Subgroups — #36334708

Order 21376.

380 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923803	C₂	1	2	yes	maximal, complemented
2	#24923671	C₂	1	1	yes	maximal, characteristic, complemented
3	#24923653	C₂	1	1	yes	maximal, characteristic, complemented
4	#24923624	C₂	1	2	yes	maximal, complemented
5	#24923510	C₂	1	1	yes	maximal, characteristic, complemented
6	#24923505	C₂	1	1	yes	maximal, characteristic
7	#24923498	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₆₆₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14581939	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₃₃₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₆₆₈	C₂²	1	1	yes	characteristic
13	C₂×#14581939	C₂²	1	2	yes	—
14	C₂×D₁₃₃₆	C₂²	1	1	yes	characteristic
15	#19128826	—	2	2	no	—
16	#19128824	C₂²	1	2	yes	—
17	#19128820	C₂²	1	2	yes	—
18	#19128815	—	2	2	no	—
19	#19128811	C₂²	1	2	yes	—
20	#19128806	C₂²	1	2	yes	—
21	C₄×Dic₃₃₄	C₂²	1	2	yes	—
22	C₄×D₆₆₈	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₁₆₇	—	2	2	no	—
24	#19128789	—	2	2	no	—
25	#19128766	C₂²	1	1	yes	characteristic
26	#19128766	C₂²	1	1	yes	characteristic
27	#19128764	C₂²	1	2	yes	—
28	#19128757	C₂²	1	2	yes	—
29	#19128757	C₂²	1	2	yes	—
30	#19128754	C₂²	1	2	yes	—
31	#19128754	C₂²	1	2	yes	—
32	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
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	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₃₄	C₂³	1	2	yes	—
40	C₂×D₆₆₈	C₂³	1	2	yes	—
41	C₂×C₂²⋊D₁₆₇	—	2	2	no	—
42	C₂×C₄×D₁₆₇	—	2	2	no	—
43	Dic₆₆₈	—	2	2	no	—
44	#14581939	—	2	4	no	—
45	D₁₃₃₆	—	2	2	no	—
46	C₄⋊Dic₁₆₇	C₂³	1	2	yes	—
47	C₄⋊Dic₁₆₇	C₂³	1	2	yes	—
48	Dic₁₆₇⋊C₄	—	2	2	no	—
49	Dic₁₆₇⋊C₄	—	2	2	no	—
50	Dic₁₆₇⋊C₄	—	2	2	no	—
51	Dic₁₆₇⋊C₄	—	2	2	no	—
52	C₂²⋊Dic₁₆₇	—	2	2	no	—
53	(C₂×D₁₆₇)⋊C₄	—	2	2	no	—
54	(C₂×D₁₆₇)⋊C₄	—	2	2	no	—
55	(C₂×D₁₆₇)⋊C₄	—	2	2	no	—
56	(C₂×D₁₆₇)⋊C₄	—	2	2	no	—
57	C₄×Dic₁₆₇	—	2	2	no	—
58	C₂²×C₄×C₁₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₆₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₆₇	—	2	2	no	—
62	C₂×C₈×C₁₆₇	—	2	2	no	—
63	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×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	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
68	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
69	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
70	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
71	C₂²×D₁₆₇	—	2	2	no	—
72	Dic₃₃₄	—	2	2	no	—
73	Dic₃₃₄	—	2	2	no	—
74	D₆₆₈	—	2	2	no	—
75	D₆₆₈	—	2	2	no	—
76	C₂²⋊D₁₆₇	—	4	4	no	—
77	C₂×Dic₁₆₇	—	2	2	no	—
78	C₂×Dic₁₆₇	—	2	2	no	—
79	C₂×Dic₁₆₇	—	2	2	no	—
80	C₄×D₁₆₇	—	4	4	no	—
81	C₂³×C₁₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₆₇	—	2	2	no	—
83	C₂×C₄×C₁₆₇	—	2	2	no	—
84	C₂×C₄×C₁₆₇	—	2	1	no	—
85	C₂×C₄×C₁₆₇	—	2	1	no	—
86	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₆₇	—	2	1	no	—
93	C₈×C₁₆₇	—	2	1	no	—
94	C₈×C₁₆₇	#41	1	2	yes	—
95	C₈×C₁₆₇	#41	1	2	yes	—
96	C₂²×C₁₆₇	—	2	1	no	—
97	C₂²×C₁₆₇	—	2	1	no	—
98	C₂²×C₁₆₇	#135	1	2	yes	—
99	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×D₁₆₇	—	4	2	no	—
101	C₂×D₁₆₇	—	4	2	no	—
102	C₂×D₁₆₇	—	4	2	no	—
103	C₄×C₁₆₇	—	2	1	no	—
104	C₄×C₁₆₇	—	2	1	no	—
105	C₄×C₁₆₇	—	2	1	no	—
106	C₄×C₁₆₇	—	2	1	no	—
107	C₄×C₁₆₇	#135	1	2	yes	—
108	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
109	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
110	Dic₁₆₇	—	4	2	no	—
111	Dic₁₆₇	—	4	2	no	—
112	Dic₁₆₇	—	4	2	no	—
113	C₂×C₁₆₇	—	2	2	no	—
114	C₂×C₁₆₇	#532	1	1	yes	characteristic
115	C₂×C₁₆₇	C₂×#135	1	1	yes	characteristic
116	C₂×C₁₆₇	C₂×#135	1	1	yes	characteristic
117	D₁₆₇	—	8	2	no	—
118	C₁₆₇	#3168	1	1	yes	characteristic, complemented, Sylow
119	#3168	—	167	1	no	maximal, Sylow
