Subgroups — #36333085

Order 21376.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 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	#24923397	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₃₃₄	C₂	1	2	yes	maximal, complemented
5	#24923297	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#24923227	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₆₇⋊C₈	C₂²	1	2	yes	—
10	#19128832	C₄	1	1	yes	characteristic
11	#19128832	C₄	1	1	yes	characteristic
12	#19128824	C₄	1	2	yes	—
13	#19128811	C₄	1	2	yes	—
14	#19128811	C₄	1	2	yes	—
15	C₄×Dic₃₃₄	C₂²	1	2	yes	—
16	C₄×Dic₃₃₄	C₂²	1	2	yes	—
17	#19128791	C₂²	1	2	yes	—
18	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	Dic₁₆₇⋊C₈	C₂²	1	4	yes	—
22	Dic₁₆₇⋊C₈	C₂²	1	4	yes	—
23	#19128742	C₂²	1	4	yes	—
24	C₈×Dic₁₆₇	C₂²	1	4	yes	—
25	C₂×M₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
28	#97×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	—	2	4	no	—
33	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₆₇⋊C₈	C₂³	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	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
39	Dic₁₆₇⋊C₄	C₂×C₄	1	4	yes	—
40	Dic₁₆₇⋊C₄	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₁₆₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₆₇	C₂³	1	4	yes	—
43	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₆₇	—	2	2	no	—
47	M₁₆×C₁₆₇	Q₈	1	4	yes	complemented
48	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
50	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
53	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
54	Dic₃₃₄	—	2	4	no	—
55	Dic₃₃₄	—	4	2	no	—
56	C₂×Dic₁₆₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₁₆₇	C₂²×C₄	1	4	yes	—
58	C₁₆₇⋊C₈	—	2	2	no	—
59	C₁₆₇⋊C₈	—	2	2	no	—
60	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆₇	—	2	2	no	—
62	C₂×C₄×C₁₆₇	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₆₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₁₆₇	#41	1	2	yes	—
70	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
71	C₈×C₁₆₇	—	2	2	no	—
72	C₈×C₁₆₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₁₆₇	—	2	1	no	—
74	C₂²×C₁₆₇	—	2	1	no	—
75	C₂²×C₁₆₇	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₁₆₇	—	2	1	no	—
78	C₄×C₁₆₇	—	2	1	no	—
79	C₄×C₁₆₇	—	2	1	no	—
80	C₄×C₁₆₇	C₄×Q₈	1	2	yes	—
81	C₄×C₁₆₇	#131	1	2	yes	—
82	C₄×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
84	Dic₁₆₇	—	2	4	no	—
85	Dic₁₆₇	—	2	4	no	—
86	C₂×C₁₆₇	—	2	2	no	—
87	C₂×C₁₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₆₇	C₂×#131	1	1	yes	characteristic
89	C₂×C₁₆₇	#568	1	1	yes	characteristic
90	C₁₆₇	#2962	1	1	yes	characteristic, complemented, Sylow
91	#2962	—	167	1	no	maximal, Sylow
92	#533	—	167	1	no	—
93	#446	—	167	4	no	—
94	#446	—	167	2	no	—
95	C₈×Q₈	—	167	2	no	—
96	#424	—	167	2	no	—
97	C₂×C₄⋊C₈	—	167	1	no	—
98	#405	—	167	2	no	—
99	C₄×M₁₆	D₁₆₇	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₁₆₇	1	1	yes	characteristic
101	C₂×M₁₆	C₂×D₁₆₇	1	1	yes	characteristic
102	C₂²×C₈	—	167	2	no	—
103	C₄⋊Q₈	—	167	1	no	—
104	C₄⋊Q₈	—	167	1	no	—
105	#125	—	167	2	no	—
106	C₂²⋊Q₈	—	167	2	no	—
107	C₂²⋊Q₈	—	167	2	no	—
108	C₄×Q₈	—	167	2	no	—
109	C₄×Q₈	—	167	2	no	—
110	#117	—	167	2	no	—
111	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
112	C₄⋊C₈	—	167	4	no	—
113	C₄⋊C₈	—	167	4	no	—
114	C₄⋊C₈	—	167	1	no	—
115	C₄⋊C₈	—	167	1	no	—
116	C₄⋊C₈	—	167	2	no	—
117	#97	—	167	4	no	—
118	#97	C₂×D₁₆₇	1	2	yes	—
119	C₄×C₈	—	167	4	no	—
120	C₄×C₈	C₂×D₁₆₇	1	2	yes	—
121	C₂×Q₈	—	167	2	no	—
122	C₂×Q₈	—	167	2	no	—
123	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₃₃₄	1	4	yes	complemented
128	C₂×C₈	—	334	4	no	—
129	C₂×C₈	—	167	2	no	—
130	C₂×C₈	—	167	2	no	—
131	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
132	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
133	C₄⋊C₄	—	167	2	no	—
134	C₄⋊C₄	—	167	2	no	—
135	C₄⋊C₄	—	167	4	no	—
136	C₄⋊C₄	—	167	4	no	—
137	C₄⋊C₄	—	167	2	no	—
138	C₄⋊C₄	—	167	2	no	—
139	C₂²⋊C₄	—	167	4	no	—
140	C₄²	—	167	4	no	—
141	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
142	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
143	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
144	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₁₆₇	1	1	yes	characteristic
146	Q₈	—	334	4	no	—
147	Q₈	—	668	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	167	4	no	—
151	C₂×C₄	—	167	4	no	—
152	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₃₃₄	1	2	yes	—
158	C₂×C₄	#14581950	1	2	yes	—
159	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	334	2	no	—
162	C₈	—	334	2	no	—
163	C₈	C₂×Dic₃₃₄	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₃₃₄	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₆₇	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	334	4	no	—
172	C₄	—	334	4	no	—
173	C₄	C₄×Dic₃₃₄	1	2	yes	—
174	C₄	#19128840	1	2	yes	—
175	C₄	C₂²×Dic₃₃₄	1	1	yes	characteristic
176	C₄	C₂×#14581950	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₃₃₄	1	1	yes	characteristic
179	C₂	C₂×#19128840	1	1	yes	characteristic
180	C₂	#24924035	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂³×C₄
Fitting	C₄×M₁₆×C₁₆₇	C₂
prime radical	C₂³×C₁₆₇	C₂²×C₄
cosocle	C₂×C₄×C₁₆₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₆₇
hypocenter	C₁₆₇	#2962
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #24923399	C₂	semidirect
3. #24923397	C₂	semidirect
4. C₈×Dic₃₃₄	C₂	semidirect
47. M₁₆×C₁₆₇	Q₈	semidirect
90. C₁₆₇	#2962	semidirect
127. M₁₆	Dic₃₃₄	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
#24923397	2
C₈×Dic₃₃₄	2
#24923297	2
C₂×C₄⋊(C₁₆₇⋊C₈)	2
#24923227	2
C₄×M₁₆×C₁₆₇	2
#2962	167