Subgroups — C₄×C₈×Q₈×C₃₇ #22641768

Order 9472.

1070 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₇	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₃₇	C₂	1	6	yes	maximal
4	C₄×C₄⋊C₈×C₃₇	C₂	1	3	yes	maximal
5	C₄²×C₈×C₃₇	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₃₇	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₃₇	C₄	1	16	yes	direct product factor
9	C₂×C₄⋊C₈×C₃₇	C₂²	1	6	yes	—
10	C₂×C₄×C₈×C₃₇	C₂²	1	3	yes	—
11	C₂×C₄×C₈×C₃₇	C₂²	1	6	yes	—
12	C₂×C₄×C₈×C₃₇	C₂²	1	6	yes	—
13	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₃₇	C₄	1	6	yes	—
15	C₄×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
16	C₄³×C₃₇	C₂²	1	3	yes	—
17	C₃₇×#336	C₂²	1	6	yes	—
18	C₂²×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₇	C₂³	1	6	yes	—
20	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
21	C₄×Q₈×C₃₇	C₈	1	16	yes	direct product factor
22	C₄×Q₈×C₃₇	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
24	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₃₇	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₃₇	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₃₇	C₂³	1	3	yes	—
28	C₂×C₄²×C₃₇	C₂³	1	3	yes	—
29	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₃₇	C₂×C₄	1	24	yes	—
31	C₄×C₈×C₃₇	—	2	12	no	—
32	C₄×C₈×C₃₇	—	2	6	no	—
33	C₄×C₈×C₃₇	C₂×C₄	1	24	yes	—
34	C₄×C₈×C₃₇	Q₈	1	4	yes	direct product factor
35	#95×C₃₇	C₂×C₄	1	6	yes	—
36	C₂×Q₈×C₃₇	C₄²	1	4	yes	—
37	C₂×Q₈×C₃₇	C₂×C₈	1	8	yes	—
38	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₃₇	C₂²×C₄	1	6	yes	—
40	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₃₇	—	2	6	no	—
44	C₂×C₈×C₃₇	—	2	6	no	—
45	C₂×C₈×C₃₇	C₂²×C₄	1	12	yes	—
46	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₃₇	#41	1	4	yes	—
49	C₄⋊C₄×C₃₇	C₄²	1	12	yes	—
50	C₄⋊C₄×C₃₇	C₂×C₈	1	24	yes	—
51	C₄²×C₃₇	—	2	12	no	—
52	C₄²×C₃₇	—	2	6	no	—
53	C₄²×C₃₇	C₂×C₈	1	24	yes	—
54	C₄²×C₃₇	C₂²×C₄	1	12	yes	—
55	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
56	C₄²×C₃₇	#41	1	2	yes	—
57	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
58	Q₈×C₃₇	C₄×C₈	1	16	yes	direct product factor
59	C₂×C₄×C₃₇	—	2	6	no	—
60	C₂×C₄×C₃₇	—	2	6	no	—
61	C₂×C₄×C₃₇	—	2	6	no	—
62	C₂×C₄×C₃₇	C₂×C₄²	1	6	yes	—
63	C₂×C₄×C₃₇	C₂×C₄²	1	6	yes	—
64	C₂×C₄×C₃₇	C₄×Q₈	1	4	yes	—
65	C₂×C₄×C₃₇	C₂²×C₈	1	12	yes	—
66	C₂×C₄×C₃₇	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₃₇	C₂²×C₈	1	4	yes	—
68	C₂×C₄×C₃₇	#131	1	4	yes	—
69	C₂×C₄×C₃₇	C₂³×C₄	1	2	yes	—
70	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
