Subgroups — C₇×#13558 #13190774

Order 1792.

290 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1401	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#131	C₄	1	1	yes	characteristic
6	C₇×#504	—	2	1	no	—
7	C₇×#504	—	2	1	no	—
8	C₇×#504	C₄	1	2	yes	—
9	C₇×#504	C₂²	1	2	yes	—
10	C₂×C₇×M₃₂	—	2	1	no	—
11	C₂×C₇×M₃₂	C₄	1	1	yes	characteristic
12	C₂²×C₇×C₁₆	—	2	1	no	—
13	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₇×M₆₄	—	2	1	no	—
15	C₇×M₆₄	—	2	1	no	—
16	C₇×M₆₄	—	2	1	no	—
17	C₇×M₆₄	C₂²	1	2	yes	complemented
18	C₂×C₇×C₃₂	—	2	1	no	—
19	C₂×C₇×C₃₂	C₂²	1	2	yes	complemented
20	#13558	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₇×#41	C₈	1	1	yes	characteristic
22	C₇×#131	—	2	1	no	—
23	C₇×#131	—	2	1	no	—
24	C₇×#131	C₈	1	2	yes	—
25	C₇×#131	C₂×C₄	1	2	yes	—
26	C₂×C₇×M₁₆	—	2	1	no	—
27	C₂×C₇×M₁₆	C₈	1	1	yes	characteristic
28	C₂²×C₇×C₈	—	2	1	no	—
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₇×M₃₂	—	2	1	no	—
31	C₇×M₃₂	—	2	2	no	—
32	C₇×M₃₂	—	2	2	no	—
33	C₇×M₃₂	C₂×C₄	1	2	yes	—
34	C₂×C₇×C₁₆	—	2	1	no	—
35	C₂×C₇×C₁₆	—	2	2	no	—
36	C₂×C₇×C₁₆	—	2	2	no	—
37	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
42	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
43	C₇×C₃₂	—	2	2	no	—
44	C₇×C₃₂	—	2	2	no	—
45	C₂×#504	C₂×C₇	1	1	yes	characteristic
46	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic, complemented
47	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic, complemented
48	#1401	C₂×C₇	1	4	yes	complemented
49	C₇×#41	—	2	1	no	—
50	C₇×#41	—	2	1	no	—
51	C₇×#41	C₁₆	1	2	yes	—
52	C₇×#41	C₂×C₈	1	2	yes	—
53	C₂×C₇×Q₈	C₁₆	1	1	yes	characteristic
54	C₂×C₇×D₄	—	2	1	no	—
55	C₂×C₇×D₄	C₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
58	C₇×M₁₆	—	2	1	no	—
59	C₇×M₁₆	—	2	2	no	—
60	C₇×M₁₆	—	2	2	no	—
61	C₇×M₁₆	C₂×C₈	1	2	yes	—
62	C₂×C₇×C₈	—	2	1	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
68	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
69	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₇×C₁₆	—	2	2	no	—
72	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
73	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
74	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
75	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
76	C₂×#131	C₄×C₇	1	1	yes	characteristic
77	#504	—	2	1	no	—
78	#504	—	2	1	no	—
79	#504	C₄×C₇	1	2	yes	—
80	#504	C₂²×C₇	1	2	yes	—
81	C₂×M₃₂	—	2	1	no	—
82	C₂×M₃₂	C₄×C₇	1	1	yes	characteristic
83	C₂²×C₁₆	—	2	1	no	—
84	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
85	M₆₄	—	2	1	no	—
86	M₆₄	—	2	1	no	—
87	M₆₄	—	2	1	no	—
88	M₆₄	C₂²×C₇	1	2	yes	complemented
89	C₂×C₃₂	—	2	1	no	—
90	C₂×C₃₂	C₂²×C₇	1	2	yes	complemented
91	C₂³×C₇	—	2	1	no	—
92	C₂³×C₇	M₃₂	1	1	yes	characteristic
93	C₇×Q₈	—	2	1	no	—
94	C₇×Q₈	C₂×C₁₆	1	2	yes	—
95	C₇×D₄	—	2	2	no	—
96	C₇×D₄	—	2	2	no	—
97	C₇×D₄	—	2	1	no	—
98	C₇×D₄	C₂×C₁₆	1	2	yes	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇	M₃₂	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
108	C₇×C₈	—	2	2	no	—
109	C₇×C₈	C₂²⋊C₈	1	1	yes	characteristic
110	C₇×C₈	C₂²⋊C₈	1	1	yes	characteristic
111	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
113	C₂×#41	C₇×C₈	1	1	yes	characteristic
114	#131	—	2	1	no	—
115	#131	—	2	1	no	—
116	#131	C₇×C₈	1	2	yes	—
117	#131	C₂×C₄×C₇	1	2	yes	—
118	C₂×M₁₆	—	2	1	no	—
119	C₂×M₁₆	C₇×C₈	1	1	yes	characteristic
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
122	M₃₂	—	2	1	no	—
123	M₃₂	—	2	2	no	—
124	M₃₂	—	2	2	no	—
125	M₃₂	C₂×C₄×C₇	1	2	yes	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
