Subgroups — C₇×#3372 #11742681

Order 896.

340 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×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#554	C₂	1	1	yes	maximal, characteristic
3	C₇×#491	C₂	1	2	yes	maximal, complemented
4	C₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#479	C₂	1	2	yes	maximal, complemented
6	C₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#461	C₂	1	2	yes	maximal
8	C₇×#461	C₂	1	1	yes	maximal, characteristic
9	C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#440	C₂	1	2	yes	maximal
11	C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₇×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
14	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#125	—	2	1	no	—
18	C₇×#125	—	2	2	no	—
19	C₇×#125	C₂²	1	2	yes	—
20	C₇×C₂²⋊Q₈	—	2	1	no	—
21	C₇×C₂²⋊Q₈	—	2	2	no	—
22	C₄×C₇×Q₈	—	2	1	no	—
23	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
24	C₄×C₇×Q₈	C₂²	1	2	yes	—
25	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
26	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
27	C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₇×C₄⋊C₄	—	2	1	no	—
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	2	yes	—
34	C₇×#103	C₂²	1	2	yes	—
35	C₇×#103	C₂²	1	2	yes	—
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#103	C₂²	1	1	yes	characteristic
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	#3372	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇×Q₈	—	2	1	no	—
43	C₂×C₇×Q₈	D₄	1	1	yes	characteristic
44	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
45	C₂×C₇×Q₈	C₂³	1	2	yes	—
46	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	—	2	2	no	—
49	C₇×SD₁₆	—	2	2	no	—
50	C₇×SD₁₆	—	2	4	no	—
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂³	1	2	yes	—
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	2	no	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	—	2	2	no	—
63	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₇×C₄⋊C₄	D₄	1	2	yes	—
65	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₇×C₄⋊C₄	C₂³	1	2	yes	—
68	C₇×C₂²⋊C₄	—	2	1	no	—
69	C₇×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₇	—	2	2	no	—
71	C₄²×C₇	—	2	1	no	—
72	C₄²×C₇	C₂³	1	1	yes	characteristic
73	C₄²×C₇	C₂³	1	2	yes	—
74	#557	C₂×C₇	1	1	yes	characteristic, complemented
75	#554	C₂×C₇	1	1	yes	characteristic
76	#491	C₂×C₇	1	2	yes	complemented
77	#487	C₂×C₇	1	1	yes	characteristic, complemented
78	#479	C₂×C₇	1	2	yes	complemented
79	#477	C₂×C₇	1	1	yes	characteristic, complemented
80	#461	C₂×C₇	1	2	yes	—
81	#461	C₂×C₇	1	1	yes	characteristic
82	#446	C₂×C₇	1	1	yes	characteristic, complemented
83	#440	C₂×C₇	1	2	yes	—
84	C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic
85	C₂³×C₇	—	2	1	no	—
86	C₇×Q₈	—	2	2	no	—
87	C₇×Q₈	—	2	1	no	—
88	C₇×Q₈	—	2	2	no	—
89	C₇×Q₈	—	2	1	no	—
90	C₇×Q₈	—	2	2	no	—
91	C₇×Q₈	—	8	1	no	—
92	C₇×D₄	—	2	1	no	—
93	C₇×D₄	#41	1	2	yes	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	2	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	—	4	2	no	—
100	C₂×C₄×C₇	—	8	2	no	—
101	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
106	C₇×C₈	—	2	1	no	—
107	C₇×C₈	—	2	2	no	—
108	C₇×C₈	—	2	1	no	—
109	C₂×SD₁₆	C₂²×C₇	1	1	yes	characteristic
110	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
111	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
112	C₄⋊Q₈	C₂²×C₇	1	2	yes	—
113	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
114	#125	—	2	1	no	—
115	#125	—	2	2	no	—
116	#125	C₂²×C₇	1	2	yes	—
117	C₂²⋊Q₈	—	2	1	no	—
118	C₂²⋊Q₈	—	2	2	no	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
121	C₄×Q₈	C₂²×C₇	1	2	yes	—
122	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
123	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
124	C₄×D₄	C₂²×C₇	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	1	no	—
126	#107	C₂²×C₇	1	2	yes	—
127	#106	C₂²×C₇	1	1	yes	characteristic
128	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
129	C₄⋊C₈	C₂²×C₇	1	2	yes	—
130	#103	C₂²×C₇	1	1	yes	characteristic
131	#103	C₂²×C₇	1	2	yes	—
132	#103	C₂²×C₇	1	2	yes	—
133	#103	C₂²×C₇	1	2	yes	—
134	#103	C₂²×C₇	1	2	yes	—
135	#102	C₂²×C₇	1	1	yes	characteristic
136	#97	C₂²×C₇	1	2	yes	—
137	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	4	1	no	—
140	C₂²×C₇	—	4	1	no	—
141	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	—	2	1	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	—	4	1	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	C₇×D₄	1	1	yes	characteristic
155	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
156	C₂×Q₈	C₂³×C₇	1	2	yes	—
157	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	2	no	—
160	SD₁₆	—	2	2	no	—
161	SD₁₆	—	2	4	no	—
162	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
164	C₂×C₈	C₂³×C₇	1	2	yes	—
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	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
175	C₄⋊C₄	C₇×D₄	1	2	yes	—
176	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₇	1	2	yes	—
179	C₂²⋊C₄	—	2	1	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₇	1	1	yes	characteristic
184	C₄²	C₂³×C₇	1	2	yes	—
185	C₂×C₇	—	4	2	no	—
186	C₂×C₇	#548	1	1	yes	characteristic
187	C₂×C₇	C₂×#137	1	1	yes	characteristic
188	C₂×C₇	#577	1	1	yes	characteristic
189	C₂³	—	2	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	8	1	no	—
196	D₄	—	2	1	no	—
197	D₄	C₇×#41	1	2	yes	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
206	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
207	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
208	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	1	no	—
213	C₇	#3372	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	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×#548	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	C₂³
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×#491	C₂	semidirect
4. C₇×#487	C₂	semidirect
5. C₇×#479	C₂	semidirect
6. C₇×#477	C₂	semidirect
9. C₇×#446	C₂	semidirect
41. #3372	C₇	direct (N × H)
74. #557	C₂×C₇	semidirect
76. #491	C₂×C₇	semidirect
77. #487	C₂×C₇	semidirect
78. #479	C₂×C₇	semidirect
79. #477	C₂×C₇	semidirect
82. #446	C₂×C₇	semidirect
213. C₇	#3372	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×#554	2
C₇×#491	2
C₇×#487	2
C₇×#479	2
C₇×#477	2
C₇×#461	2
C₇×#446	2
C₇×#440	2
C₄×C₇×SD₁₆	2
#3372	7

Subgroups — C7×#3372 #11742681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3372 #11742681