Subgroups — C₇×#3368 #11742677

Order 896.

382 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#550	C₂	1	1	yes	maximal, characteristic
3	C₇×#490	C₂	1	2	yes	maximal, complemented
4	C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#478	C₂	1	2	yes	maximal, complemented
6	C₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#460	C₂	1	2	yes	maximal, complemented
8	C₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#440	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#41	—	2	1	no	—
13	C₂×C₇×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
15	C₇×C₄⋊Q₈	—	2	1	no	—
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#127	C₂²	1	1	yes	characteristic
18	C₇×#127	C₂²	1	2	yes	—
19	C₇×#125	—	2	2	no	—
20	C₇×#125	C₂²	1	2	yes	—
21	C₇×#121	—	2	1	no	—
22	C₇×#121	—	2	2	no	—
23	C₄×C₇×Q₈	—	2	1	no	—
24	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
25	C₄×C₇×Q₈	C₂²	1	2	yes	—
26	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
27	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
28	C₄×C₇×D₄	C₂²	1	2	yes	—
29	C₇×#107	C₂²	1	2	yes	—
30	C₇×#106	C₂²	1	1	yes	characteristic
31	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₇×C₄⋊C₈	C₂²	1	2	yes	—
33	C₇×#103	C₂²	1	1	yes	characteristic
34	C₇×#102	C₂²	1	2	yes	—
35	C₇×#102	C₂²	1	2	yes	—
36	C₇×#102	C₂²	1	2	yes	—
37	C₇×#102	C₂²	1	2	yes	—
38	C₇×#102	C₂²	1	1	yes	characteristic
39	C₇×#97	C₂²	1	2	yes	—
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	#3368	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₇×#41	—	4	4	no	—
43	C₂×C₇×Q₈	—	2	1	no	—
44	C₂×C₇×Q₈	D₄	1	1	yes	characteristic
45	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
46	C₂×C₇×D₄	—	2	2	no	—
47	C₂×C₇×D₄	—	2	1	no	—
48	C₂×C₇×D₄	—	2	2	no	—
49	C₂×C₇×D₄	—	2	1	no	—
50	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
51	C₂×C₇×D₄	C₂³	1	2	yes	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₇×SD₁₆	—	2	2	no	—
55	C₇×SD₁₆	—	2	4	no	—
56	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂³	1	2	yes	—
59	C₇×C₄⋊C₄	—	2	2	no	—
60	C₇×C₄⋊C₄	—	2	1	no	—
61	C₇×C₄⋊C₄	—	2	2	no	—
62	C₇×C₄⋊C₄	—	2	1	no	—
63	C₇×C₄⋊C₄	—	2	2	no	—
64	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₇×C₄⋊C₄	D₄	1	2	yes	—
66	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₇×C₄⋊C₄	C₂³	1	2	yes	—
68	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₇×C₂²⋊C₄	—	2	2	no	—
70	C₇×C₂²⋊C₄	—	2	1	no	—
71	C₄²×C₇	—	2	2	no	—
72	C₄²×C₇	—	2	1	no	—
73	C₄²×C₇	C₂³	1	1	yes	characteristic
74	C₄²×C₇	C₂³	1	2	yes	—
75	#557	C₂×C₇	1	1	yes	characteristic, complemented
76	#550	C₂×C₇	1	1	yes	characteristic
77	#490	C₂×C₇	1	2	yes	complemented
78	#486	C₂×C₇	1	1	yes	characteristic, complemented
79	#478	C₂×C₇	1	2	yes	complemented
80	#474	C₂×C₇	1	1	yes	characteristic, complemented
81	#460	C₂×C₇	1	2	yes	complemented
82	#460	C₂×C₇	1	1	yes	characteristic, complemented
83	#446	C₂×C₇	1	1	yes	characteristic, complemented
84	#440	C₂×C₇	1	2	yes	complemented
85	C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	—	2	1	no	—
88	C₇×Q₈	—	2	2	no	—
89	C₇×Q₈	—	2	1	no	—
90	C₇×Q₈	—	4	2	no	—
91	C₇×Q₈	#41	1	2	yes	complemented
92	C₇×D₄	—	2	1	no	—
93	C₇×D₄	—	2	2	no	—
94	C₇×D₄	—	2	1	no	—
95	C₇×D₄	—	2	2	no	—
96	C₇×D₄	—	4	4	no	—
97	C₇×D₄	—	4	2	no	—
98	C₇×D₄	—	8	2	no	—
99	C₇×D₄	—	8	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	4	4	no	—
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
111	C₇×C₈	—	2	1	no	—
112	C₇×C₈	—	2	2	no	—
113	C₇×C₈	—	2	1	no	—
114	C₂×#41	—	2	1	no	—
115	C₂×SD₁₆	C₂²×C₇	1	1	yes	characteristic
116	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
119	#127	C₂²×C₇	1	1	yes	characteristic
120	#127	C₂²×C₇	1	2	yes	—
121	#125	—	2	2	no	—
122	#125	C₂²×C₇	1	2	yes	—