72	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
73	C₈×C₃₇	—	2	12	no	—
74	C₈×C₃₇	C₄×Q₈	1	8	yes	direct product factor
75	C₄×C₈×Q₈	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
77	C₂²×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
78	C₂²×C₃₇	C₄×#41	1	1	yes	characteristic
79	C₂²×C₃₇	C₂³×C₈	1	2	yes	—
80	C₂²×C₃₇	C₂×#131	1	2	yes	—
81	C₄×C₃₇	—	2	6	no	—
82	C₄×C₃₇	—	2	12	no	—
83	C₄×C₃₇	C₂×C₄×C₈	1	12	yes	—
84	C₄×C₃₇	C₈×Q₈	1	8	yes	direct product factor
85	C₄×C₃₇	C₂×C₄×Q₈	1	2	yes	—
86	C₄×C₃₇	C₄×#41	1	2	yes	—
87	C₂×C₈×Q₈	C₂×C₃₇	1	2	yes	—
88	C₄²×Q₈	C₂×C₃₇	1	1	yes	characteristic
89	C₈×C₄⋊C₄	C₂×C₃₇	1	6	yes	—
90	C₄×C₄⋊C₈	C₂×C₃₇	1	3	yes	—
91	C₄²×C₈	C₂×C₃₇	1	3	yes	—
92	C₂×C₃₇	C₄²×Q₈	1	1	yes	characteristic
93	C₂×C₃₇	C₂²×C₄×C₈	1	1	yes	characteristic
94	C₂×C₃₇	C₄×#131	1	1	yes	characteristic
95	C₂×C₃₇	C₂×C₈×Q₈	1	2	yes	—
96	C₂×C₃₇	C₈×#41	1	2	yes	—
97	C₂×C₄×Q₈	C₄×C₃₇	1	2	yes	—
98	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
99	C₈×Q₈	C₄×C₃₇	1	16	yes	direct product factor
100	C₂×C₄⋊C₈	C₂²×C₃₇	1	6	yes	—
101	C₂×C₄×C₈	C₂²×C₃₇	1	3	yes	—
102	C₂×C₄×C₈	C₂²×C₃₇	1	6	yes	—
103	C₂×C₄×C₈	C₂²×C₃₇	1	6	yes	—
104	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₄×C₃₇	1	6	yes	—
106	C₄×C₄⋊C₄	C₂²×C₃₇	1	3	yes	—
107	C₄³	C₂²×C₃₇	1	3	yes	—
108	#336	C₂²×C₃₇	1	6	yes	—
109	C₃₇	C₄×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×Q₈	C₂×C₄×C₃₇	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₃₇	1	6	yes	—
112	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
113	C₄×Q₈	C₈×C₃₇	1	16	yes	direct product factor
114	C₄×Q₈	C₂×C₄×C₃₇	1	8	yes	—
115	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	3	yes	—
116	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	6	yes	—
117	C₂×C₄²	C₂×C₄×C₃₇	1	6	yes	—
118	C₂×C₄²	C₂×C₄×C₃₇	1	6	yes	—
119	C₂×C₄²	C₂³×C₃₇	1	3	yes	—
120	C₂×C₄²	C₂³×C₃₇	1	3	yes	—
121	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
122	C₄⋊C₈	C₂×C₄×C₃₇	1	24	yes	—
123	C₄×C₈	—	2	12	no	—
124	C₄×C₈	—	2	6	no	—
125	C₄×C₈	C₂×C₄×C₃₇	1	24	yes	—
126	C₄×C₈	Q₈×C₃₇	1	4	yes	direct product factor
127	#95	C₂×C₄×C₃₇	1	6	yes	—
128	C₂×Q₈	C₄²×C₃₇	1	4	yes	—
129	C₂×Q₈	C₂×C₈×C₃₇	1	8	yes	—
130	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
131	C₂²×C₄	C₂²×C₄×C₃₇	1	6	yes	—
132	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
133	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
134	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	C₂²×C₄×C₃₇	1	12	yes	—
138	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
