Subgroups — C₇×C₈×C₂²⋊C₈ #13185359

Order 1792.

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	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₇×C₈²	C₂	1	2	yes	maximal, complemented
4	C₇×#1273	C₂	1	2	yes	maximal, complemented
5	C₂³×C₇×C₈	C₄	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₂²⋊C₈	C₄	1	4	yes	—
8	C₂×C₄×C₇×C₈	—	2	4	no	—
9	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₇×#336	C₄	1	4	yes	—
14	C₇×C₈²	—	2	4	no	—
15	C₈×C₂²⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	4	no	—
19	C₂²×C₇×C₈	—	2	4	no	—
20	C₂²×C₇×C₈	C₈	1	8	yes	complemented
21	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇	—	2	2	no	—
25	C₂×C₄²×C₇	—	2	2	no	—
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₇×C₂²⋊C₈	C₈	1	16	yes	direct product factor
30	C₄×C₇×C₈	—	2	4	no	—
31	C₄×C₇×C₈	—	2	4	no	—
32	C₄×C₇×C₈	—	2	2	no	—
33	C₄×C₇×C₈	—	2	4	no	—
34	C₄×C₇×C₈	D₄	1	4	yes	—
35	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
36	C₄×C₂²⋊C₈	C₂×C₇	1	2	yes	—
37	C₂×C₈²	C₂×C₇	1	2	yes	complemented
38	#1273	C₂×C₇	1	2	yes	complemented
39	C₂⁴×C₇	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₇	—	2	1	no	—
41	C₂²×C₄×C₇	—	2	1	no	—
42	C₂²×C₄×C₇	—	2	4	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	2	4	no	—
45	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
48	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
51	C₂×C₇×C₈	—	2	2	no	—
52	C₂×C₇×C₈	—	2	8	no	—
53	C₂×C₇×C₈	—	2	2	no	—
54	C₂×C₇×C₈	—	2	8	no	—
55	C₂×C₇×C₈	—	2	16	no	—
56	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
57	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
58	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
59	C₂×C₇×C₈	C₂×C₈	1	16	yes	complemented
60	C₂×C₇×C₈	M₁₆	1	4	yes	—
61	C₄²×C₇	—	2	1	no	—
62	C₄²×C₇	—	2	1	no	—
63	C₄²×C₇	—	2	4	no	—
64	C₄²×C₇	C₂×D₄	1	2	yes	—
65	C₄²×C₇	#41	1	2	yes	—
66	C₂³×C₈	C₄×C₇	1	2	yes	—
67	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
68	C₂×C₂²⋊C₈	C₄×C₇	1	4	yes	—
69	C₂×C₄×C₈	—	2	4	no	—
70	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
71	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
73	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
74	#336	C₄×C₇	1	4	yes	—
75	C₈²	—	2	4	no	—
76	C₂³×C₇	—	2	1	no	—
77	C₂³×C₇	—	2	1	no	—
78	C₂³×C₇	—	2	2	no	—
79	C₂³×C₇	C₄×C₈	1	4	yes	—
80	C₂³×C₇	C₄×C₈	1	2	yes	—
81	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇	—	2	1	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	2	16	no	—
89	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
90	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
91	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
96	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₇	#131	1	4	yes	—
98	C₇×C₈	—	2	4	no	—
99	C₇×C₈	—	2	16	no	—
100	C₇×C₈	C₂²⋊C₈	1	8	yes	direct product factor
101	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
102	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
103	C₂²×C₈	—	2	4	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	C₇×C₈	1	8	yes	complemented
106	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
107	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
108	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
112	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
113	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
114	C₂²⋊C₈	C₇×C₈	1	16	yes	direct product factor
115	C₄×C₈	—	2	4	no	—
116	C₄×C₈	—	2	4	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	4	no	—
