Subgroups — #37547540

Order 23168.

1060 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₆₂	C₂	1	2	yes	maximal, complemented
2	#26042008	C₂	1	2	yes	maximal, complemented
3	#26041986	C₂	1	6	yes	maximal, complemented
4	D₄×Dic₃₆₂	C₂	1	6	yes	maximal, complemented
5	#26041907	C₂	1	6	yes	maximal, complemented
6	#26041836	C₂	1	3	yes	maximal, complemented
7	C₂×#19153816	C₂	1	3	yes	maximal, complemented
8	#26041762	C₂	1	2	yes	maximal
9	C₄×#41×C₁₈₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₆₂	C₂²	1	6	yes	—
11	#19153816	C₂²	1	6	yes	—
12	#19153816	C₂²	1	6	yes	—
13	#19153816	C₂²	1	6	yes	—
14	#19153816	C₂²	1	3	yes	—
15	#19153816	C₂²	1	1	yes	characteristic
16	#19153815	C₂²	1	12	yes	—
17	#19153808	C₂²	1	6	yes	—
18	#19153807	C₂²	1	12	yes	—
19	#19153795	C₂²	1	6	yes	—
20	#19153795	C₂²	1	6	yes	—
21	#19153793	C₂²	1	24	yes	—
22	Q₈×Dic₁₈₁	C₂²	1	4	yes	—
23	C₄×Dic₃₆₂	C₂²	1	6	yes	—
24	C₄×Dic₃₆₂	C₂²	1	6	yes	—
25	D₄×Dic₁₈₁	C₂²	1	12	yes	—
26	#19153775	C₂²	1	6	yes	—
27	C₂×C₄⋊Dic₁₈₁	C₂²	1	6	yes	—
28	C₂×C₄⋊Dic₁₈₁	C₂²	1	6	yes	—
29	C₂×#41×C₁₈₁	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₈₁	C₂²	1	3	yes	—
33	C₄×D₄×C₁₈₁	C₂²	1	3	yes	—
34	#117×C₁₈₁	C₂²	1	3	yes	—
35	C₂×C₄²×C₁₈₁	C₂²	1	3	yes	—
36	C₂×Dic₃₆₂	—	4	12	no	—
37	C₂×Dic₃₆₂	C₂³	1	2	yes	—
38	C₂×Dic₃₆₂	C₂³	1	6	yes	—
39	C₂×Dic₃₆₂	C₂³	1	12	yes	—
40	C₂²×Dic₁₈₁	C₂³	1	12	yes	—
41	C₄⋊Dic₁₈₁	C₂³	1	6	yes	—
42	C₄⋊Dic₁₈₁	C₂³	1	6	yes	—
43	C₄⋊Dic₁₈₁	C₂³	1	12	yes	—
44	C₄⋊Dic₁₈₁	C₂³	1	12	yes	—
45	C₄⋊Dic₁₈₁	C₂³	1	2	yes	—
46	C₄⋊Dic₁₈₁	C₂³	1	2	yes	—
47	#15668428	C₂³	1	12	yes	—
48	#15668428	C₂³	1	12	yes	—
49	C₂²⋊Dic₁₈₁	C₂³	1	12	yes	—
50	C₄×Dic₁₈₁	C₂³	1	12	yes	—
51	#41×C₁₈₁	Q₈	1	8	yes	complemented
52	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₈₁	C₂³	1	3	yes	—
54	C₂²×C₄×C₁₈₁	C₂³	1	3	yes	—
55	C₂²×C₄×C₁₈₁	C₂³	1	3	yes	—
56	C₂²×C₄×C₁₈₁	C₂³	1	3	yes	—
57	C₄⋊C₄×C₁₈₁	C₂³	1	3	yes	—
58	C₄⋊C₄×C₁₈₁	C₂³	1	3	yes	—
59	C₂²⋊C₄×C₁₈₁	C₂³	1	6	yes	—
60	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
61	C₄²×C₁₈₁	C₂³	1	3	yes	—
62	C₄²×C₁₈₁	C₂³	1	3	yes	—
63	C₄²×C₁₈₁	C₂³	1	3	yes	—
64	Dic₃₆₂	—	2	12	no	—
65	Dic₃₆₂	—	2	4	no	—
66	Dic₃₆₂	—	4	12	no	—
67	C₂×Dic₁₈₁	—	4	12	no	—
68	C₂×Dic₁₈₁	C₂⁴	1	4	yes	—
69	C₂×Dic₁₈₁	C₂⁴	1	12	yes	—
70	C₂³×C₁₈₁	C₂⁴	1	3	yes	—
