Subgroups — #36333300

Order 21376.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923803	C₂	1	1	yes	maximal, characteristic
2	#24923399	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₃₃₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#24923348	C₂	1	1	yes	maximal, characteristic, complemented
5	#24923337	C₂	1	2	yes	maximal, complemented
6	C₈×D₆₆₈	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₁₆₇)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#24923329	C₂	1	2	yes	maximal, complemented
9	#24923294	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#431×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14581933	C₂²	1	2	yes	—
13	C₂²×C₁₆₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×D₁₆₇	C₂²	1	2	yes	—
15	#19128826	C₄	1	2	yes	—
16	#19128824	C₄	1	1	yes	characteristic
17	#19128820	C₄	1	1	yes	characteristic
18	#19128815	C₄	1	2	yes	—
19	#19128811	C₄	1	1	yes	characteristic
20	#19128806	C₄	1	1	yes	characteristic
21	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
22	C₄×D₆₆₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₆₇	C₂²	1	2	yes	—
24	#19128789	C₂²	1	2	yes	—
25	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
28	Dic₁₆₇⋊C₈	C₂²	1	2	yes	—
29	Dic₁₆₇⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₆₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₁₆₇)⋊C₈	C₂²	1	2	yes	—
32	#19128742	C₂²	1	2	yes	—
33	C₈×Dic₁₆₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₃₄	C₂×C₄	1	1	yes	characteristic
42	C₂×D₆₆₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₆₇	C₂³	1	2	yes	—
45	#14581933	—	4	2	no	—
46	C₂×C₁₆₇⋊C₈	—	4	2	no	—
47	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
49	C₈×D₁₆₇	—	4	2	no	—
50	C₄⋊Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
52	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
57	(C₂×D₁₆₇)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₆₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₆₇)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₆₇)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₆₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
69	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
73	C₂²×D₁₆₇	C₂²×C₄	1	2	yes	—
74	Dic₃₃₄	—	2	2	no	—
75	D₆₆₈	—	2	2	no	—
76	C₂²⋊D₁₆₇	—	2	4	no	—
77	C₂×Dic₁₆₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₆₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₆₇	C₂²×C₄	1	2	yes	—
80	C₄×D₁₆₇	—	2	2	no	—
81	C₄×D₁₆₇	—	2	2	no	—
82	C₁₆₇⋊C₈	—	2	2	no	—
83	C₁₆₇⋊C₈	—	2	2	no	—
84	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₆₇	—	2	2	no	—
86	C₂×C₄×C₁₆₇	—	2	2	no	—
87	C₂×C₄×C₁₆₇	—	2	1	no	—
88	C₂×C₄×C₁₆₇	—	2	1	no	—
89	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₆₇	—	2	1	no	—
96	C₈×C₁₆₇	—	2	1	no	—
97	C₈×C₁₆₇	—	2	1	no	—
98	C₈×C₁₆₇	—	2	1	no	—
99	C₂²×C₁₆₇	—	2	1	no	—
100	C₂²×C₁₆₇	—	2	1	no	—
101	C₂²×C₁₆₇	#131	1	2	yes	—
102	C₂²×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₆₇	—	2	2	no	—
104	C₂×D₁₆₇	—	2	2	no	—
105	C₂×D₁₆₇	—	2	2	no	—
106	C₄×C₁₆₇	—	2	1	no	—
107	C₄×C₁₆₇	—	2	1	no	—
108	C₄×C₁₆₇	—	2	1	no	—
109	C₄×C₁₆₇	—	2	1	no	—
110	C₄×C₁₆₇	#131	1	2	yes	—
111	C₄×C₁₆₇	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₁₆₇	—	2	2	no	—
114	Dic₁₆₇	—	2	2	no	—
115	Dic₁₆₇	—	2	2	no	—
116	C₂×C₁₆₇	—	2	2	no	—
117	C₂×C₁₆₇	#520	1	1	yes	characteristic
118	C₂×C₁₆₇	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₆₇	C₂×#131	1	1	yes	characteristic
120	D₁₆₇	—	4	2	no	—
121	C₁₆₇	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	167	1	no	maximal, Sylow
123	#529	—	167	1	no	—
