Subgroups — Q₈×#100×C₃₇ #22643767

Order 9472.

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×C₃₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₃₇	C₂	1	2	yes	maximal, complemented
3	C₃₇×#1997	C₂	1	6	yes	maximal, complemented
4	C₃₇×#1800	C₂	1	3	yes	maximal
5	C₄×#100×C₃₇	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₃₇	C₄	1	2	yes	—
7	D₄×Q₈×C₃₇	—	2	2	no	—
8	D₄×Q₈×C₃₇	—	2	6	no	—
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	3	yes	—
10	C₂×C₂²⋊Q₈×C₃₇	C₄	1	6	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₇	C₂²	1	3	yes	—
13	C₃₇×#446	—	2	6	no	—
14	C₈×Q₈×C₃₇	—	2	2	no	—
15	C₃₇×#423	C₂²	1	6	yes	—
16	C₂×#100×C₃₇	C₂²	1	3	yes	—
17	C₂×#100×C₃₇	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₃₇	C₂²	1	6	yes	—
19	C₃₇×#343	C₂²	1	6	yes	—
20	C₃₇×#343	C₂²	1	6	yes	—
21	C₂²×Q₈×C₃₇	—	2	6	no	—
22	C₂²×Q₈×C₃₇	—	2	2	no	—
23	C₂²×Q₈×C₃₇	—	4	8	no	—
24	C₂²×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₃₇	C₂×C₄	1	6	yes	—
26	C₂²×D₄×C₃₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₃₇	C₂×C₄	1	6	yes	—
28	C₂×M₁₆×C₃₇	C₂³	1	6	yes	—
29	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₃₇	—	2	6	no	—
31	C₄⋊Q₈×C₃₇	—	2	6	no	—
32	C₂²⋊Q₈×C₃₇	—	2	12	no	—
33	C₂²⋊Q₈×C₃₇	—	2	12	no	—
34	C₄×Q₈×C₃₇	D₄	1	2	yes	—
35	C₄×Q₈×C₃₇	D₄	1	6	yes	—
36	C₄×D₄×C₃₇	—	2	6	no	—
37	C₄×D₄×C₃₇	—	2	6	no	—
38	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	6	yes	—
39	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
40	C₂×C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
41	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₃₇	C₂³	1	3	yes	—
43	C₄⋊C₈×C₃₇	—	2	6	no	—
44	C₄⋊C₈×C₃₇	—	2	6	no	—
45	#100×C₃₇	—	2	6	no	—
46	#100×C₃₇	Q₈	1	4	yes	direct product factor
47	#97×C₃₇	—	2	6	no	—
48	C₄×C₈×C₃₇	—	2	6	no	—
49	C₂⁴×C₃₇	C₂²×C₄	1	2	yes	—
50	C₂×Q₈×C₃₇	—	2	4	no	—
51	C₂×Q₈×C₃₇	—	2	6	no	—
52	C₂×Q₈×C₃₇	—	2	6	no	—
53	C₂×Q₈×C₃₇	—	4	12	no	—
54	C₂×Q₈×C₃₇	—	4	24	no	—
55	C₂×Q₈×C₃₇	—	4	8	no	—
56	C₂×Q₈×C₃₇	C₂²⋊C₄	1	3	yes	—
57	C₂×Q₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×D₄×C₃₇	—	2	2	no	—
59	C₂×D₄×C₃₇	—	2	2	no	—
60	C₂×D₄×C₃₇	C₂×Q₈	1	2	yes	—
61	C₂×D₄×C₃₇	#41	1	2	yes	—
62	C₂²×C₄×C₃₇	—	2	6	no	—
63	C₂²×C₄×C₃₇	—	2	6	no	—
64	C₂²×C₄×C₃₇	—	4	12	no	—
65	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₃₇	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₃₇	—	2	2	no	—
70	M₁₆×C₃₇	—	2	6	no	—
71	M₁₆×C₃₇	—	2	6	no	—
72	M₁₆×C₃₇	C₂×Q₈	1	4	yes	complemented
73	C₂×C₈×C₃₇	—	2	2	no	—
74	C₂×C₈×C₃₇	—	2	6	no	—
75	C₄⋊C₄×C₃₇	—	2	12	no	—
76	C₄⋊C₄×C₃₇	—	2	6	no	—
77	C₄⋊C₄×C₃₇	C₂²⋊C₄	1	6	yes	—
78	C₄⋊C₄×C₃₇	C₂²⋊C₄	1	6	yes	—
79	C₂²⋊C₄×C₃₇	—	2	12	no	—
80	C₄²×C₃₇	C₂×D₄	1	6	yes	—
81	C₄²×C₃₇	C₂×D₄	1	6	yes	—
