Subgroups — #36333239

Order 21376.

330 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923826	C₂	1	1	yes	maximal, characteristic
2	#24923399	C₂	1	2	yes	maximal, complemented
3	#24923396	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₃₃₄	C₂	1	2	yes	maximal, complemented
5	#24923393	C₂	1	2	yes	maximal, complemented
6	#24923327	C₂	1	4	yes	maximal
7	#24923322	C₂	1	1	yes	maximal, characteristic
8	#431×C₁₆₇	C₂	1	1	yes	maximal, characteristic
9	#19128832	C₄	1	1	yes	characteristic
10	#19128832	C₄	1	1	yes	characteristic
11	#19128824	C₄	1	2	yes	—
12	#19128811	C₄	1	2	yes	—
13	#19128811	C₄	1	2	yes	—
14	C₄×Dic₃₃₄	C₂²	1	2	yes	—
15	C₄×Dic₃₃₄	C₂²	1	2	yes	—
16	#19128791	C₂²	1	2	yes	—
17	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₁₆₇⋊C₈	C₂²	1	4	yes	—
20	Dic₁₆₇⋊C₈	C₂²	1	4	yes	—
21	C₂²⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
22	#19128742	C₂²	1	4	yes	—
23	C₄×C₁₆₇⋊C₈	C₂²	1	2	yes	—
24	C₈×Dic₁₆₇	C₂²	1	4	yes	—
25	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₁₆₇	C₂²	1	2	yes	—
29	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
30	C₂×Dic₃₃₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₃₃₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
33	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
36	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
38	Dic₁₆₇⋊C₄	C₂×C₄	1	4	yes	—
39	Dic₁₆₇⋊C₄	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₁₆₇	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₆₇	C₂³	1	4	yes	—
42	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
46	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
47	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
50	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
51	Dic₃₃₄	—	2	4	no	—
52	Dic₃₃₄	—	2	4	no	—
53	C₂×Dic₁₆₇	C₂²×C₄	1	4	yes	—
54	C₂×Dic₁₆₇	C₂²×C₄	1	4	yes	—
55	C₁₆₇⋊C₈	—	2	2	no	—
56	C₁₆₇⋊C₈	—	2	2	no	—
57	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₆₇	—	2	2	no	—
59	C₂×C₄×C₁₆₇	—	2	2	no	—
60	C₂×C₄×C₁₆₇	—	2	1	no	—
61	C₂×C₄×C₁₆₇	—	2	1	no	—
62	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
68	C₈×C₁₆₇	—	2	2	no	—
69	C₈×C₁₆₇	—	2	2	no	—
70	C₂²×C₁₆₇	—	2	1	no	—
71	C₂²×C₁₆₇	—	2	1	no	—
72	C₂²×C₁₆₇	—	2	1	no	—
73	C₂²×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₆₇	—	2	1	no	—
75	C₄×C₁₆₇	—	2	1	no	—
76	C₄×C₁₆₇	—	2	1	no	—
77	C₄×C₁₆₇	#131	1	2	yes	—
78	C₄×C₁₆₇	#131	1	2	yes	—
79	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
80	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
81	Dic₁₆₇	—	2	4	no	—
82	Dic₁₆₇	—	2	4	no	—
83	C₂×C₁₆₇	—	4	1	no	—
84	C₂×C₁₆₇	#519	1	1	yes	characteristic
85	C₂×C₁₆₇	C₂×#131	1	1	yes	characteristic
86	C₂×C₁₆₇	C₂×#131	1	1	yes	characteristic
87	C₁₆₇	#2986	1	1	yes	characteristic, complemented, Sylow
88	#2986	—	167	1	no	maximal, Sylow
89	#533	—	167	1	no	—
90	#446	—	167	2	no	—
91	#446	—	167	2	no	—
92	C₈×Q₈	—	167	2	no	—
93	C₈×Q₈	—	167	2	no	—
94	#433	—	167	4	no	—
95	#431	—	167	1	no	—
96	#431	D₁₆₇	1	1	yes	characteristic
97	C₄⋊Q₈	—	167	1	no	—
98	C₄⋊Q₈	—	167	1	no	—
99	#125	—	167	2	no	—
100	C₂²⋊Q₈	—	167	2	no	—
101	C₂²⋊Q₈	—	167	2	no	—
102	C₄×Q₈	—	167	2	no	—
103	C₄×Q₈	—	167	2	no	—
104	#117	—	167	2	no	—
105	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
106	C₄⋊C₈	—	167	4	no	—
107	C₄⋊C₈	—	167	4	no	—
108	C₄⋊C₈	—	167	1	no	—
109	C₄⋊C₈	—	167	1	no	—
110	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
112	C₂²⋊C₈	—	167	2	no	—
113	C₂²⋊C₈	C₂×D₁₆₇	1	2	yes	—
114	#97	—	167	4	no	—
115	C₄×C₈	—	167	4	no	—
116	C₄×C₈	—	167	2	no	—
117	C₄×C₈	C₂×D₁₆₇	1	2	yes	—
118	C₂×Q₈	—	167	2	no	—
119	C₂×Q₈	—	167	2	no	—
120	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
123	C₂×C₈	—	167	2	no	—
124	C₂×C₈	—	167	2	no	—
125	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
126	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
127	C₄⋊C₄	—	167	2	no	—
128	C₄⋊C₄	—	167	2	no	—
129	C₄⋊C₄	—	167	4	no	—
130	C₄⋊C₄	—	167	4	no	—
131	C₄⋊C₄	—	167	2	no	—
132	C₄⋊C₄	—	167	2	no	—
133	C₂²⋊C₄	—	167	4	no	—
134	C₄²	—	167	4	no	—
135	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
136	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
137	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
138	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
139	C₂³	C₂×C₄×D₁₆₇	1	1	yes	characteristic
140	Q₈	—	334	4	no	—
141	Q₈	—	334	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	167	4	no	—
147	C₂×C₄	—	167	4	no	—
148	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	334	2	no	—
157	C₈	—	334	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂²×C₄×D₁₆₇	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	334	4	no	—
166	C₄	—	334	4	no	—
167	C₄	#19128838	1	2	yes	—
168	C₄	#19128840	1	2	yes	—
169	C₄	#19128879	1	1	yes	characteristic
170	C₄	#19128879	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#24923746	1	1	yes	characteristic
173	C₂	C₂×#19128838	1	1	yes	characteristic
174	C₂	C₂×#19128840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂³×C₄
Fitting	#431×C₁₆₇	C₂
prime radical	C₂³×C₁₆₇	C₂²×C₄
cosocle	C₂×C₄×C₁₆₇	C₂⁴
hypercenter	#431	D₁₆₇
hypocenter	C₁₆₇	#2986
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #24923399	C₂	semidirect
3. #24923396	C₂	semidirect
4. C₈×Dic₃₃₄	C₂	semidirect
5. #24923393	C₂	semidirect
87. C₁₆₇	#2986	semidirect

Maximal subgroups

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

subgroup	index
#24923826	2
#24923399	2
#24923396	2
C₈×Dic₃₃₄	2
#24923393	2
#24923327	2
#24923322	2
#431×C₁₆₇	2
#2986	167