123	#121	—	2	1	no	—
124	#121	—	2	2	no	—
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
127	C₄×Q₈	C₂²×C₇	1	2	yes	—
128	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
129	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
130	C₄×D₄	C₂²×C₇	1	2	yes	—
131	#107	C₂²×C₇	1	2	yes	—
132	#106	C₂²×C₇	1	1	yes	characteristic
133	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
134	C₄⋊C₈	C₂²×C₇	1	2	yes	—
135	#103	C₂²×C₇	1	1	yes	characteristic
136	#102	C₂²×C₇	1	1	yes	characteristic
137	#102	C₂²×C₇	1	2	yes	—
138	#102	C₂²×C₇	1	2	yes	—
139	#102	C₂²×C₇	1	2	yes	—
140	#102	C₂²×C₇	1	2	yes	—
141	#97	C₂²×C₇	1	2	yes	—
142	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
143	C₂²×C₇	—	4	2	no	—
144	C₂²×C₇	—	4	1	no	—
145	C₂²×C₇	—	4	2	no	—
146	C₂²×C₇	—	4	1	no	—
147	C₂²×C₇	—	4	2	no	—
148	C₂²×C₇	—	4	1	no	—
149	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	4	1	no	—
155	C₄×C₇	—	4	2	no	—
156	C₄×C₇	—	4	1	no	—
157	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
158	C₄×C₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
160	#41	—	4	4	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	C₇×D₄	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
169	C₂×D₄	C₂³×C₇	1	2	yes	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	1	no	—
172	SD₁₆	—	2	2	no	—
173	SD₁₆	—	2	4	no	—
174	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₇	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
183	C₄⋊C₄	C₇×D₄	1	2	yes	—
184	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₇	1	2	yes	—
186	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	1	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	1	no	—
191	C₄²	C₂³×C₇	1	1	yes	characteristic
192	C₄²	C₂³×C₇	1	2	yes	—
193	C₂×C₇	—	8	2	no	—
194	C₂×C₇	—	8	1	no	—
195	C₂×C₇	#548	1	1	yes	characteristic
196	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂×C₇	#577	1	1	yes	characteristic
198	C₂³	—	2	2	no	—
199	C₂³	—	2	1	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	4	2	no	—
203	Q₈	C₇×#41	1	2	yes	complemented
204	D₄	—	2	1	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	1	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	2	no	—
210	D₄	—	8	2	no	—
211	D₄	—	8	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
219	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
221	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	1	no	—
226	C₇	#3368	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	4	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
242	C₄	C₂×C₇×#41	1	1	yes	characteristic
243	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
244	C₂	—	8	2	no	—
245	C₂	—	8	1	no	—
246	C₂	C₇×#548	1	1	yes	characteristic
247	C₂	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂	C₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#557	C₂	semidirect
3. C₇×#490	C₂	semidirect
4. C₇×#486	C₂	semidirect
5. C₇×#478	C₂	semidirect
6. C₇×#474	C₂	semidirect
7. C₇×#460	C₂	semidirect
8. C₇×#460	C₂	semidirect
9. C₇×#446	C₂	semidirect
10. C₇×#440	C₂	semidirect
11. C₄×C₇×SD₁₆	C₂	semidirect
41. #3368	C₇	direct (N × H)
75. #557	C₂×C₇	semidirect
77. #490	C₂×C₇	semidirect
78. #486	C₂×C₇	semidirect
79. #478	C₂×C₇	semidirect
80. #474	C₂×C₇	semidirect
81. #460	C₂×C₇	semidirect
82. #460	C₂×C₇	semidirect
83. #446	C₂×C₇	semidirect
84. #440	C₂×C₇	semidirect
85. C₄×SD₁₆	C₂×C₇	semidirect
91. C₇×Q₈	#41	semidirect
203. Q₈	C₇×#41	semidirect
226. C₇	#3368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#557	2
C₇×#550	2
C₇×#490	2
C₇×#486	2
C₇×#478	2
C₇×#474	2
C₇×#460	2
C₇×#446	2
C₇×#440	2
C₄×C₇×SD₁₆	2
#3368	7

Subgroups — C7×#3368 #11742677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3368 #11742677