130	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
131	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
132	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
133	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
134	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
135	C₃₂	—	2	2	no	—
136	C₃₂	—	2	2	no	—
137	C₂²×C₇	—	2	2	no	—
138	C₂²×C₇	—	4	1	no	—
139	C₂²×C₇	—	4	2	no	—
140	C₂²×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
141	C₂²×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
142	C₂²×C₇	C₂×M₃₂	1	1	yes	characteristic
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
145	C₄×C₇	C₂²⋊C₁₆	1	1	yes	characteristic
146	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₄×C₇	C₂²×C₁₆	1	1	yes	characteristic
148	#41	—	2	1	no	—
149	#41	—	2	1	no	—
150	#41	C₇×C₁₆	1	2	yes	—
151	#41	C₂×C₇×C₈	1	2	yes	—
152	C₂×Q₈	C₇×C₁₆	1	1	yes	characteristic
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	C₇×C₁₆	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
157	M₁₆	—	2	1	no	—
158	M₁₆	—	2	2	no	—
159	M₁₆	—	2	2	no	—
160	M₁₆	C₂×C₇×C₈	1	2	yes	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
167	C₂×C₈	C₇×M₁₆	1	1	yes	characteristic
168	C₂×C₈	C₇×M₁₆	1	1	yes	characteristic
169	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
170	C₁₆	—	2	2	no	—
171	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
172	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
173	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
174	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
175	C₂×C₇	—	2	1	no	—
176	C₂×C₇	—	2	1	no	—
177	C₂×C₇	—	2	1	no	—
178	C₂×C₇	—	4	2	no	—
179	C₂×C₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
180	C₂³	—	2	1	no	—
181	C₂³	C₇×M₃₂	1	1	yes	characteristic
182	Q₈	—	2	1	no	—
183	Q₈	C₂×C₇×C₁₆	1	2	yes	—
184	D₄	—	2	2	no	—
185	D₄	—	2	2	no	—
186	D₄	—	2	1	no	—
187	D₄	C₂×C₇×C₁₆	1	2	yes	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
192	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
194	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
195	C₂×C₄	C₇×M₃₂	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	C₇×C₂²⋊C₈	1	1	yes	characteristic
199	C₈	C₇×C₂²⋊C₈	1	1	yes	characteristic
200	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
201	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
202	C₇	#13558	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²	—	2	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
207	C₂²	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
208	C₂²	C₂×C₇×M₃₂	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
211	C₄	C₇×C₂²⋊C₁₆	1	1	yes	characteristic
212	C₄	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
213	C₄	C₂²×C₇×C₁₆	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	4	2	no	—
218	C₂	C₂×C₇×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₂×C₂²⋊C₁₆
prime radical	C₂×C₇×#41	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₁₆ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₆₄	C₂	semidirect
3. C₂×C₇×M₆₄	C₂	semidirect
4. C₇×#1401	C₂	semidirect
17. C₇×M₆₄	C₂²	semidirect
19. C₂×C₇×C₃₂	C₂²	semidirect
20. #13558	C₇	direct (N × H)
46. C₂×M₆₄	C₂×C₇	semidirect
47. C₂×M₆₄	C₂×C₇	semidirect
48. #1401	C₂×C₇	semidirect
88. M₆₄	C₂²×C₇	semidirect
90. C₂×C₃₂	C₂²×C₇	semidirect
202. C₇	#13558	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×#504	2
C₂×C₇×M₆₄	2
C₇×#1401	2
#13558	7

Subgroups — C7×#13558 #13190774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13558 #13190774