Subgroups — C₇×#9709 #13186925

Order 1792.

466 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2862	C₂	1	1	yes	maximal, characteristic
2	C₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₇×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#333	C₂	1	2	yes	maximal
5	C₂×C₇×#332	C₂	1	2	yes	maximal
6	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₇×#97	—	2	1	no	—
11	C₇×#391	—	2	1	no	—
12	C₇×#391	—	2	1	no	—
13	C₇×#391	C₄	1	4	yes	—
14	C₇×#382	—	2	2	no	—
15	C₄³×C₇	C₂²	1	1	yes	characteristic
16	C₇×#336	—	2	1	no	—
17	C₇×#336	—	2	2	no	—
18	C₇×#333	—	2	4	no	—
19	C₇×#332	—	2	4	no	—
20	#9709	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₇×C₄⋊Q₈	—	2	2	no	—
26	C₇×C₄⋊Q₈	—	2	2	no	—
27	C₇×C₄⋊Q₈	—	2	2	no	—
28	C₇×C₄⋊Q₈	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₄⋊C₄	—	2	1	no	—
31	C₂×C₇×C₄⋊C₄	—	2	1	no	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇	D₄	1	2	yes	—
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₇×C₄⋊C₈	—	2	2	no	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₄⋊C₈	D₄	1	4	yes	—
42	C₇×#97	—	4	2	no	—
43	C₇×#95	—	2	4	no	—
44	C₇×#95	—	2	4	no	—
45	#2862	C₂×C₇	1	1	yes	characteristic
46	#1831	C₂×C₇	1	1	yes	characteristic
47	#1767	C₂×C₇	1	1	yes	characteristic
48	C₂×#333	C₂×C₇	1	2	yes	—
49	C₂×#332	C₂×C₇	1	2	yes	—
50	C₂×C₇×Q₈	—	2	4	no	—
51	C₂×C₇×Q₈	—	4	4	no	—
52	C₂×C₇×Q₈	—	4	2	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂×C₇×Q₈	—	4	1	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₇×C₈	—	2	4	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	—	4	2	no	—
67	C₂×C₇×C₈	—	4	4	no	—
68	C₇×C₄⋊C₄	—	2	4	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	1	no	—
71	C₇×C₄⋊C₄	—	4	1	no	—
72	C₇×C₄⋊C₄	—	4	1	no	—
73	C₇×C₄⋊C₄	—	4	1	no	—
74	C₄²×C₇	—	2	2	no	—
75	C₄²×C₇	—	2	1	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	4	no	—
78	C₄²×C₇	—	4	2	no	—
79	C₄²×C₇	C₂×D₄	1	2	yes	—
80	C₄²×C₇	#41	1	2	yes	—
81	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
82	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
83	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
84	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
85	C₂×#97	—	2	1	no	—
86	#391	—	2	1	no	—
87	#391	—	2	1	no	—
88	#391	C₄×C₇	1	4	yes	—
89	#382	—	2	2	no	—
90	C₄³	C₂²×C₇	1	1	yes	characteristic
91	#336	—	2	2	no	—
92	#336	—	2	1	no	—
93	#333	—	2	4	no	—
94	#332	—	2	4	no	—
95	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₇×Q₈	—	8	4	no	—
97	C₇×Q₈	—	8	2	no	—
98	C₇×Q₈	—	8	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₇	#121	1	2	yes	—
113	C₂×C₄×C₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₇	#135	1	2	yes	—
116	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₇	#137	1	1	yes	characteristic
118	C₂×C₄×C₇	#137	1	4	yes	—
119	C₇×C₈	—	4	4	no	—
120	C₇×C₈	—	8	2	no	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
133	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
134	C₂×C₄²	C₇×D₄	1	2	yes	—
135	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
136	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	C₇×D₄	1	4	yes	—
142	#97	—	4	2	no	—
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	C₂²×C₇	#390	1	1	yes	characteristic
146	C₂²×C₇	#413	1	2	yes	—
147	C₂²×C₇	#416	1	1	yes	characteristic
148	C₂²×C₇	#417	1	1	yes	characteristic
149	C₂²×C₇	#419	1	2	yes	—
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	4	2	no	—
152	C₄×C₇	—	4	2	no	—
153	C₄×C₇	—	8	2	no	—
154	C₄×C₇	—	8	2	no	—
155	C₄×C₇	#463	1	2	yes	—
156	C₄×C₇	#464	1	2	yes	—
157	C₄×C₇	#491	1	2	yes	—
158	C₄×C₇	#497	1	2	yes	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
168	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	2	no	—
188	C₄²	C₂×C₇×D₄	1	2	yes	—
189	C₄²	C₇×#41	1	2	yes	—
190	C₂×C₇	#1554	1	2	yes	—
191	C₂×C₇	#1559	1	2	yes	—
192	C₂×C₇	#1960	1	1	yes	characteristic
193	C₂×C₇	#1967	1	1	yes	characteristic
194	C₂×C₇	#1983	1	1	yes	characteristic
195	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	Q₈	—	8	2	no	—
198	Q₈	—	8	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
212	C₂×C₄	C₇×#121	1	2	yes	—
213	C₂×C₄	C₇×#124	1	1	yes	characteristic
214	C₂×C₄	C₇×#127	1	1	yes	characteristic
215	C₂×C₄	C₇×#135	1	2	yes	—
216	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂×C₄	C₇×#137	1	4	yes	—
218	C₂×C₄	C₇×#137	1	1	yes	characteristic
219	C₈	—	4	4	no	—
220	C₈	—	8	2	no	—
221	C₇	#9709	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	C₇×#390	1	1	yes	characteristic
223	C₂²	C₇×#413	1	2	yes	—
224	C₂²	C₇×#416	1	1	yes	characteristic
225	C₂²	C₇×#417	1	1	yes	characteristic
226	C₂²	C₇×#419	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	2	no	—
232	C₄	C₇×#463	1	2	yes	—
233	C₄	C₇×#464	1	2	yes	—
234	C₄	C₇×#491	1	2	yes	—
235	C₄	C₇×#497	1	2	yes	—
236	C₂	C₇×#1554	1	2	yes	—
237	C₂	C₇×#1559	1	2	yes	—
238	C₂	C₇×#1960	1	1	yes	characteristic
239	C₂	C₇×#1967	1	1	yes	characteristic
240	C₂	C₇×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
20. #9709	C₇	direct (N × H)
221. C₇	#9709	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×#2862	2
C₇×#1831	2
C₇×#1767	2
C₂×C₇×#333	2
C₂×C₇×#332	2
#9709	7

Subgroups — C7×#9709 #13186925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#9709 #13186925