120	#529	—	167	2	no	—
121	#499	—	167	1	no	—
122	#495	—	167	1	no	—
123	#488	—	167	2	no	—
124	#467	—	167	1	no	—
125	#466	—	167	1	no	—
126	#465	—	167	2	no	—
127	C₄×Q₁₆	—	167	1	no	—
128	C₄×SD₁₆	—	167	2	no	—
129	C₄×D₈	—	167	1	no	—
130	C₂×C₄×C₈	D₁₆₇	1	1	yes	characteristic, complemented
131	C₂×Q₁₆	—	167	1	no	—
132	C₂×SD₁₆	—	167	2	no	—
133	C₂×D₈	—	167	1	no	—
134	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
136	#126	—	334	2	no	—
137	#125	—	167	2	no	—
138	#124	—	167	2	no	—
139	#123	—	334	2	no	—
140	C₂²⋊Q₈	—	167	2	no	—
141	#121	—	167	2	no	—
142	C₄×Q₈	—	167	2	no	—
143	C₄×D₄	—	334	2	no	—
144	C₄×D₄	—	167	2	no	—
145	#117	—	334	2	no	—
146	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
147	#107	—	167	1	no	—
148	#107	—	167	1	no	—
149	#106	—	167	2	no	—
150	#103	—	167	2	no	—
151	#103	—	167	2	no	—
152	#102	—	167	2	no	—
153	#102	—	167	2	no	—
154	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
155	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
158	C₂×Q₈	—	167	2	no	—
159	C₂×D₄	—	334	2	no	—
160	C₂×D₄	—	167	2	no	—
161	C₂²×C₄	—	334	2	no	—
162	C₂²×C₄	D₆₆₈	1	1	yes	characteristic
163	C₂²×C₄	D₆₆₈	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
165	Q₁₆	—	334	2	no	—
166	SD₁₆	—	334	4	no	—
167	D₈	—	334	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
174	C₄⋊C₄	—	334	2	no	—
175	C₄⋊C₄	—	334	2	no	—
176	C₄⋊C₄	—	334	2	no	—
177	C₄⋊C₄	—	334	2	no	—
178	C₄⋊C₄	—	167	2	no	—
179	C₄⋊C₄	—	167	2	no	—
180	C₂²⋊C₄	—	334	2	no	—
181	C₂²⋊C₄	—	334	2	no	—
182	C₂²⋊C₄	—	334	2	no	—
183	C₂²⋊C₄	—	334	2	no	—
184	C₂²⋊C₄	—	334	2	no	—
185	C₄²	—	334	2	no	—
186	C₄²	D₆₆₈	1	1	yes	characteristic
187	C₄²	D₆₆₈	1	1	yes	characteristic
188	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
189	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
190	C₂³	—	334	2	no	—
191	C₂³	C₂×D₆₆₈	1	1	yes	characteristic
192	Q₈	—	334	2	no	—
193	Q₈	—	334	2	no	—
194	D₄	—	334	2	no	—
195	D₄	—	334	2	no	—
196	D₄	—	668	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	334	2	no	—
202	C₂×C₄	—	334	2	no	—
203	C₂×C₄	—	334	2	no	—
204	C₂×C₄	—	668	4	no	—
205	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
206	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
207	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
209	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	#14581950	1	2	yes	—
214	C₈	#14581950	1	2	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	668	2	no	—
218	C₂²	—	668	2	no	—
219	C₂²	—	668	2	no	—
220	C₂²	#19128854	1	2	yes	—
221	C₂²	C₂²×D₆₆₈	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	668	2	no	—
227	C₄	—	668	2	no	—
228	C₄	—	668	2	no	—
229	C₄	#19128854	1	2	yes	—
230	C₄	C₂×#14581950	1	1	yes	characteristic
231	C₄	C₂×#14581950	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	1336	2	no	—
234	C₂	#24923819	1	1	yes	characteristic
235	C₂	C₂×#19128854	1	1	yes	characteristic
236	C₂	C₂×#19128854	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₈
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting	C₂×C₄×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
prime radical	#24923505	C₂
hypercenter	C₂×C₄×C₈	D₁₆₇
hypocenter	C₁₆₇	#3168
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #24923803	C₂	semidirect
2. #24923671	C₂	semidirect
3. #24923653	C₂	semidirect
4. #24923624	C₂	semidirect
5. #24923510	C₂	semidirect
7. #24923498	C₂	semidirect
8. C₄×Dic₆₆₈	C₂	semidirect
9. C₄×#14581939	C₂	semidirect
10. C₄×D₁₃₃₆	C₂	semidirect
11. C₂×C₄×C₈×C₁₆₇	C₂	semidirect
118. C₁₆₇	#3168	semidirect
130. C₂×C₄×C₈	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
#24923671	2
#24923653	2
#24923624	2
#24923510	2
#24923505	2
#24923498	2
C₄×Dic₆₆₈	2
C₄×#14581939	2
C₄×D₁₃₃₆	2
C₂×C₄×C₈×C₁₆₇	2
#3168	167