Subgroups — C₇×#67⋊C₈ #13055761

Order 1344.

268 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#67	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×C₈×D₄	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×#67	C₄	1	1	yes	characteristic
11	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×#217	C₄	1	1	yes	characteristic
15	C₇×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
16	C₇×Dic₃⋊C₄	C₄	1	1	yes	characteristic
17	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₃⋊C₈	—	2	2	no	—
19	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	D₃×C₇×C₈	—	2	2	no	—
22	C₂²×C₇×C₈	—	3	1	no	—
23	C₂²×C₇×C₈	—	3	1	no	—
24	C₄×C₇×D₄	—	3	1	no	—
25	C₇×C₄⋊C₈	—	3	1	no	—
26	C₇×C₂²⋊C₈	—	3	1	no	—
27	C₇×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
28	C₄×C₇×C₈	—	3	1	no	—
29	#67⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₃×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	—	2	1	no	—
33	C₂×C₃×C₄×C₇	—	2	1	no	—
34	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₇×#67	C₈	1	4	yes	complemented
37	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₄×D₃×C₇	—	2	1	no	—
41	C₄×D₃×C₇	—	2	1	no	—
42	C₃×C₇×C₈	—	2	1	no	—
43	C₃×C₇×C₈	—	2	1	no	—
44	C₇×C₃⋊C₈	—	2	1	no	—
45	C₇×C₃⋊C₈	D₄	1	2	yes	complemented
46	C₂×C₇×D₄	—	3	1	no	—
47	C₂²×C₄×C₇	—	3	1	no	—
48	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
49	C₂×C₇×C₈	—	3	1	no	—
50	C₂×C₇×C₈	—	3	1	no	—
51	C₂×C₇×C₈	—	6	2	no	—
52	C₂×C₇×C₈	—	6	2	no	—
53	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
55	C₇×C₄⋊C₄	—	3	1	no	—
56	C₇×C₂²⋊C₄	—	3	1	no	—
57	C₇×C₂²⋊C₄	—	3	1	no	—
58	C₄²×C₇	—	3	1	no	—
59	C₄×#67	C₂×C₇	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂×D₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
62	C₃×C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
63	(C₂×D₃)⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
64	Dic₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
65	C₈×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂²×C₃×C₇	—	2	1	no	—
67	C₂²×C₃×C₇	—	2	1	no	—
68	C₂²×C₃×C₇	C₂×C₈	1	2	yes	complemented
69	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×D₃×C₇	—	2	1	no	—
71	C₂×D₃×C₇	—	2	1	no	—
72	C₂×D₃×C₇	C₂×C₈	1	2	yes	complemented
73	C₃×C₄×C₇	—	2	1	no	—
74	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×C₇	#41	1	1	yes	characteristic
76	C₇×Dic₃	—	2	1	no	—
77	C₇×Dic₃	—	2	1	no	—
78	C₇×Dic₃	C₂×C₈	1	2	yes	complemented
79	C₈×D₄	—	3	1	no	Sylow
80	C₂³×C₇	—	3	1	no	—
81	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
82	C₇×D₄	—	3	4	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	3	1	no	—
86	C₂×C₄×C₇	—	3	1	no	—
87	C₂×C₄×C₇	—	3	1	no	—
88	C₂×C₄×C₇	—	6	1	no	—
89	C₂×C₄×C₇	—	6	1	no	—
90	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
92	C₇×C₈	—	2	1	no	—
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	—	3	2	no	—
95	C₇×C₈	—	6	1	no	—
96	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
97	C₂×#67	C₄×C₇	1	1	yes	characteristic
98	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
101	#217	C₄×C₇	1	1	yes	characteristic
102	(C₂×D₃)⋊C₄	C₄×C₇	1	1	yes	characteristic
103	Dic₃⋊C₄	C₄×C₇	1	1	yes	characteristic
104	C₄×Dic₃	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₃⋊C₈	—	2	2	no	—
106	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
107	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
108	D₃×C₈	—	2	2	no	—
109	C₂×C₃×C₇	—	2	2	no	—
110	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₃×C₇	#131	1	1	yes	characteristic
113	D₃×C₇	—	2	2	no	—
114	C₂²×C₈	—	3	1	no	—
115	C₂²×C₈	—	3	1	no	—
116	C₄×D₄	—	3	1	no	—
117	C₄⋊C₈	—	3	1	no	—
118	C₂²⋊C₈	—	3	1	no	—
119	C₂²⋊C₈	D₃×C₇	1	1	yes	characteristic, complemented
120	C₄×C₈	—	3	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	3	2	no	—
124	C₂²×C₇	—	6	1	no	—
125	C₂²×C₇	—	6	1	no	—
126	C₂²×C₇	D₃×C₈	1	2	yes	complemented
127	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	—	3	2	no	—
130	C₄×C₇	—	6	1	no	—
131	C₄×C₇	—	6	1	no	—
132	C₄×C₇	D₃×D₄	1	1	yes	characteristic
133	C₄×C₇	#237	1	1	yes	characteristic
