Subgroups — C₇×#17837 #13195053

Order 1792.

612 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2991	C₂	1	2	yes	maximal
2	C₇×C₈×#41	C₂	1	1	yes	maximal, characteristic
3	C₇×#1574	C₂	1	6	yes	maximal, complemented
4	C₇×#1567	C₂	1	3	yes	maximal, complemented
5	C₇×#1564	C₂	1	3	yes	maximal, complemented
6	C₄×C₇×#41	C₂²	1	1	yes	characteristic
7	C₇×C₈×Q₈	—	2	2	no	—
8	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₇×#436	—	2	6	no	—
11	C₇×C₈×D₄	C₄	1	3	yes	—
12	C₇×C₈×D₄	C₄	1	3	yes	—
13	C₇×#431	C₂²	1	6	yes	—
14	C₇×#431	C₂²	1	3	yes	—
15	C₇×#423	C₂²	1	6	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
17	C₇×#335	C₂²	1	1	yes	characteristic
18	C₇×#335	C₂²	1	3	yes	—
19	C₇×#335	C₂²	1	6	yes	—
20	C₇×#334	C₂²	1	6	yes	—
21	#17837	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×#41	Q₈	1	1	yes	characteristic
23	C₂×C₇×M₁₆	—	2	6	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
28	C₄×C₇×D₄	C₂×C₄	1	3	yes	—
29	C₄×C₇×D₄	D₄	1	3	yes	—
30	C₇×#117	Q₈	1	3	yes	—
31	C₂×C₄²×C₇	C₂³	1	3	yes	—
32	C₇×C₄⋊C₈	—	2	6	no	—
33	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
34	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₇×C₄⋊C₈	C₂³	1	2	yes	—
36	C₇×C₄⋊C₈	C₂³	1	6	yes	—
37	C₇×C₂²⋊C₈	—	2	6	no	—
38	C₇×C₂²⋊C₈	C₂×C₄	1	6	yes	—
39	C₄×C₇×C₈	—	2	6	no	—
40	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
41	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
42	C₄×C₇×C₈	C₂³	1	3	yes	—
43	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
44	#2991	C₂×C₇	1	2	yes	—
45	C₈×#41	C₂×C₇	1	1	yes	characteristic
46	#1574	C₂×C₇	1	6	yes	complemented
47	#1567	C₂×C₇	1	3	yes	complemented
48	#1564	C₂×C₇	1	3	yes	complemented
49	C₇×#41	—	2	4	no	—
50	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇×D₄	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
55	C₇×M₁₆	—	8	6	no	—
56	C₂×C₇×C₈	—	2	6	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	6	no	—
59	C₂×C₇×C₈	—	4	3	no	—
60	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
63	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
65	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
66	C₇×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
67	C₄²×C₇	C₂²×C₄	1	3	yes	—
68	C₄²×C₇	C₂×D₄	1	3	yes	—
69	C₄²×C₇	C₂×Q₈	1	3	yes	—
70	C₄²×C₇	C₂⁴	1	1	yes	characteristic
71	C₄×#41	C₂²×C₇	1	1	yes	characteristic
72	C₈×Q₈	—	2	2	no	—
73	C₈×Q₈	C₄×C₇	1	1	yes	characteristic
74	C₈×Q₈	C₄×C₇	1	1	yes	characteristic
75	#436	—	2	6	no	—
76	C₈×D₄	C₄×C₇	1	3	yes	—
77	C₈×D₄	C₄×C₇	1	3	yes	—
78	#431	C₂²×C₇	1	6	yes	—
79	#431	C₂²×C₇	1	3	yes	—
80	#423	C₂²×C₇	1	6	yes	—
81	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
