Subgroups — C₇×#3356 #11742665

Order 896.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#556	C₂	1	1	yes	maximal, characteristic
2	C₇×#554	C₂	1	1	yes	maximal, characteristic
3	C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#485	C₂	1	2	yes	maximal, complemented
5	C₇×#483	C₂	1	2	yes	maximal, complemented
6	C₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#470	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#436	C₂	1	1	yes	maximal, characteristic
11	C₇×#428	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₇×M₁₆	C₂²	1	2	yes	—
14	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×#126	—	2	2	no	—
17	C₇×#125	—	2	1	no	—
18	C₇×#124	—	2	1	no	—
19	C₇×C₂²⋊Q₈	—	2	2	no	—
20	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×C₇×Q₈	—	2	1	no	—
23	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
24	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
25	C₄×C₇×D₄	—	2	1	no	—
26	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
27	C₇×#117	C₂²	1	2	yes	—
28	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
29	C₇×#107	C₂²	1	2	yes	—
30	C₇×#107	C₂²	1	1	yes	characteristic
31	C₇×#106	C₂²	1	2	yes	—
32	C₇×#106	C₂²	1	2	yes	—
33	C₇×#106	C₂²	1	2	yes	—
34	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₇×#103	C₂²	1	2	yes	—
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#103	C₂²	1	1	yes	characteristic
38	C₇×#103	C₂²	1	1	yes	characteristic
39	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	#3356	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇×Q₈	—	2	1	no	—
43	C₂×C₇×Q₈	—	2	1	no	—
44	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
45	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
46	C₂×C₇×D₄	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	2	no	—
48	C₂²×C₄×C₇	C₂³	1	2	yes	—
49	C₇×Q₁₆	—	2	2	no	—
50	C₇×M₁₆	—	2	4	no	—
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂³	1	2	yes	—
53	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	—	2	1	no	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₇×C₄⋊C₄	—	2	1	no	—
58	C₇×C₄⋊C₄	—	2	1	no	—
59	C₇×C₄⋊C₄	—	2	2	no	—
60	C₇×C₄⋊C₄	—	2	1	no	—
61	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₇×C₄⋊C₄	C₂³	1	2	yes	—
63	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₇×C₄⋊C₄	C₂³	1	2	yes	—
65	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₇×C₂²⋊C₄	—	2	2	no	—
67	C₇×C₂²⋊C₄	—	2	2	no	—
68	C₇×C₂²⋊C₄	—	2	2	no	—
69	C₇×C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²×C₇	—	2	2	no	—
71	C₄²×C₇	—	2	1	no	—
72	C₄²×C₇	—	2	1	no	—
73	C₄²×C₇	C₂³	1	1	yes	characteristic
74	#556	C₂×C₇	1	1	yes	characteristic
75	#554	C₂×C₇	1	1	yes	characteristic
76	#498	C₂×C₇	1	1	yes	characteristic, complemented
77	#485	C₂×C₇	1	2	yes	complemented
78	#483	C₂×C₇	1	2	yes	complemented
79	#477	C₂×C₇	1	1	yes	characteristic, complemented
80	#475	C₂×C₇	1	1	yes	characteristic, complemented
81	#470	C₂×C₇	1	2	yes	complemented
82	C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
83	#436	C₂×C₇	1	1	yes	characteristic
84	#428	C₂×C₇	1	2	yes	complemented
85	C₂³×C₇	C₂×D₄	1	2	yes	—
86	C₇×Q₈	—	2	1	no	—
87	C₇×Q₈	—	2	1	no	—
88	C₇×Q₈	—	2	1	no	—
89	C₇×Q₈	—	2	1	no	—
90	C₇×Q₈	—	4	2	no	—
91	C₇×Q₈	—	8	1	no	—
92	C₇×D₄	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	8	2	no	—
102	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
106	C₇×C₈	—	2	2	no	—
107	C₇×C₈	—	2	1	no	—
108	C₇×C₈	#41	1	2	yes	—
109	C₂×Q₁₆	C₂²×C₇	1	1	yes	characteristic
