Subgroups — C₇×#22538 #13199754

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#557	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×M₁₆	C₂	1	2	yes	maximal
3	C₇×#1966	C₂	1	6	yes	maximal
4	C₇×#1801	C₂	1	3	yes	maximal
5	C₄×C₇×#101	C₂	1	3	yes	maximal
6	C₇×#557	—	2	2	no	—
7	C₇×#557	—	2	6	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₂×C₇×#125	C₄	1	6	yes	—
10	C₂×C₄×C₇×Q₈	C₄	1	2	yes	—
11	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
12	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₇×#446	—	2	6	no	—
14	C₇×C₈×Q₈	—	2	2	no	—
15	C₇×#423	C₂²	1	6	yes	—
16	C₂×C₇×#101	C₂²	1	6	yes	—
17	C₂×C₇×#101	C₂²	1	6	yes	—
18	C₄×C₇×M₁₆	C₂²	1	6	yes	—
19	C₇×#343	C₂²	1	1	yes	characteristic
20	C₇×#343	C₂²	1	3	yes	—
21	#22538	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×Q₈	C₂³	1	3	yes	—
24	C₂×C₇×M₁₆	C₂³	1	6	yes	—
25	C₂×C₇×M₁₆	C₂³	1	2	yes	—
26	C₇×C₄⋊Q₈	—	2	6	no	—
27	C₇×C₄⋊Q₈	—	2	6	no	—
28	C₇×#125	—	2	12	no	—
29	C₇×#125	—	2	12	no	—
30	C₄×C₇×Q₈	—	2	6	no	—
31	C₄×C₇×Q₈	—	2	2	no	—
32	C₄×C₇×Q₈	—	2	6	no	—
33	C₄×C₇×Q₈	—	2	6	no	—
34	C₄×C₇×Q₈	D₄	1	2	yes	—
35	C₄×C₇×Q₈	D₄	1	6	yes	—
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₇	C₂³	1	3	yes	—
43	C₇×C₄⋊C₈	—	2	6	no	—
44	C₇×C₄⋊C₈	—	2	6	no	—
45	C₇×#101	—	2	12	no	—
46	C₇×#101	—	4	6	no	—
47	C₇×#97	—	2	6	no	—
48	C₄×C₇×C₈	—	2	6	no	—
49	C₂×#557	C₂×C₇	1	1	yes	characteristic
50	Q₈×M₁₆	C₂×C₇	1	2	yes	—
51	#1966	C₂×C₇	1	6	yes	—
52	#1801	C₂×C₇	1	3	yes	—
53	C₄×#101	C₂×C₇	1	3	yes	—
54	C₂×C₇×Q₈	—	2	4	no	—
55	C₂×C₇×Q₈	—	2	6	no	—
56	C₂×C₇×Q₈	—	2	6	no	—
57	C₂×C₇×Q₈	—	2	3	no	—
58	C₂×C₇×Q₈	—	2	3	no	—
59	C₂×C₇×Q₈	C₂²⋊C₄	1	3	yes	—
60	C₂×C₇×Q₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
66	C₇×M₁₆	—	2	2	no	—
67	C₇×M₁₆	—	2	6	no	—
68	C₇×M₁₆	—	2	6	no	—
69	C₇×M₁₆	#41	1	4	yes	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	—	2	6	no	—
72	C₇×C₄⋊C₄	—	2	12	no	—
73	C₇×C₄⋊C₄	—	2	6	no	—
74	C₇×C₄⋊C₄	—	2	6	no	—
75	C₇×C₄⋊C₄	—	2	12	no	—
76	C₇×C₄⋊C₄	—	2	2	no	—
77	C₇×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
78	C₇×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
79	C₄²×C₇	—	2	6	no	—
80	C₄²×C₇	—	2	6	no	—
81	C₄²×C₇	C₂×D₄	1	6	yes	—
82	C₄²×C₇	C₂×D₄	1	6	yes	—
83	#557	—	2	2	no	—
84	#557	—	2	6	no	—
85	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
86	C₂×#125	C₄×C₇	1	6	yes	—
87	C₂×C₄×Q₈	C₄×C₇	1	2	yes	—
88	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
89	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
90	#446	—	2	6	no	—
91	C₈×Q₈	—	2	2	no	—
92	#423	C₂²×C₇	1	6	yes	—
93	C₂×#101	C₂²×C₇	1	6	yes	—
94	C₂×#101	C₂²×C₇	1	6	yes	—
95	C₄×M₁₆	C₂²×C₇	1	6	yes	—
96	#343	C₂²×C₇	1	3	yes	—
97	#343	C₂²×C₇	1	1	yes	characteristic
98	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₇×Q₈	—	2	6	no	—
100	C₇×Q₈	—	4	6	no	—
101	C₇×Q₈	—	4	6	no	—
102	C₇×Q₈	#101	1	4	yes	complemented
103	C₂×C₄×C₇	—	2	6	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	6	no	—
106	C₂×C₄×C₇	—	2	6	no	—
107	C₂×C₄×C₇	—	2	6	no	—
108	C₂×C₄×C₇	—	4	3	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	3	yes	—
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
112	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	3	yes	—
113	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₇×C₈	—	2	4	no	—
118	C₇×C₈	—	4	6	no	—
119	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
120	C₂²×Q₈	C₂³×C₇	1	3	yes	—
121	C₂×M₁₆	C₂³×C₇	1	6	yes	—
122	C₂×M₁₆	C₂³×C₇	1	2	yes	—
