₂²	1	6	yes	—
20	C₃₁×#343	C₂²	1	6	yes	—
21	C₂²×Q₈×C₃₁	—	2	6	no	—
22	C₂²×Q₈×C₃₁	—	2	2	no	—
23	C₂²×Q₈×C₃₁	—	4	8	no	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₃₁	C₂×C₄	1	6	yes	—
26	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂×C₄	1	6	yes	—
28	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
29	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
30	C₃₁×C₄⋊Q₈	—	2	6	no	—
31	C₃₁×C₄⋊Q₈	—	2	6	no	—
32	C₃₁×C₂²⋊Q₈	—	2	12	no	—
33	C₃₁×C₂²⋊Q₈	—	2	12	no	—
34	C₄×Q₈×C₃₁	D₄	1	2	yes	—
35	C₄×Q₈×C₃₁	D₄	1	6	yes	—
36	C₄×D₄×C₃₁	—	2	6	no	—
37	C₄×D₄×C₃₁	—	2	6	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
43	C₃₁×C₄⋊C₈	—	2	6	no	—
44	C₃₁×C₄⋊C₈	—	2	6	no	—
45	C₃₁×#100	—	2	6	no	—
46	C₃₁×#100	Q₈	1	4	yes	direct product factor
47	C₃₁×#97	—	2	6	no	—
48	C₄×C₈×C₃₁	—	2	6	no	—
49	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
50	C₂×Q₈×C₃₁	—	2	4	no	—
51	C₂×Q₈×C₃₁	—	2	6	no	—
52	C₂×Q₈×C₃₁	—	2	6	no	—
53	C₂×Q₈×C₃₁	—	4	12	no	—
54	C₂×Q₈×C₃₁	—	4	24	no	—
55	C₂×Q₈×C₃₁	—	4	8	no	—
56	C₂×Q₈×C₃₁	C₂²⋊C₄	1	3	yes	—
57	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂×D₄×C₃₁	#41	1	2	yes	—
62	C₂²×C₄×C₃₁	—	2	6	no	—
63	C₂²×C₄×C₃₁	—	2	6	no	—
64	C₂²×C₄×C₃₁	—	4	12	no	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₃₁	—	2	2	no	—
70	M₁₆×C₃₁	—	2	6	no	—
71	M₁₆×C₃₁	—	2	6	no	—
72	M₁₆×C₃₁	C₂×Q₈	1	4	yes	complemented
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	6	no	—
75	C₄⋊C₄×C₃₁	—	2	12	no	—
76	C₄⋊C₄×C₃₁	—	2	6	no	—
77	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
78	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
79	C₂²⋊C₄×C₃₁	—	2	12	no	—
80	C₄²×C₃₁	C₂×D₄	1	6	yes	—
81	C₄²×C₃₁	C₂×D₄	1	6	yes	—
82	Q₈×#100	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	2	2	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	4	2	no	—
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	C₄×Q₈	1	4	yes	—
88	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
89	Q₈×C₃₁	—	2	6	no	—
90	Q₈×C₃₁	—	4	12	no	—
91	Q₈×C₃₁	—	4	12	no	—
92	Q₈×C₃₁	#100	1	4	yes	direct product factor
93	D₄×C₃₁	—	2	4	no	—
94	D₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	2	6	no	—
96	C₂×C₄×C₃₁	—	2	6	no	—
97	C₂×C₄×C₃₁	—	4	3	no	—
98	C₂×C₄×C₃₁	—	4	12	no	—
99	C₂×C₄×C₃₁	—	4	12	no	—
100	C₂×C₄×C₃₁	—	4	6	no	—
101	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
102	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
103	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
104	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₃₁	—	2	4	no	—
109	C₈×C₃₁	—	4	6	no	—
110	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
111	Q₈×M₁₆	C₂×C₃₁	1	2	yes	complemented
112	#1997	C₂×C₃₁	1	6	yes	complemented
113	#1800	C₂×C₃₁	1	3	yes	—
114	C₄×#100	C₂×C₃₁	1	3	yes	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	4	no	—
118	C₂²×C₃₁	—	4	4	no	—
119	C₂²×C₃₁	—	4	4	no	—
120	C₂²×C₃₁	—	4	2	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₂²×C₃₁	#519	1	1	yes	characteristic
125	C₄×C₃₁	—	2	3	no	—
126	C₄×C₃₁	—	2	6	no	—
127	C₄×C₃₁	—	4	6	no	—
128	C₄×C₃₁	C₂×#100	1	6	yes	—
129	C₄×C₃₁	#548	1	2	yes	—
130	C₄×C₃₁	D₄×Q₈	1	2	yes	—
131	C₂³×Q₈	C₄×C₃₁	1	2	yes	—
132	D₄×Q₈	—	2	2	no	—
133	D₄×Q₈	—	2	6	no	—
134	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
135	C₂×C₂²⋊Q₈	C₄×C₃₁	1	6	yes	—
136	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
138	#446	—	2	6	no	—
139	C₈×Q₈	—	2	2	no	—
140	#423	C₂²×C₃₁	1	6	yes	—
141	C₂×#100	C₂²×C₃₁	1	3	yes	—
142	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
143	C₄×M₁₆	C₂²×C₃₁	1	6	yes	—
144	#343	C₂²×C₃₁	1	6	yes	—
145	#343	C₂²×C₃₁	1	6	yes	—
