Subgroups — C₇×#9693 #13186909

Order 1792.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₄×C₇×C₄⋊C₈	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×C₄×C₇×C₈	—	2	1	no	—
11	C₇×#391	—	2	2	no	—
12	C₇×#391	C₄	1	4	yes	—
13	C₇×#382	—	2	1	no	—
14	C₇×#382	—	2	1	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	#9693	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	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	2	yes	—
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	C₄×C₇×C₈	—	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	C₄×C₄⋊C₈	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	1	no	—
53	C₂×C₇×Q₈	—	4	1	no	—
54	C₂×C₇×Q₈	—	4	4	no	—
55	C₂²×C₄×C₇	—	2	2	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₇×C₈	—	2	4	no	—
63	C₂×C₇×C₈	—	4	2	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	—	4	4	no	—
67	C₇×C₄⋊C₄	—	2	4	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	1	no	—
71	C₇×C₄⋊C₄	—	4	1	no	—
72	C₄²×C₇	—	2	4	no	—
73	C₄²×C₇	—	2	2	no	—
74	C₄²×C₇	—	4	2	no	—
75	C₄²×C₇	SD₁₆	1	2	yes	—
76	C₄²×C₇	Q₁₆	1	2	yes	—
77	C₄²×C₇	C₂×D₄	1	2	yes	—
78	C₄²×C₇	#41	1	2	yes	—
79	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
81	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
82	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
83	C₂×C₄×C₈	—	2	1	no	—
84	#391	—	2	2	no	—
85	#391	C₄×C₇	1	4	yes	—
86	#382	—	2	1	no	—
87	#382	—	2	1	no	—
88	C₄³	C₂²×C₇	1	1	yes	characteristic
89	#336	—	2	2	no	—
90	#336	—	2	1	no	—
91	#333	—	2	4	no	—
92	#332	—	2	4	no	—
93	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₇×Q₈	—	8	4	no	—
95	C₇×Q₈	—	8	2	no	—
96	C₇×Q₈	—	8	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	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	—	4	2	no	—
104	C₂×C₄×C₇	—	4	2	no	—
105	C₂×C₄×C₇	—	4	4	no	—
106	C₂×C₄×C₇	—	4	4	no	—
107	C₂×C₄×C₇	#103	1	4	yes	—
108	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇	#121	1	2	yes	—
110	C₂×C₄×C₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₇	#127	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇	#135	1	4	yes	—
115	C₂×C₄×C₇	#137	1	2	yes	—
116	C₇×C₈	—	4	4	no	—
117	C₇×C₈	—	8	2	no	—
118	C₂²×Q₈	—	2	1	no	—
119	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
130	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
131	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₇×D₄	1	2	yes	—
133	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	C₇×D₄	1	4	yes	—
139	C₄×C₈	—	4	2	no	—
140	#95	—	2	4	no	—
141	#95	—	2	4	no	—
142	C₂²×C₇	#390	1	1	yes	characteristic
143	C₂²×C₇	#413	1	2	yes	—
144	C₂²×C₇	C₂×#103	1	1	yes	characteristic
145	C₂²×C₇	#416	1	2	yes	—
146	C₂²×C₇	#419	1	1	yes	characteristic
147	C₄×C₇	—	2	4	no	—
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	—	4	2	no	—
150	C₄×C₇	—	8	2	no	—
151	C₄×C₇	—	8	2	no	—
152	C₄×C₇	#461	1	2	yes	—
153	C₄×C₇	#462	1	2	yes	—
154	C₄×C₇	#488	1	2	yes	—
155	C₄×C₇	#495	1	2	yes	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
164	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	C₇×SD₁₆	1	2	yes	—
182	C₄²	C₇×Q₁₆	1	2	yes	—
183	C₄²	C₂×C₇×D₄	1	2	yes	—
184	C₄²	C₇×#41	1	2	yes	—
185	C₂×C₇	#1542	1	2	yes	—
186	C₂×C₇	#1554	1	2	yes	—
187	C₂×C₇	#1958	1	1	yes	characteristic
188	C₂×C₇	#1967	1	1	yes	characteristic
189	C₂×C₇	#1981	1	1	yes	characteristic
190	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
191	Q₈	—	8	4	no	—
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₇×#103	1	4	yes	—
205	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
206	C₂×C₄	C₇×#121	1	2	yes	—
207	C₂×C₄	C₇×#124	1	1	yes	characteristic
208	C₂×C₄	C₇×#127	1	1	yes	characteristic
209	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
210	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
211	C₂×C₄	C₇×#135	1	4	yes	—
212	C₂×C₄	C₇×#137	1	2	yes	—
213	C₈	—	4	4	no	—
214	C₈	—	8	2	no	—
215	C₇	#9693	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	C₇×#390	1	1	yes	characteristic
217	C₂²	C₇×#413	1	2	yes	—
218	C₂²	C₂×C₇×#103	1	1	yes	characteristic
219	C₂²	C₇×#416	1	2	yes	—
220	C₂²	C₇×#419	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	C₇×#461	1	2	yes	—
227	C₄	C₇×#462	1	2	yes	—
228	C₄	C₇×#488	1	2	yes	—
229	C₄	C₇×#495	1	2	yes	—
230	C₂	C₇×#1542	1	2	yes	—
231	C₂	C₇×#1554	1	2	yes	—
232	C₂	C₇×#1958	1	1	yes	characteristic
233	C₂	C₇×#1967	1	1	yes	characteristic
234	C₂	C₇×#1981	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. #9693	C₇	direct (N × H)
215. C₇	#9693	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
C₄×C₇×C₄⋊C₈	2
C₂×C₇×#333	2
C₂×C₇×#332	2
#9693	7

Subgroups — C7×#9693 #13186909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#9693 #13186909