110	C₂×M₁₆	C₂²×C₇	1	2	yes	—
111	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
112	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
113	#126	—	2	2	no	—
114	#125	—	2	1	no	—
115	#124	—	2	1	no	—
116	C₂²⋊Q₈	—	2	2	no	—
117	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
118	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
121	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
122	C₄×D₄	—	2	1	no	—
123	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
124	#117	C₂²×C₇	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
126	#107	C₂²×C₇	1	1	yes	characteristic
127	#107	C₂²×C₇	1	2	yes	—
128	#106	C₂²×C₇	1	2	yes	—
129	#106	C₂²×C₇	1	2	yes	—
130	#106	C₂²×C₇	1	2	yes	—
131	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
132	#103	C₂²×C₇	1	1	yes	characteristic
133	#103	C₂²×C₇	1	1	yes	characteristic
134	#103	C₂²×C₇	1	2	yes	—
135	#103	C₂²×C₇	1	2	yes	—
136	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
137	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	4	1	no	—
145	C₄×C₇	—	4	1	no	—
146	C₄×C₇	—	4	2	no	—
147	C₄×C₇	—	4	1	no	—
148	C₄×C₇	—	4	1	no	—
149	C₄×C₇	—	4	2	no	—
150	C₄×C₇	#137	1	2	yes	—
151	C₄×C₇	C₂×#41	1	1	yes	characteristic
152	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	1	no	—
155	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
156	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
157	C₂×D₄	C₇×D₄	1	1	yes	characteristic
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	C₂³×C₇	1	2	yes	—
160	Q₁₆	—	2	2	no	—
161	M₁₆	—	2	4	no	—
162	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₇	1	2	yes	—
164	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
173	C₄⋊C₄	C₂³×C₇	1	2	yes	—
174	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
175	C₄⋊C₄	C₂³×C₇	1	2	yes	—
176	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	C₇×D₄	1	2	yes	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	C₂³×C₇	1	1	yes	characteristic
185	C₂×C₇	—	4	2	no	—
186	C₂×C₇	#547	1	1	yes	characteristic
187	C₂×C₇	C₂×#137	1	1	yes	characteristic
188	C₂×C₇	#577	1	1	yes	characteristic
189	C₂³	C₂×C₇×D₄	1	2	yes	—
190	Q₈	—	2	1	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	8	1	no	—
196	D₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
207	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	C₇×#41	1	2	yes	—
213	C₇	#3356	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	C₇×#137	1	2	yes	—
227	C₄	C₂×C₇×#41	1	1	yes	characteristic
228	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	C₇×#547	1	1	yes	characteristic
231	C₂	C₂×C₇×#137	1	1	yes	characteristic
232	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	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
3. C₇×#498	C₂	semidirect
4. C₇×#485	C₂	semidirect
5. C₇×#483	C₂	semidirect
6. C₇×#477	C₂	semidirect
7. C₇×#475	C₂	semidirect
8. C₇×#470	C₂	semidirect
9. C₄×C₇×Q₁₆	C₂	semidirect
11. C₇×#428	C₂	semidirect
41. #3356	C₇	direct (N × H)
76. #498	C₂×C₇	semidirect
77. #485	C₂×C₇	semidirect
78. #483	C₂×C₇	semidirect
79. #477	C₂×C₇	semidirect
80. #475	C₂×C₇	semidirect
81. #470	C₂×C₇	semidirect
82. C₄×Q₁₆	C₂×C₇	semidirect
84. #428	C₂×C₇	semidirect
213. C₇	#3356	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×#556	2
C₇×#554	2
C₇×#498	2
C₇×#485	2
C₇×#483	2
C₇×#477	2
C₇×#475	2
C₇×#470	2
C₄×C₇×Q₁₆	2
C₇×#436	2
C₇×#428	2
#3356	7

Subgroups — C7×#3356 #11742665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3356 #11742665