Subgroups — #36333255

Order 21376.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923400	C₂	1	2	yes	maximal, complemented
2	#24923347	C₂	1	2	yes	maximal, complemented
3	#24923341	C₂	1	4	yes	maximal, complemented
4	#24923326	C₂	1	2	yes	maximal, complemented
5	#24923322	C₂	1	1	yes	maximal, characteristic, complemented
6	#24923293	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₁₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14581934	C₂²	1	4	yes	—
10	C₂²×C₁₆₇⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	4	yes	—
13	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂²⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂²⋊(C₁₆₇⋊C₈)	C₂²	1	4	yes	—
18	#19128743	C₂²	1	4	yes	—
19	C₄×C₁₆₇⋊C₈	C₂²	1	2	yes	—
20	C₂×#41×C₁₆₇	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₆₇	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₆₇	C₄	1	2	yes	—
24	C₄×D₄×C₁₆₇	C₄	1	1	yes	characteristic
25	C₄×D₄×C₁₆₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
27	#117×C₁₆₇	C₄	1	2	yes	—
28	#117×C₁₆₇	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
31	#14581934	—	4	4	no	—
32	C₂×C₁₆₇⋊C₈	—	4	2	no	—
33	C₂×C₁₆₇⋊C₈	C₂³	1	4	yes	—
34	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
36	#41×C₁₆₇	—	2	4	no	—
37	C₂×Q₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₆₇	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₆₇	C₂×C₄	1	2	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	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
53	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₆₇	C₂×C₄	1	2	yes	—
55	C₄²×C₁₆₇	C₂×C₄	1	2	yes	—
56	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
58	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
59	C₄²×C₁₆₇	C₂³	1	2	yes	—
60	C₁₆₇⋊C₈	—	2	4	no	—
61	C₁₆₇⋊C₈	—	2	2	no	—
62	C₁₆₇⋊C₈	—	2	2	no	—
63	C₂³×C₁₆₇	C₂²×C₄	1	2	yes	—
64	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
65	Q₈×C₁₆₇	—	2	2	no	—
66	D₄×C₁₆₇	—	2	4	no	—
67	D₄×C₁₆₇	—	2	2	no	—
68	C₂×C₄×C₁₆₇	—	2	4	no	—
69	C₂×C₄×C₁₆₇	—	2	1	no	—
70	C₂×C₄×C₁₆₇	—	2	2	no	—
71	C₂×C₄×C₁₆₇	—	2	1	no	—
72	C₂×C₄×C₁₆₇	—	2	2	no	—
73	C₂×C₄×C₁₆₇	—	4	1	no	—
74	C₂×C₄×C₁₆₇	—	4	2	no	—
75	C₂×C₄×C₁₆₇	M₁₆	1	4	yes	complemented
76	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
85	C₂²×C₁₆₇	—	2	1	no	—
86	C₂²×C₁₆₇	—	2	1	no	—
87	C₂²×C₁₆₇	—	2	2	no	—
88	C₂²×C₁₆₇	—	2	2	no	—
89	C₂²×C₁₆₇	—	2	2	no	—
90	C₂²×C₁₆₇	C₂×M₁₆	1	2	yes	—
91	C₂²×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₁₆₇	—	2	1	no	—
93	C₄×C₁₆₇	—	2	2	no	—
94	C₄×C₁₆₇	—	2	2	no	—
95	C₄×C₁₆₇	—	2	1	no	—
96	C₄×C₁₆₇	—	2	1	no	—
97	C₄×C₁₆₇	—	2	2	no	—
98	C₄×C₁₆₇	C₂×M₁₆	1	2	yes	—
99	C₄×C₁₆₇	C₂×M₁₆	1	2	yes	—
100	C₄×C₁₆₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₁₆₇	—	2	2	no	—
103	C₂×C₁₆₇	—	4	2	no	—
104	C₂×C₁₆₇	#520	1	1	yes	characteristic
105	C₂×C₁₆₇	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₁₆₇	#568	1	1	yes	characteristic
107	C₁₆₇	#2989	1	1	yes	characteristic, complemented, Sylow
108	#2989	—	167	1	no	maximal, Sylow
109	C₄×#41	D₁₆₇	1	1	yes	characteristic
110	#446	—	167	2	no	—
111	#436	—	167	2	no	—
112	#435	—	167	4	no	—
113	#432	—	167	2	no	—
114	#431	—	167	1	no	—
115	#423	—	167	2	no	—
116	C₂×C₄⋊C₈	—	167	1	no	—
117	C₂×#41	Dic₁₆₇	1	1	yes	characteristic
118	C₂×M₁₆	—	167	4	no	—
119	C₂²×C₈	—	167	2	no	—