134	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
135	C₂²×D₃	C₂×C₄×C₇	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
140	#67	C₇×C₈	1	4	yes	complemented
141	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
142	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
143	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
144	C₄×D₃	—	2	1	no	—
145	C₄×D₃	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	2	1	no	—
148	C₃⋊C₈	—	2	1	no	—
149	C₃⋊C₈	C₇×D₄	1	2	yes	complemented
150	C₃×C₇	C₈×D₄	1	1	yes	characteristic, complemented, Hall
151	C₂×D₄	—	3	1	no	—
152	C₂²×C₄	—	3	1	no	—
153	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
154	C₂×C₈	—	3	1	no	—
155	C₂×C₈	—	3	1	no	—
156	C₂×C₈	—	6	2	no	—
157	C₂×C₈	—	6	2	no	—
158	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
159	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
160	C₄⋊C₄	—	3	1	no	—
161	C₂²⋊C₄	—	3	1	no	—
162	C₂²⋊C₄	—	3	1	no	—
163	C₄²	—	3	1	no	—
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	—	6	2	no	—
166	C₂×C₇	#67⋊C₄	1	1	yes	characteristic
167	C₂×C₇	C₂×D₃×C₈	1	1	yes	characteristic
168	C₂×C₇	#907	1	1	yes	characteristic
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	C₂×C₇×C₈	1	2	yes	complemented
172	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
173	C₂×D₃	—	2	1	no	—
174	C₂×D₃	—	2	1	no	—
175	C₂×D₃	C₂×C₇×C₈	1	2	yes	complemented
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
178	C₃×C₄	C₇×#41	1	1	yes	characteristic
179	Dic₃	—	2	1	no	—
180	Dic₃	—	2	1	no	—
181	Dic₃	C₂×C₇×C₈	1	2	yes	complemented
182	C₂³	—	3	1	no	—
183	C₂³	C₄×D₃×C₇	1	1	yes	characteristic
184	D₄	—	3	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	3	1	no	—
188	C₂×C₄	—	3	1	no	—
189	C₂×C₄	—	3	1	no	—
190	C₂×C₄	—	6	1	no	—
191	C₂×C₄	—	6	1	no	—
192	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	3	2	no	—
197	C₈	—	6	1	no	—
198	C₇	#67⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×C₃	—	2	2	no	—
200	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
201	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
202	C₂×C₃	C₇×#131	1	1	yes	characteristic
203	D₃	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	3	2	no	—
207	C₂²	—	6	1	no	—
208	C₂²	—	6	1	no	—
209	C₂²	D₃×C₇×C₈	1	2	yes	complemented
210	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	3	2	no	—
213	C₄	—	6	1	no	—
214	C₄	—	6	1	no	—
215	C₄	D₃×C₇×D₄	1	1	yes	characteristic
216	C₄	C₇×#237	1	1	yes	characteristic
217	C₃	C₇×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	6	2	no	—
220	C₂	C₇×#67⋊C₄	1	1	yes	characteristic
221	C₂	C₂×D₃×C₇×C₈	1	1	yes	characteristic
222	C₂	C₇×#907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×C₂²⋊C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₇×#67	C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×C₂²⋊C₈	D₃
hypocenter	C₃	C₇×C₈×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₃⋊C₈	C₂	semidirect
3. C₂×D₃×C₇×C₈	C₂	semidirect
4. C₃×C₇×C₂²⋊C₈	C₂	semidirect
5. C₇×(C₂×D₃)⋊C₈	C₂	semidirect
6. C₇×Dic₃⋊C₈	C₂	semidirect
7. C₇×C₈×Dic₃	C₂	semidirect
27. C₇×C₂²⋊C₈	D₃	semidirect
29. #67⋊C₈	C₇	direct (N × H)
36. C₇×#67	C₈	semidirect
45. C₇×C₃⋊C₈	D₄	semidirect
60. C₂²×C₃⋊C₈	C₂×C₇	semidirect
61. C₂×D₃×C₈	C₂×C₇	semidirect
62. C₃×C₂²⋊C₈	C₂×C₇	semidirect
63. (C₂×D₃)⋊C₈	C₂×C₇	semidirect
64. Dic₃⋊C₈	C₂×C₇	semidirect
65. C₈×Dic₃	C₂×C₇	semidirect
68. C₂²×C₃×C₇	C₂×C₈	semidirect
72. C₂×D₃×C₇	C₂×C₈	semidirect
78. C₇×Dic₃	C₂×C₈	semidirect
119. C₂²⋊C₈	D₃×C₇	semidirect
126. C₂²×C₇	D₃×C₈	semidirect
140. #67	C₇×C₈	semidirect
149. C₃⋊C₈	C₇×D₄	semidirect
150. C₃×C₇	C₈×D₄	semidirect
171. C₂²×C₃	C₂×C₇×C₈	semidirect
175. C₂×D₃	C₂×C₇×C₈	semidirect
181. Dic₃	C₂×C₇×C₈	semidirect
198. C₇	#67⋊C₈	direct (N × H)
209. C₂²	D₃×C₇×C₈	semidirect
217. C₃	C₇×C₈×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#67	2
C₂²×C₇×C₃⋊C₈	2
C₂×D₃×C₇×C₈	2
C₃×C₇×C₂²⋊C₈	2
C₇×(C₂×D₃)⋊C₈	2
C₇×Dic₃⋊C₈	2
C₇×C₈×Dic₃	2
C₇×C₈×D₄	3
#67⋊C₈	7

Subgroups — C7×#67⋊C8 #13055761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#67⋊C₈ #13055761