Subgroups — C₈×C₂²⋊(C₆₇⋊C₈) #30746296

Order 17152.

726 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	C₄×C₂²⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₆₇⋊#1273	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	4	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
7	#16863151	C₄	1	4	yes	—
8	C₈×C₆₇⋊C₈	—	2	4	no	—
9	C₂³×C₈×C₆₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	—	2	4	no	—
12	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²⋊(C₆₇⋊C₈)	C₈	1	16	yes	direct product factor
17	C₄×C₆₇⋊C₈	—	2	4	no	—
18	C₄×C₆₇⋊C₈	—	2	4	no	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	—	2	4	no	—
23	C₂²×C₈×C₆₇	C₈	1	8	yes	complemented
24	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₇	—	2	2	no	—
27	C₂×C₄²×C₆₇	—	2	2	no	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₇	—	2	4	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	D₄	1	4	yes	—
34	C₂×C₆₇⋊C₈	—	2	8	no	—
35	C₂×C₆₇⋊C₈	—	2	8	no	—
36	C₂×C₆₇⋊C₈	C₂×C₈	1	16	yes	complemented
37	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	2	1	no	—
39	C₂²×C₄×C₆₇	—	2	1	no	—
40	C₂²×C₄×C₆₇	—	2	4	no	—
41	C₂²×C₄×C₆₇	—	2	2	no	—
42	C₂²×C₄×C₆₇	—	2	4	no	—
43	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
46	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	16	no	—
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
56	C₄²×C₆₇	—	2	1	no	—
57	C₄²×C₆₇	—	2	1	no	—
58	C₄²×C₆₇	—	2	4	no	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	—	2	1	no	—
62	C₂³×C₆₇	—	2	1	no	—
63	C₂³×C₆₇	—	2	2	no	—
64	C₂³×C₆₇	C₄×C₈	1	4	yes	—
65	C₂³×C₆₇	C₄×C₈	1	2	yes	—
66	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	16	no	—
74	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
75	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
80	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	#131	1	4	yes	—
83	C₈×C₆₇	—	2	4	no	—
84	C₈×C₆₇	C₂²⋊C₈	1	8	yes	complemented
85	C₆₇⋊C₈	—	2	16	no	—
86	C₂²×C₆₇	—	2	4	no	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	—	2	4	no	—
90	C₂²×C₆₇	C₈²	1	4	yes	complemented
91	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
93	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
95	C₂²×C₆₇	#405	1	2	yes	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	2	4	no	—
98	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
99	C₄×C₆₇	#431	1	2	yes	—
100	C₄×C₆₇	C₈×D₄	1	8	yes	—
101	C₈×C₂²⋊C₈	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	—	2	4	no	—
103	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
104	C₂×C₆₇	C₈×M₁₆	1	2	yes	—
105	C₂×C₆₇	#1450	1	1	yes	characteristic
106	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₆₇	C₈×C₂²⋊C₄	1	2	yes	—
108	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
109	C₄×C₂²⋊C₈	—	67	2	no	—
110	C₂×C₈²	—	67	2	no	—
111	#1273	—	67	2	no	—
112	C₆₇	C₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₈	Dic₆₇	1	2	yes	—
114	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
115	C₂×C₂²⋊C₈	—	67	4	no	—
116	C₂×C₄×C₈	—	2	4	no	—
117	C₂×C₄×C₈	—	67	4	no	—
118	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
119	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
121	#336	—	67	4	no	—
122	C₈²	—	134	4	no	—
123	C₂³×C₄	C₄×D₆₇	1	2	yes	—
124	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	67	8	no	—
128	C₂²×C₈	C₆₇⋊C₈	1	8	yes	complemented
129	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
130	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₄×D₆₇	1	2	yes	—
134	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
136	C₂²⋊C₈	—	67	16	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	134	4	no	—
140	C₄×C₈	—	134	4	no	—
141	C₄×C₈	C₂²⋊D₆₇	1	4	yes	—
142	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
149	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
150	C₂²×C₄	C₈×D₆₇	1	8	yes	—
151	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
152	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	16	no	—
157	C₂×C₈	—	67	16	no	—
158	C₂×C₈	—	134	8	no	—
159	C₂×C₈	—	134	8	no	—
160	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
161	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
162	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
163	C₂×C₈	#11761612	1	4	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	4	no	—
167	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
168	C₄²	#11761628	1	2	yes	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₈×Dic₆₇	1	4	yes	—
173	C₂³	C₄×C₆₇⋊C₈	1	2	yes	—
174	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
183	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₂×C₄	#14524002	1	1	yes	characteristic
186	C₂×C₄	C₄×C₂²⋊D₆₇	1	4	yes	—
187	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
188	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
189	C₂×C₄	C₂×#11761612	1	2	yes	—
190	C₂×C₄	#14524049	1	4	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	134	16	no	—
193	C₈	C₂²⋊(C₆₇⋊C₈)	1	8	yes	direct product factor
194	C₂²	—	2	4	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₈×C₆₇⋊C₈	1	4	yes	complemented
199	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₂²	C₂×C₈×Dic₆₇	1	2	yes	—
201	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
202	C₂²	C₄×#11761612	1	1	yes	characteristic
203	C₂²	#16863351	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
207	C₄	#16863449	1	2	yes	—
208	C₄	C₈×C₂²⋊D₆₇	1	8	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₂×C₈×C₆₇⋊C₈	1	1	yes	characteristic
211	C₂	C₈×#11761612	1	2	yes	—
212	C₂	#22558492	1	1	yes	characteristic
213	C₂	C₄×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
214	C₂	C₈×C₂²⋊Dic₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₆₇
commutator	C₂×C₆₇	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄²
prime radical	C₂⁴×C₆₇	C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₈×C₂²⋊C₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
3. C₆₇⋊#1273	C₂	semidirect
16. C₂²⋊(C₆₇⋊C₈)	C₈	direct (N × H)
23. C₂²×C₈×C₆₇	C₈	semidirect
36. C₂×C₆₇⋊C₈	C₂×C₈	semidirect
84. C₈×C₆₇	C₂²⋊C₈	semidirect
90. C₂²×C₆₇	C₈²	semidirect
112. C₆₇	C₈×C₂²⋊C₈	semidirect
128. C₂²×C₈	C₆₇⋊C₈	semidirect
193. C₈	C₂²⋊(C₆₇⋊C₈)	direct (N × H)
198. C₂²	C₈×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₆₇⋊C₈)	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇⋊#1273	2
C₂²×C₄×C₈×C₆₇	2
C₈×C₂²⋊C₈	67

Subgroups — C8×C22⋊(C67⋊C8) #30746296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂²⋊(C₆₇⋊C₈) #30746296