146	C₂×C₃₁	—	2	1	no	—
147	C₂×C₃₁	—	2	1	no	—
148	C₂×C₃₁	—	4	4	no	—
149	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
151	C₂×C₃₁	#2889	1	1	yes	characteristic
152	C₂²×Q₈	—	2	6	no	—
153	C₂²×Q₈	—	2	2	no	—
154	C₂²×Q₈	—	4	8	no	—
155	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₂²×Q₈	C₂×C₄×C₃₁	1	6	yes	—
157	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
158	C₂³×C₄	C₂×C₄×C₃₁	1	6	yes	—
159	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
160	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
161	C₄⋊Q₈	—	2	6	no	—
162	C₄⋊Q₈	—	2	6	no	—
163	C₂²⋊Q₈	—	2	12	no	—
164	C₂²⋊Q₈	—	2	12	no	—
165	C₄×Q₈	D₄×C₃₁	1	2	yes	—
166	C₄×Q₈	D₄×C₃₁	1	6	yes	—
167	C₄×D₄	—	2	6	no	—
168	C₄×D₄	—	2	6	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
171	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
173	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
174	C₄⋊C₈	—	2	6	no	—
175	C₄⋊C₈	—	2	6	no	—
176	#100	—	2	6	no	—
177	#100	Q₈×C₃₁	1	4	yes	direct product factor
178	#97	—	2	6	no	—
179	C₄×C₈	—	2	6	no	—
180	C₃₁	Q₈×#100	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	6	no	—
184	C₂×Q₈	—	2	6	no	—
185	C₂×Q₈	—	4	12	no	—
186	C₂×Q₈	—	4	24	no	—
187	C₂×Q₈	—	4	8	no	—
188	C₂×Q₈	C₂²⋊C₄×C₃₁	1	3	yes	—
189	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
193	C₂×D₄	#41×C₃₁	1	2	yes	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	2	6	no	—
196	C₂²×C₄	—	4	12	no	—
197	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
198	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
200	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	6	no	—
203	M₁₆	—	2	6	no	—
204	M₁₆	C₂×Q₈×C₃₁	1	4	yes	complemented
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	6	no	—
207	C₄⋊C₄	—	2	12	no	—
208	C₄⋊C₄	—	2	6	no	—
209	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
210	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
211	C₂²⋊C₄	—	2	12	no	—
212	C₄²	C₂×D₄×C₃₁	1	6	yes	—
213	C₄²	C₂×D₄×C₃₁	1	6	yes	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₄×Q₈×C₃₁	1	4	yes	—
219	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
220	Q₈	—	2	6	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	12	no	—
223	Q₈	C₃₁×#100	1	4	yes	direct product factor
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	C₂×C₄	—	2	6	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	4	12	no	—
230	C₂×C₄	—	4	12	no	—
231	C₂×C₄	—	4	6	no	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
235	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
236	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	4	6	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
250	C₂²	C₃₁×#519	1	1	yes	characteristic
251	C₄	—	2	3	no	—
252	C₄	—	2	6	no	—
253	C₄	—	4	6	no	—
254	C₄	C₂×C₃₁×#100	1	6	yes	—
255	C₄	C₃₁×#548	1	2	yes	—
256	C₄	D₄×Q₈×C₃₁	1	2	yes	—
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	4	no	—
260	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
262	C₂	C₃₁×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₃₁	C₂	semidirect
3. C₃₁×#1997	C₂	semidirect
46. C₃₁×#100	Q₈	direct (N × H)
72. M₁₆×C₃₁	C₂×Q₈	semidirect
82. Q₈×#100	C₃₁	direct (N × H)
92. Q₈×C₃₁	#100	direct (N × H)
111. Q₈×M₁₆	C₂×C₃₁	semidirect
112. #1997	C₂×C₃₁	semidirect
177. #100	Q₈×C₃₁	direct (N × H)
180. C₃₁	Q₈×#100	direct (N × H)
204. M₁₆	C₂×Q₈×C₃₁	semidirect
223. Q₈	C₃₁×#100	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×D₄×Q₈×C₃₁	2
Q₈×M₁₆×C₃₁	2
C₃₁×#1997	2
C₃₁×#1800	2
C₄×C₃₁×#100	2
Q₈×#100	31

Subgroups — Q8×C31×#100 #21479273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃₁×#100 #21479273