Subgroups — C₇×#3336 #11742645

Order 896.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#552	C₂	1	2	yes	maximal
2	C₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#485	C₂	1	2	yes	maximal, complemented
4	C₇×#483	C₂	1	2	yes	maximal, complemented
5	C₇×#479	C₂	1	2	yes	maximal, complemented
6	C₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
10	C₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#126	—	2	2	no	—
16	C₇×#125	—	2	2	no	—
17	C₇×#125	C₂²	1	2	yes	—
18	C₇×#123	—	2	2	no	—
19	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₄×C₇×Q₈	C₂²	1	2	yes	—
22	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
23	C₇×#117	—	2	2	no	—
24	C₇×#117	C₂²	1	2	yes	—
25	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×#107	C₂²	1	1	yes	characteristic
27	C₇×#107	C₂²	1	1	yes	characteristic
28	C₇×#107	C₂²	1	2	yes	—
29	C₇×#107	C₂²	1	1	yes	characteristic
30	C₇×#106	C₂²	1	2	yes	—
31	C₇×#106	C₂²	1	2	yes	—
32	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₇×#103	C₂²	1	2	yes	—
34	C₇×#103	C₂²	1	2	yes	—
35	C₇×#103	C₂²	1	2	yes	—
36	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
39	#3336	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇×Q₈	C₂³	1	2	yes	—
41	C₂×C₇×D₄	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₇×Q₁₆	—	2	2	no	—
46	C₂×C₇×C₈	—	2	2	no	—
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	2	no	—
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	—	2	2	no	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₇×C₄⋊C₄	—	2	2	no	—
58	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	C₂³	1	2	yes	—
60	C₇×C₄⋊C₄	C₂³	1	2	yes	—
61	C₇×C₄⋊C₄	C₂³	1	2	yes	—
62	C₇×C₂²⋊C₄	—	2	2	no	—
63	C₇×C₂²⋊C₄	—	2	2	no	—
64	C₇×C₂²⋊C₄	—	2	2	no	—
65	C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
66	C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	C₂³	1	1	yes	characteristic
70	#552	C₂×C₇	1	2	yes	—
71	#499	C₂×C₇	1	1	yes	characteristic, complemented
72	#485	C₂×C₇	1	2	yes	complemented
73	#483	C₂×C₇	1	2	yes	complemented
74	#479	C₂×C₇	1	2	yes	complemented
75	#467	C₂×C₇	1	1	yes	characteristic, complemented
76	#467	C₂×C₇	1	1	yes	characteristic, complemented
77	C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
78	C₈×D₄	C₂×C₇	1	1	yes	characteristic
79	#427	C₂×C₇	1	1	yes	characteristic, complemented
80	C₂×#107	C₂×C₇	1	1	yes	characteristic, complemented
81	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
83	C₇×Q₈	—	2	2	no	—
84	C₇×Q₈	—	2	2	no	—
85	C₇×D₄	—	2	2	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	—	2	1	no	—
102	C₇×C₈	—	2	1	no	—
103	C₇×C₈	#41	1	2	yes	—
104	C₂×Q₁₆	C₂²×C₇	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
107	#126	—	2	2	no	—
108	#125	—	2	2	no	—
109	#125	C₂²×C₇	1	2	yes	—
110	#123	—	2	2	no	—
111	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
112	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
113	C₄×Q₈	C₂²×C₇	1	2	yes	—
114	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
115	#117	—	2	2	no	—
116	#117	C₂²×C₇	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
118	#107	C₂²×C₇	1	1	yes	characteristic
119	#107	C₂²×C₇	1	1	yes	characteristic
120	#107	C₂²×C₇	1	2	yes	—
121	#107	C₂²×C₇	1	1	yes	characteristic
122	#106	C₂²×C₇	1	2	yes	—
123	#106	C₂²×C₇	1	2	yes	—
124	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
125	#103	C₂²×C₇	1	2	yes	—
126	#103	C₂²×C₇	1	2	yes	—
127	#103	C₂²×C₇	1	2	yes	—
128	C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
129	C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
131	C₂²×C₇	—	2	1	no	—
132	C₂²×C₇	—	2	1	no	—
133	C₂²×C₇	—	2	1	no	—
134	C₂²×C₇	—	2	1	no	—
135	C₂²×C₇	—	2	1	no	—
136	C₂²×C₇	#135	1	2	yes	—
137	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₇	—	2	1	no	—
139	C₄×C₇	—	2	1	no	—
140	C₄×C₇	—	2	1	no	—
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	—	4	2	no	—
143	C₄×C₇	—	4	2	no	—
144	C₄×C₇	—	4	2	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	C₂×#41	1	1	yes	characteristic
147	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×Q₈	C₂³×C₇	1	2	yes	—
149	C₂×D₄	C₇×D₄	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
153	Q₁₆	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
167	C₄⋊C₄	C₂³×C₇	1	2	yes	—
168	C₄⋊C₄	C₂³×C₇	1	2	yes	—
169	C₄⋊C₄	C₂³×C₇	1	2	yes	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
174	C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂³×C₇	1	1	yes	characteristic
178	C₂×C₇	—	2	2	no	—
179	C₂×C₇	—	4	1	no	—
180	C₂×C₇	#547	1	1	yes	characteristic
181	C₂×C₇	C₂×#135	1	1	yes	characteristic
182	C₂×C₇	#578	1	1	yes	characteristic
183	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
184	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	Q₈	—	2	2	no	—
187	D₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	C₇×#41	1	2	yes	—
206	C₇	#3336	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₇×#135	1	2	yes	—
213	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂×C₇×#41	1	1	yes	characteristic
223	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	C₇×#547	1	1	yes	characteristic
227	C₂	C₂×C₇×#135	1	1	yes	characteristic
228	C₂	C₇×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₇×#499	C₂	semidirect
3. C₇×#485	C₂	semidirect
4. C₇×#483	C₂	semidirect
5. C₇×#479	C₂	semidirect
6. C₇×#467	C₂	semidirect
7. C₇×#467	C₂	semidirect
8. C₄×C₇×Q₁₆	C₂	semidirect
10. C₇×#427	C₂	semidirect
11. C₂×C₇×#107	C₂	semidirect
39. #3336	C₇	direct (N × H)
71. #499	C₂×C₇	semidirect
72. #485	C₂×C₇	semidirect
73. #483	C₂×C₇	semidirect
74. #479	C₂×C₇	semidirect
75. #467	C₂×C₇	semidirect
76. #467	C₂×C₇	semidirect
77. C₄×Q₁₆	C₂×C₇	semidirect
79. #427	C₂×C₇	semidirect
80. C₂×#107	C₂×C₇	semidirect
206. C₇	#3336	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×#552	2
C₇×#499	2
C₇×#485	2
C₇×#483	2
C₇×#479	2
C₇×#467	2
C₄×C₇×Q₁₆	2
C₇×C₈×D₄	2
C₇×#427	2
C₂×C₇×#107	2
#3336	7

Subgroups — C7×#3336 #11742645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3336 #11742645