71	Q₈×C₁₈₁	C₂×Q₈	1	4	yes	complemented
72	D₄×C₁₈₁	C₂×Q₈	1	12	yes	complemented
73	C₂×C₄×C₁₈₁	—	2	6	no	—
74	C₂×C₄×C₁₈₁	—	2	6	no	—
75	C₂×C₄×C₁₈₁	C₂×Q₈	1	12	yes	complemented
76	C₂×C₄×C₁₈₁	C₂⁴	1	3	yes	—
77	C₂×C₄×C₁₈₁	C₂⁴	1	3	yes	—
78	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₄×C₁₈₁	C₂⁴	1	3	yes	—
80	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
82	C₂²×C₁₈₁	—	2	3	no	—
83	C₂²×C₁₈₁	—	2	3	no	—
84	C₂²×C₁₈₁	C₂²×Q₈	1	6	yes	—
85	C₂²×C₁₈₁	C₂⁵	1	1	yes	characteristic
86	C₄×C₁₈₁	—	2	3	no	—
87	C₄×C₁₈₁	—	2	3	no	—
88	C₄×C₁₈₁	C₂²×Q₈	1	2	yes	—
89	C₄×C₁₈₁	C₂²×Q₈	1	6	yes	—
90	C₄×C₁₈₁	H(5,𝔽₂)	1	2	yes	—
91	C₄×C₁₈₁	2₋¹⁺⁴	1	2	yes	—
92	Dic₁₈₁	—	2	12	no	—
93	Dic₁₈₁	—	2	4	no	—
94	C₂×C₁₈₁	—	2	6	no	—
95	C₂×C₁₈₁	C₂³×Q₈	1	1	yes	characteristic
96	C₂×C₁₈₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
97	C₂×C₁₈₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
98	C₁₈₁	#3503	1	1	yes	characteristic, complemented, Sylow
99	#3503	—	181	1	no	maximal, Sylow
100	Q₈²	—	181	2	no	—
101	#557	—	181	2	no	—
102	#554	—	181	6	no	—
103	D₄×Q₈	—	181	6	no	—
104	#544	—	181	6	no	—
105	#533	—	181	3	no	—
106	C₂×C₄⋊Q₈	—	181	3	no	—
107	#520	—	181	2	no	—
108	C₄×#41	D₁₈₁	1	1	yes	characteristic
109	C₂×#41	C₂×D₁₈₁	1	1	yes	characteristic
110	C₂²×Q₈	—	181	6	no	—
111	C₄⋊Q₈	—	181	12	no	—
112	C₄⋊Q₈	—	181	3	no	—
113	C₄⋊Q₈	—	181	6	no	—
114	C₄⋊Q₈	—	181	6	no	—
115	C₄⋊Q₈	—	181	6	no	—
116	C₄⋊Q₈	—	181	1	no	—
117	#125	—	181	12	no	—
118	#125	—	181	6	no	—
119	C₂²⋊Q₈	—	181	6	no	—
120	C₂²⋊Q₈	—	181	6	no	—
121	C₂²⋊Q₈	—	181	24	no	—
122	C₄×Q₈	—	181	4	no	—
123	C₄×Q₈	—	181	6	no	—
124	C₄×Q₈	—	181	6	no	—
125	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
126	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
127	C₄×D₄	—	181	12	no	—
128	C₄×D₄	C₂×D₁₈₁	1	3	yes	—
129	C₄×D₄	C₂×D₁₈₁	1	3	yes	—
130	#117	—	181	6	no	—
131	#117	C₂×D₁₈₁	1	3	yes	—
132	C₂×C₄⋊C₄	—	181	6	no	—
133	C₂×C₄⋊C₄	—	181	6	no	—
134	C₂×C₄²	C₂×D₁₈₁	1	3	yes	—
135	#41	Dic₃₆₂	1	8	yes	complemented
136	C₂×Q₈	—	181	2	no	—
137	C₂×Q₈	—	181	6	no	—
138	C₂×Q₈	—	181	12	no	—
139	C₂×Q₈	—	724	12	no	—
140	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
141	C₂×D₄	C₂²×D₁₈₁	1	3	yes	—
142	C₂²×C₄	—	181	12	no	—
143	C₂²×C₄	C₂²×D₁₈₁	1	3	yes	—
144	C₂²×C₄	C₂²×D₁₈₁	1	3	yes	—
145	C₂²×C₄	C₂²×D₁₈₁	1	3	yes	—