120	C₄×Q₈	Dic₁₆₇	1	1	yes	characteristic
121	C₄×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
122	C₄×D₄	Dic₁₆₇	1	2	yes	—
123	C₄×D₄	Dic₁₆₇	1	1	yes	characteristic
124	C₄×D₄	C₂×D₁₆₇	1	2	yes	—
125	C₄×D₄	C₂×D₁₆₇	1	1	yes	characteristic
126	#117	Dic₁₆₇	1	2	yes	—
127	#117	Dic₁₆₇	1	1	yes	characteristic
128	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₁₆₇	1	2	yes	—
130	C₄⋊C₈	—	167	2	no	—
131	C₄⋊C₈	—	167	2	no	—
132	C₄⋊C₈	—	167	1	no	—
133	C₄⋊C₈	—	167	4	no	—
134	C₄⋊C₈	—	167	1	no	—
135	C₂²⋊C₈	—	167	2	no	—
136	C₂²⋊C₈	—	167	4	no	—
137	#97	—	167	4	no	—
138	C₄×C₈	—	167	2	no	—
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂×Dic₁₆₇	1	1	yes	characteristic
141	C₂×D₄	C₂×Dic₁₆₇	1	2	yes	—
142	C₂×D₄	C₂×Dic₁₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₆₇	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₆₇	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₆₇	1	2	yes	—
149	M₁₆	—	668	4	no	—
150	C₂×C₈	—	668	2	no	—
151	C₂×C₈	—	167	2	no	—
152	C₂×C₈	—	167	2	no	—
153	C₂×C₈	—	167	4	no	—
154	C₄⋊C₄	C₂×Dic₁₆₇	1	2	yes	—
155	C₄⋊C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
156	C₄⋊C₄	C₂×Dic₁₆₇	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
158	C₂²⋊C₄	C₂×Dic₁₆₇	1	2	yes	—
159	C₂²⋊C₄	C₂×Dic₁₆₇	1	2	yes	—
160	C₂²⋊C₄	C₂×Dic₁₆₇	1	2	yes	—
161	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
162	C₄²	C₂×Dic₁₆₇	1	2	yes	—
163	C₄²	C₂×Dic₁₆₇	1	2	yes	—
164	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
165	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
166	C₄²	C₂²×D₁₆₇	1	2	yes	—
167	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
168	C₂³	C₂²×Dic₁₆₇	1	2	yes	—
169	C₂³	C₂²×Dic₁₆₇	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	#14581934	1	4	yes	complemented
181	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₆₇	1	2	yes	—
184	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₆₇	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₆₇	1	2	yes	—
187	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
190	C₈	—	334	4	no	—
191	C₈	—	334	2	no	—
192	C₈	—	334	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂×#14581934	1	2	yes	—
199	C₂²	C₂³×Dic₁₆₇	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂×#14581934	1	2	yes	—
207	C₄	C₂×#14581934	1	2	yes	—
208	C₄	#19128878	1	1	yes	characteristic
209	C₄	#19128879	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	4	2	no	—
212	C₂	#24923752	1	1	yes	characteristic
213	C₂	C₂²×#14581934	1	1	yes	characteristic
214	C₂	#24924036	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂³×C₄
Fitting	C₄×#41×C₁₆₇	C₂
prime radical	C₂×#41×C₁₆₇	C₄
cosocle	C₂×C₄×C₁₆₇	C₂⁴
hypercenter	C₄×#41	D₁₆₇
hypocenter	C₁₆₇	#2989
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #24923400	C₂	semidirect
2. #24923347	C₂	semidirect
3. #24923341	C₂	semidirect
4. #24923326	C₂	semidirect
5. #24923322	C₂	semidirect
6. #24923293	C₂	semidirect
7. C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	semidirect
75. C₂×C₄×C₁₆₇	M₁₆	semidirect
107. C₁₆₇	#2989	semidirect
180. C₂×C₄	#14581934	semidirect

Maximal subgroups

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

subgroup	index
#24923400	2
#24923347	2
#24923341	2
#24923326	2
#24923322	2
#24923293	2
C₂×C₄⋊(C₁₆₇⋊C₈)	2
C₄×#41×C₁₆₇	2
#2989	167