123	C₄⋊Q₈	—	2	6	no	—
124	C₄⋊Q₈	—	2	6	no	—
125	#125	—	2	12	no	—
126	#125	—	2	12	no	—
127	C₄×Q₈	—	2	6	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₄×Q₈	—	2	6	no	—
130	C₄×Q₈	—	2	6	no	—
131	C₄×Q₈	C₇×D₄	1	2	yes	—
132	C₄×Q₈	C₇×D₄	1	6	yes	—
133	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
134	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
135	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
137	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
138	C₂×C₄²	C₂×C₄×C₇	1	6	yes	—
139	C₂×C₄²	C₂³×C₇	1	3	yes	—
140	C₄⋊C₈	—	2	6	no	—
141	C₄⋊C₈	—	2	6	no	—
142	#101	—	2	12	no	—
143	#101	—	4	6	no	—
144	#97	—	2	6	no	—
145	C₄×C₈	—	2	6	no	—
146	C₂²×C₇	—	2	1	no	—
147	C₂²×C₇	—	2	1	no	—
148	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₇	C₄×#41	1	1	yes	characteristic
150	C₂²×C₇	#520	1	1	yes	characteristic
151	C₄×C₇	—	2	3	no	—
152	C₄×C₇	—	2	6	no	—
153	C₄×C₇	—	4	6	no	—
154	C₄×C₇	—	4	2	no	—
155	C₄×C₇	C₂×#101	1	6	yes	—
156	C₄×C₇	#548	1	2	yes	—
157	C₄×C₇	#550	1	2	yes	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	2	6	no	—
160	C₂×Q₈	—	2	6	no	—
161	C₂×Q₈	—	2	3	no	—
162	C₂×Q₈	—	2	3	no	—
163	C₂×Q₈	C₇×C₂²⋊C₄	1	3	yes	—
164	C₂×Q₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
166	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
167	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
169	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
170	M₁₆	—	2	2	no	—
171	M₁₆	—	2	6	no	—
172	M₁₆	—	2	6	no	—
173	M₁₆	C₇×#41	1	4	yes	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	6	no	—
176	C₄⋊C₄	—	2	12	no	—
177	C₄⋊C₄	—	2	6	no	—
178	C₄⋊C₄	—	2	6	no	—
179	C₄⋊C₄	—	2	12	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₇×C₂²⋊C₄	1	6	yes	—
182	C₄⋊C₄	C₇×C₂²⋊C₄	1	6	yes	—
183	C₄²	—	2	6	no	—
184	C₄²	—	2	6	no	—
185	C₄²	C₂×C₇×D₄	1	6	yes	—
186	C₄²	C₂×C₇×D₄	1	6	yes	—
187	C₂×C₇	—	2	1	no	—
188	C₂×C₇	—	2	1	no	—
189	C₂×C₇	#2341	1	1	yes	characteristic
190	C₂×C₇	C₂²×#101	1	1	yes	characteristic
191	C₂×C₇	#2889	1	1	yes	characteristic
192	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
193	Q₈	—	2	6	no	—
194	Q₈	—	4	6	no	—
195	Q₈	—	4	6	no	—
196	Q₈	C₇×#101	1	4	yes	complemented
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	4	3	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	3	yes	—
205	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
206	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	3	yes	—
207	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
208	C₂×C₄	C₂×C₇×#41	1	2	yes	—
209	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
211	C₈	—	2	4	no	—
212	C₈	—	4	6	no	—
213	C₇	#22538	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²	C₄×C₇×#41	1	1	yes	characteristic
218	C₂²	C₇×#520	1	1	yes	characteristic
219	C₄	—	2	3	no	—
220	C₄	—	2	6	no	—
221	C₄	—	4	6	no	—
222	C₄	—	4	2	no	—
223	C₄	C₂×C₇×#101	1	6	yes	—
224	C₄	C₇×#548	1	2	yes	—
225	C₄	C₇×#550	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	C₇×#2341	1	1	yes	characteristic
229	C₂	C₂²×C₇×#101	1	1	yes	characteristic
230	C₂	C₇×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
21. #22538	C₇	direct (N × H)
102. C₇×Q₈	#101	semidirect
196. Q₈	C₇×#101	semidirect
213. C₇	#22538	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#557	2
C₇×Q₈×M₁₆	2
C₇×#1966	2
C₇×#1801	2
C₄×C₇×#101	2
#22538	7

Subgroups — C7×#22538 #13199754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22538 #13199754