82	C₂³×C₃₇	—	2	2	no	—
83	C₂³×C₃₇	—	2	2	no	—
84	C₂³×C₃₇	—	4	2	no	—
85	C₂³×C₃₇	—	4	2	no	—
86	C₂³×C₃₇	C₄×Q₈	1	4	yes	—
87	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₃₇	—	2	6	no	—
89	Q₈×C₃₇	—	4	12	no	—
90	Q₈×C₃₇	—	4	12	no	—
91	Q₈×C₃₇	#100	1	4	yes	direct product factor
92	D₄×C₃₇	—	2	4	no	—
93	D₄×C₃₇	—	2	4	no	—
94	C₂×C₄×C₃₇	—	2	6	no	—
95	C₂×C₄×C₃₇	—	2	6	no	—
96	C₂×C₄×C₃₇	—	4	3	no	—
97	C₂×C₄×C₃₇	—	4	12	no	—
98	C₂×C₄×C₃₇	—	4	12	no	—
99	C₂×C₄×C₃₇	—	4	6	no	—
100	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	3	yes	—
102	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	3	yes	—
103	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
107	C₈×C₃₇	—	2	4	no	—
108	C₈×C₃₇	—	4	6	no	—
109	Q₈×#100	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₃₇	—	2	1	no	—
111	C₂²×C₃₇	—	2	1	no	—
112	C₂²×C₃₇	—	2	4	no	—
113	C₂²×C₃₇	—	4	4	no	—
114	C₂²×C₃₇	—	4	4	no	—
115	C₂²×C₃₇	—	4	2	no	—
116	C₂²×C₃₇	—	4	2	no	—
117	C₂²×C₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₂²×C₃₇	#519	1	1	yes	characteristic
120	C₄×C₃₇	—	2	3	no	—
121	C₄×C₃₇	—	2	6	no	—
122	C₄×C₃₇	—	4	6	no	—
123	C₄×C₃₇	C₂×#100	1	6	yes	—
124	C₄×C₃₇	#548	1	2	yes	—
125	C₄×C₃₇	D₄×Q₈	1	2	yes	—
126	C₂×D₄×Q₈	C₂×C₃₇	1	1	yes	characteristic
127	Q₈×M₁₆	C₂×C₃₇	1	2	yes	complemented
128	#1997	C₂×C₃₇	1	6	yes	complemented
129	#1800	C₂×C₃₇	1	3	yes	—
130	C₄×#100	C₂×C₃₇	1	3	yes	—
131	C₂×C₃₇	—	2	1	no	—
132	C₂×C₃₇	—	2	1	no	—
133	C₂×C₃₇	—	4	4	no	—
134	C₂×C₃₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₃₇	C₂²×#100	1	1	yes	characteristic
136	C₂×C₃₇	#2889	1	1	yes	characteristic
137	C₂³×Q₈	C₄×C₃₇	1	2	yes	—
138	D₄×Q₈	—	2	2	no	—
139	D₄×Q₈	—	2	6	no	—
140	C₂×C₄⋊Q₈	C₂²×C₃₇	1	3	yes	—
141	C₂×C₂²⋊Q₈	C₄×C₃₇	1	6	yes	—
142	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
143	C₂×C₄×D₄	C₂²×C₃₇	1	3	yes	—
144	#446	—	2	6	no	—
145	C₈×Q₈	—	2	2	no	—
146	#423	C₂²×C₃₇	1	6	yes	—
147	C₂×#100	C₂²×C₃₇	1	3	yes	—
148	C₂×#100	C₂²×C₃₇	1	1	yes	characteristic
149	C₄×M₁₆	C₂²×C₃₇	1	6	yes	—
150	#343	C₂²×C₃₇	1	6	yes	—
151	#343	C₂²×C₃₇	1	6	yes	—
152	C₃₇	Q₈×#100	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×Q₈	—	2	6	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×Q₈	—	4	8	no	—
156	C₂²×Q₈	C₂×C₄×C₃₇	1	1	yes	characteristic
157	C₂²×Q₈	C₂×C₄×C₃₇	1	6	yes	—
158	C₂²×D₄	C₂³×C₃₇	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₃₇	1	6	yes	—
160	C₂×M₁₆	C₂³×C₃₇	1	6	yes	—
161	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
162	C₄⋊Q₈	—	2	6	no	—
163	C₄⋊Q₈	—	2	6	no	—
164	C₂²⋊Q₈	—	2	12	no	—
165	C₂²⋊Q₈	—	2	12	no	—
166	C₄×Q₈	D₄×C₃₇	1	2	yes	—
167	C₄×Q₈	D₄×C₃₇	1	6	yes	—
168	C₄×D₄	—	2	6	no	—