119	C₄×C₈	C₇×D₄	1	4	yes	—
120	C₂²×C₇	—	2	4	no	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	—	2	4	no	—
124	C₂²×C₇	C₈²	1	4	yes	complemented
125	C₂²×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₇	C₂×C₄×C₈	1	2	yes	—
127	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
129	C₂²×C₇	#405	1	2	yes	—
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	C₂×C₂²⋊C₈	1	2	yes	—
133	C₄×C₇	#431	1	2	yes	—
134	C₄×C₇	C₈×D₄	1	8	yes	—
135	C₂⁴	C₄²×C₇	1	1	yes	characteristic
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
142	C₂²×C₄	C₄²×C₇	1	2	yes	—
143	C₂²×C₄	C₂×C₇×C₈	1	8	yes	—
144	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
153	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
154	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
155	C₂×C₈	C₂×C₇×C₈	1	16	yes	complemented
156	C₂×C₈	C₇×M₁₆	1	4	yes	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	4	no	—
160	C₄²	C₂×C₇×D₄	1	2	yes	—
161	C₄²	C₇×#41	1	2	yes	—
162	C₂×C₇	—	2	4	no	—
163	C₂×C₇	C₂×C₈²	1	1	yes	characteristic
164	C₂×C₇	C₈×M₁₆	1	2	yes	—
165	C₂×C₇	#1450	1	1	yes	characteristic
166	C₂×C₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₇	C₈×C₂²⋊C₄	1	2	yes	—
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₄×C₇×C₈	1	2	yes	—
172	C₂³	C₄×C₇×C₈	1	4	yes	—
173	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
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	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	C₄×C₇×C₈	1	2	yes	—
182	C₂×C₄	C₄×C₇×C₈	1	4	yes	—
183	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₇×#117	1	1	yes	characteristic
185	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
186	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
187	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
188	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
189	C₂×C₄	C₇×#131	1	4	yes	—
190	C₈	—	2	4	no	—
191	C₈	—	2	16	no	—
192	C₈	C₇×C₂²⋊C₈	1	8	yes	direct product factor
193	C₇	C₈×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₇×C₈²	1	4	yes	complemented
199	C₂²	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²	C₂×C₄×C₇×C₈	1	1	yes	characteristic
201	C₂²	C₂×C₄×C₇×C₈	1	2	yes	—
202	C₂²	C₄×C₇×M₁₆	1	1	yes	characteristic
203	C₂²	C₇×#405	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	C₂×C₇×C₂²⋊C₈	1	2	yes	—
207	C₄	C₇×#431	1	2	yes	—
208	C₄	C₇×C₈×D₄	1	8	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₂×C₇×C₈²	1	1	yes	characteristic
211	C₂	C₇×C₈×M₁₆	1	2	yes	—
212	C₂	C₇×#1450	1	1	yes	characteristic
213	C₂	C₄×C₇×C₂²⋊C₈	1	1	yes	characteristic
214	C₂	C₇×C₈×C₂²⋊C₄	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇×C₈	C₂²
commutator	C₂	C₂×C₇×C₈²
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂³×C₇	C₂×C₄²
prime radical	C₂⁴×C₇	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×C₈²	C₂	semidirect
4. C₇×#1273	C₂	semidirect
15. C₈×C₂²⋊C₈	C₇	direct (N × H)
20. C₂²×C₇×C₈	C₈	semidirect
29. C₇×C₂²⋊C₈	C₈	direct (N × H)
37. C₂×C₈²	C₂×C₇	semidirect
38. #1273	C₂×C₇	semidirect
59. C₂×C₇×C₈	C₂×C₈	semidirect
100. C₇×C₈	C₂²⋊C₈	direct (N × H)
105. C₂²×C₈	C₇×C₈	semidirect
114. C₂²⋊C₈	C₇×C₈	direct (N × H)
124. C₂²×C₇	C₈²	semidirect
155. C₂×C₈	C₂×C₇×C₈	semidirect
192. C₈	C₇×C₂²⋊C₈	direct (N × H)
193. C₇	C₈×C₂²⋊C₈	direct (N × H)
198. C₂²	C₇×C₈²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₄×C₇×C₂²⋊C₈	2
C₂×C₇×C₈²	2
C₇×#1273	2
C₈×C₂²⋊C₈	7

Subgroups — C7×C8×C22⋊C8 #13185359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₈×C₂²⋊C₈ #13185359