139	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
140	C₂×C₈	#41×C₃₇	1	4	yes	—
141	C₄⋊C₄	C₄²×C₃₇	1	12	yes	—
142	C₄⋊C₄	C₂×C₈×C₃₇	1	24	yes	—
143	C₄²	—	2	12	no	—
144	C₄²	—	2	6	no	—
145	C₄²	C₂×C₈×C₃₇	1	24	yes	—
146	C₄²	C₂²×C₄×C₃₇	1	12	yes	—
147	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
148	C₄²	#41×C₃₇	1	2	yes	—
149	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
150	Q₈	C₄×C₈×C₃₇	1	16	yes	direct product factor
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	C₂×C₄²×C₃₇	1	6	yes	—
155	C₂×C₄	C₂×C₄²×C₃₇	1	6	yes	—
156	C₂×C₄	C₄×Q₈×C₃₇	1	4	yes	—
157	C₂×C₄	C₂²×C₈×C₃₇	1	12	yes	—
158	C₂×C₄	C₂²×C₈×C₃₇	1	12	yes	—
159	C₂×C₄	C₂²×C₈×C₃₇	1	4	yes	—
160	C₂×C₄	#131×C₃₇	1	4	yes	—
161	C₂×C₄	C₂³×C₄×C₃₇	1	2	yes	—
162	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
163	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
164	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
165	C₈	—	2	12	no	—
166	C₈	C₄×Q₈×C₃₇	1	8	yes	direct product factor
167	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
168	C₂²	C₂×C₄×Q₈×C₃₇	1	1	yes	characteristic
169	C₂²	C₄×#41×C₃₇	1	1	yes	characteristic
170	C₂²	C₂³×C₈×C₃₇	1	2	yes	—
171	C₂²	C₂×#131×C₃₇	1	2	yes	—
172	C₄	—	2	6	no	—
173	C₄	—	2	12	no	—
174	C₄	C₂×C₄×C₈×C₃₇	1	12	yes	—
175	C₄	C₈×Q₈×C₃₇	1	8	yes	direct product factor
176	C₄	C₂×C₄×Q₈×C₃₇	1	2	yes	—
177	C₄	C₄×#41×C₃₇	1	2	yes	—
178	C₂	C₄²×Q₈×C₃₇	1	1	yes	characteristic
179	C₂	C₂²×C₄×C₈×C₃₇	1	1	yes	characteristic
180	C₂	C₄×#131×C₃₇	1	1	yes	characteristic
181	C₂	C₂×C₈×Q₈×C₃₇	1	2	yes	—
182	C₂	C₈×#41×C₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₃₇	C₂²
commutator	C₂	C₂²×C₄×C₈×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle, prime radical	C₂³×C₃₇	C₂³×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₃₇	C₄	direct (N × H)
21. C₄×Q₈×C₃₇	C₈	direct (N × H)
34. C₄×C₈×C₃₇	Q₈	direct (N × H)
58. Q₈×C₃₇	C₄×C₈	direct (N × H)
74. C₈×C₃₇	C₄×Q₈	direct (N × H)
75. C₄×C₈×Q₈	C₃₇	direct (N × H)
84. C₄×C₃₇	C₈×Q₈	direct (N × H)
99. C₈×Q₈	C₄×C₃₇	direct (N × H)
109. C₃₇	C₄×C₈×Q₈	direct (N × H)
113. C₄×Q₈	C₈×C₃₇	direct (N × H)
126. C₄×C₈	Q₈×C₃₇	direct (N × H)
150. Q₈	C₄×C₈×C₃₇	direct (N × H)
166. C₈	C₄×Q₈×C₃₇	direct (N × H)
175. C₄	C₈×Q₈×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×C₈×Q₈×C₃₇	2
C₄²×Q₈×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
C₄×C₄⋊C₈×C₃₇	2
C₄²×C₈×C₃₇	2
C₄×C₈×Q₈	37

Subgroups — C4×C8×Q8×C37 #22641768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×Q₈×C₃₇ #22641768