124	#446	—	167	1	no	—
125	C₈×Q₈	—	167	1	no	—
126	#436	—	167	1	no	—
127	#435	—	167	2	no	—
128	C₈×D₄	—	167	2	no	—
129	C₈×D₄	—	167	1	no	—
130	#433	—	167	2	no	—
131	#431	D₁₆₇	1	1	yes	characteristic, complemented
132	#424	—	167	2	no	—
133	C₂×C₄⋊C₈	—	167	1	no	—
134	C₂×M₁₆	—	167	2	no	—
135	C₂²×C₈	—	167	2	no	—
136	C₂²×C₈	—	167	2	no	—
137	#126	—	167	2	no	—
138	#125	—	167	1	no	—
139	#124	—	167	1	no	—
140	#123	—	167	2	no	—
141	C₂²⋊Q₈	—	167	1	no	—
142	#121	—	167	1	no	—
143	C₄×Q₈	—	167	1	no	—
144	C₄×D₄	—	167	2	no	—
145	C₄×D₄	—	167	1	no	—
146	#117	—	167	2	no	—
147	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
148	C₄⋊C₈	—	167	2	no	—
149	C₄⋊C₈	—	167	2	no	—
150	C₄⋊C₈	—	167	2	no	—
151	C₄⋊C₈	—	167	1	no	—
152	C₄⋊C₈	—	167	1	no	—
153	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
155	C₂²⋊C₈	—	167	2	no	—
156	C₂²⋊C₈	—	167	2	no	—
157	C₂²⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
159	#97	—	167	2	no	—
160	C₄×C₈	—	167	2	no	—
161	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
162	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
163	C₂×Q₈	—	167	1	no	—
164	C₂×D₄	—	167	2	no	—
165	C₂×D₄	—	167	1	no	—
166	C₂²×C₄	—	167	2	no	—
167	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
170	M₁₆	—	668	2	no	—
171	C₂×C₈	—	668	2	no	—
172	C₂×C₈	—	668	2	no	—
173	C₂×C₈	—	167	2	no	—
174	C₂×C₈	—	167	2	no	—
175	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
179	C₄⋊C₄	—	167	2	no	—
180	C₄⋊C₄	—	167	2	no	—
181	C₄⋊C₄	—	167	1	no	—
182	C₄⋊C₄	—	167	2	no	—
183	C₄⋊C₄	—	167	1	no	—
184	C₄⋊C₄	—	167	2	no	—
185	C₂²⋊C₄	—	167	2	no	—
186	C₂²⋊C₄	—	167	2	no	—
187	C₂²⋊C₄	—	167	2	no	—
188	C₂²⋊C₄	—	167	2	no	—
189	C₂²⋊C₄	—	167	2	no	—
190	C₄²	—	167	2	no	—
191	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
192	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
193	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
194	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
195	C₂³	—	167	2	no	—
196	C₂³	C₂×C₄×D₁₆₇	1	1	yes	characteristic
197	Q₈	—	334	2	no	—
198	D₄	—	334	4	no	—
199	D₄	—	334	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	334	2	no	—
205	C₂×C₄	—	334	2	no	—
206	C₂×C₄	—	167	2	no	—
207	C₂×C₄	—	167	2	no	—
208	C₂×C₄	—	167	2	no	—
209	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	334	2	no	—
220	C₈	—	334	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	334	2	no	—
224	C₂²	—	334	2	no	—
225	C₂²	—	334	2	no	—
226	C₂²	#19128838	1	2	yes	—
227	C₂²	C₂²×C₄×D₁₆₇	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	334	2	no	—
233	C₄	—	334	2	no	—
234	C₄	—	334	2	no	—
235	C₄	#19128840	1	2	yes	—
236	C₄	#19128878	1	1	yes	characteristic
237	C₄	#19128879	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	668	2	no	—
240	C₂	#24923749	1	1	yes	characteristic
241	C₂	C₂×#19128838	1	1	yes	characteristic
242	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	#19128806	C₄
cosocle	C₂×C₄×C₁₆₇	C₂⁴
hypercenter	#431	D₁₆₇
hypocenter	C₁₆₇	#2994
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. C₈×Dic₃₃₄	C₂	semidirect
4. #24923348	C₂	semidirect
5. #24923337	C₂	semidirect
6. C₈×D₆₆₈	C₂	semidirect
7. (C₂²⋊D₁₆₇)⋊C₈	C₂	semidirect
8. #24923329	C₂	semidirect
9. #24923294	C₂	semidirect
10. C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	semidirect
11. #431×C₁₆₇	C₂	semidirect
121. C₁₆₇	#2994	semidirect
131. #431	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923803	2
#24923399	2
C₈×Dic₃₃₄	2
#24923348	2
#24923337	2
C₈×D₆₆₈	2
(C₂²⋊D₁₆₇)⋊C₈	2
#24923329	2
#24923294	2
C₂×C₄⋊(C₁₆₇⋊C₈)	2
#431×C₁₆₇	2
#2994	167