146	C₄⋊C₄	—	181	6	no	—
147	C₄⋊C₄	—	181	6	no	—
148	C₄⋊C₄	—	181	12	no	—
149	C₄⋊C₄	—	181	12	no	—
150	C₄⋊C₄	—	181	2	no	—
151	C₄⋊C₄	—	181	2	no	—
152	C₄⋊C₄	—	181	12	no	—
153	C₄⋊C₄	—	181	12	no	—
154	C₄⋊C₄	C₂²×D₁₈₁	1	3	yes	—
155	C₄⋊C₄	C₂²×D₁₈₁	1	3	yes	—
156	C₂²⋊C₄	—	181	12	no	—
157	C₂²⋊C₄	C₂²×D₁₈₁	1	6	yes	—
158	C₄²	—	181	12	no	—
159	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
160	C₄²	C₂²×D₁₈₁	1	3	yes	—
161	C₄²	C₂²×D₁₈₁	1	3	yes	—
162	C₄²	C₂²×D₁₈₁	1	3	yes	—
163	C₂³	C₂³×D₁₈₁	1	3	yes	—
164	Q₈	—	362	12	no	—
165	Q₈	—	362	4	no	—
166	Q₈	—	724	12	no	—
167	Q₈	C₂×Dic₃₆₂	1	4	yes	complemented
168	D₄	C₂×Dic₃₆₂	1	12	yes	complemented
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	181	4	no	—
172	C₂×C₄	—	181	12	no	—
173	C₂×C₄	—	724	12	no	—
174	C₂×C₄	C₂×Dic₃₆₂	1	12	yes	complemented
175	C₂×C₄	C₂³×D₁₈₁	1	3	yes	—
176	C₂×C₄	C₂³×D₁₈₁	1	3	yes	—
177	C₂×C₄	C₂³×D₁₈₁	1	3	yes	—
178	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
181	C₂²	—	2	3	no	—
182	C₂²	—	2	3	no	—
183	C₂²	C₂²×Dic₃₆₂	1	6	yes	—
184	C₂²	C₂⁴×D₁₈₁	1	1	yes	characteristic
185	C₄	—	2	3	no	—
186	C₄	—	2	3	no	—
187	C₄	—	362	4	no	—
188	C₄	—	362	12	no	—
189	C₄	C₂²×Dic₃₆₂	1	2	yes	—
190	C₄	C₂²×Dic₃₆₂	1	6	yes	—
191	C₄	#19153862	1	2	yes	—
192	C₄	#19153863	1	2	yes	—
193	C₂	—	2	6	no	—
194	C₂	C₂³×Dic₃₆₂	1	1	yes	characteristic
195	C₂	C₂×#19153862	1	1	yes	characteristic
196	C₂	C₂×#19153863	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₈₁
commutator, socle, cosocle	C₂²×C₁₈₁	C₂⁵
Fitting	C₄×#41×C₁₈₁	C₂
prime radical	C₂×#41×C₁₈₁	C₂²
hypercenter	C₄×#41	D₁₈₁
hypocenter	C₁₈₁	#3503
LCS	G ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₆₂	C₂	semidirect
2. #26042008	C₂	semidirect
3. #26041986	C₂	semidirect
4. D₄×Dic₃₆₂	C₂	semidirect
5. #26041907	C₂	semidirect
6. #26041836	C₂	semidirect
7. C₂×#19153816	C₂	semidirect
51. #41×C₁₈₁	Q₈	semidirect
71. Q₈×C₁₈₁	C₂×Q₈	semidirect
72. D₄×C₁₈₁	C₂×Q₈	semidirect
75. C₂×C₄×C₁₈₁	C₂×Q₈	semidirect
98. C₁₈₁	#3503	semidirect
135. #41	Dic₃₆₂	semidirect
167. Q₈	C₂×Dic₃₆₂	semidirect
168. D₄	C₂×Dic₃₆₂	semidirect
174. C₂×C₄	C₂×Dic₃₆₂	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₆₂	2
#26042008	2
#26041986	2
D₄×Dic₃₆₂	2
#26041907	2
#26041836	2
C₂×#19153816	2
#26041762	2
C₄×#41×C₁₈₁	2
#3503	181