169	C₄×D₄	—	2	6	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	6	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	3	yes	—
172	C₂×C₄⋊C₄	C₂³×C₃₇	1	3	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	6	yes	—
174	C₂×C₄²	C₂³×C₃₇	1	3	yes	—
175	C₄⋊C₈	—	2	6	no	—
176	C₄⋊C₈	—	2	6	no	—
177	#100	—	2	6	no	—
178	#100	Q₈×C₃₇	1	4	yes	direct product factor
179	#97	—	2	6	no	—
180	C₄×C₈	—	2	6	no	—
181	C₂⁴	C₂²×C₄×C₃₇	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×C₃₇	1	3	yes	—
189	C₂×Q₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	C₂×Q₈×C₃₇	1	2	yes	—
193	C₂×D₄	#41×C₃₇	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×C₃₇	1	3	yes	—
198	C₂²×C₄	C₂²×C₄×C₃₇	1	6	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
200	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	6	no	—
203	M₁₆	—	2	6	no	—
204	M₁₆	C₂×Q₈×C₃₇	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×C₃₇	1	6	yes	—
210	C₄⋊C₄	C₂²⋊C₄×C₃₇	1	6	yes	—
211	C₂²⋊C₄	—	2	12	no	—
212	C₄²	C₂×D₄×C₃₇	1	6	yes	—
213	C₄²	C₂×D₄×C₃₇	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×C₃₇	1	4	yes	—
219	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
220	Q₈	—	2	6	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	12	no	—
223	Q₈	#100×C₃₇	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×C₃₇	1	6	yes	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	3	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	3	yes	—
235	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
236	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
238	C₂×C₄	2₋¹⁺⁴×C₃₇	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×C₃₇	1	1	yes	characteristic
249	C₂²	C₂×C₄×Q₈×C₃₇	1	1	yes	characteristic
250	C₂²	C₃₇×#519	1	1	yes	characteristic
251	C₄	—	2	3	no	—
252	C₄	—	2	6	no	—
253	C₄	—	4	6	no	—
254	C₄	C₂×#100×C₃₇	1	6	yes	—
255	C₄	C₃₇×#548	1	2	yes	—
256	C₄	D₄×Q₈×C₃₇	1	2	yes	—
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	4	no	—
260	C₂	Q₈×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
261	C₂	C₂²×#100×C₃₇	1	1	yes	characteristic
262	C₂	C₃₇×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₃₇	C₂	semidirect
3. C₃₇×#1997	C₂	semidirect
46. #100×C₃₇	Q₈	direct (N × H)
72. M₁₆×C₃₇	C₂×Q₈	semidirect
91. Q₈×C₃₇	#100	direct (N × H)
109. Q₈×#100	C₃₇	direct (N × H)
127. Q₈×M₁₆	C₂×C₃₇	semidirect
128. #1997	C₂×C₃₇	semidirect
152. C₃₇	Q₈×#100	direct (N × H)
178. #100	Q₈×C₃₇	direct (N × H)
204. M₁₆	C₂×Q₈×C₃₇	semidirect
223. Q₈	#100×C₃₇	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×C₃₇	2
Q₈×M₁₆×C₃₇	2
C₃₇×#1997	2
C₃₇×#1800	2
C₄×#100×C₃₇	2
Q₈×#100	37

Subgroups — Q8×#100×C37 #22643767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#100×C₃₇ #22643767