82	#335	C₂²×C₇	1	1	yes	characteristic
83	#335	C₂²×C₇	1	3	yes	—
84	#335	C₂²×C₇	1	6	yes	—
85	#334	C₂²×C₇	1	6	yes	—
86	C₂³×C₇	C₂×C₄⋊C₄	1	3	yes	—
87	C₇×Q₈	#108	1	4	yes	—
88	C₇×D₄	—	2	6	no	—
89	C₂×C₄×C₇	—	2	3	no	—
90	C₂×C₄×C₇	—	2	3	no	—
91	C₂×C₄×C₇	—	2	3	no	—
92	C₂×C₄×C₇	—	2	3	no	—
93	C₂×C₄×C₇	—	2	6	no	—
94	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
97	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
100	C₇×C₈	—	2	3	no	—
101	C₇×C₈	—	2	1	no	—
102	C₇×C₈	—	2	3	no	—
103	C₇×C₈	—	4	6	no	—
104	C₇×C₈	—	4	2	no	—
105	C₇×C₈	#131	1	2	yes	—
106	C₂×#41	C₇×Q₈	1	1	yes	characteristic
107	C₂×M₁₆	—	2	6	no	—
108	C₂²×C₈	C₂×C₄×C₇	1	3	yes	—
109	C₂²×C₈	C₂×C₄×C₇	1	3	yes	—
110	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
111	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
112	C₄×D₄	C₂×C₄×C₇	1	3	yes	—
113	C₄×D₄	C₇×D₄	1	3	yes	—
114	#117	C₇×Q₈	1	3	yes	—
115	C₂×C₄²	C₂³×C₇	1	3	yes	—
116	C₄⋊C₈	—	2	6	no	—
117	C₄⋊C₈	C₂×C₄×C₇	1	3	yes	—
118	C₄⋊C₈	C₂×C₄×C₇	1	3	yes	—
119	C₄⋊C₈	C₂³×C₇	1	2	yes	—
120	C₄⋊C₈	C₂³×C₇	1	6	yes	—
121	C₂²⋊C₈	—	2	6	no	—
122	C₂²⋊C₈	C₂×C₄×C₇	1	6	yes	—
123	C₄×C₈	—	2	6	no	—
124	C₄×C₈	C₂×C₄×C₇	1	3	yes	—
125	C₄×C₈	C₂×C₄×C₇	1	3	yes	—
126	C₄×C₈	C₂³×C₇	1	3	yes	—
127	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
128	C₂²×C₇	—	2	3	no	—
129	C₂²×C₇	—	2	3	no	—
130	C₂²×C₇	—	2	3	no	—
131	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₇	—	2	3	no	—
133	C₄×C₇	—	2	3	no	—
134	C₄×C₇	—	2	1	no	—
135	C₄×C₇	C₂×#108	1	6	yes	—
136	C₄×C₇	C₂×#131	1	1	yes	characteristic
137	C₄×C₇	#568	1	1	yes	characteristic
138	C₄×C₇	#576	1	1	yes	characteristic
139	C₄×C₇	#578	1	1	yes	characteristic
140	#41	—	2	4	no	—
141	C₂×Q₈	C₇×C₄⋊C₄	1	1	yes	characteristic
142	C₂×D₄	C₇×C₄⋊C₄	1	3	yes	—
143	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
144	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
145	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
146	M₁₆	—	8	6	no	—
147	C₂×C₈	—	2	6	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	6	no	—
150	C₂×C₈	—	4	3	no	—
151	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
152	C₂×C₈	C₂²×C₄×C₇	1	3	yes	—
153	C₂×C₈	C₂²×C₄×C₇	1	3	yes	—
154	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
155	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
156	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
157	C₂²⋊C₄	C₇×C₄⋊C₄	1	6	yes	—
158	C₄²	C₂²×C₄×C₇	1	3	yes	—
159	C₄²	C₂×C₇×D₄	1	3	yes	—
160	C₄²	C₂×C₇×Q₈	1	3	yes	—
161	C₄²	C₂⁴×C₇	1	1	yes	characteristic
162	C₂×C₇	—	4	3	no	—
163	C₂×C₇	#2906	1	1	yes	characteristic
164	C₂×C₇	#2913	1	1	yes	characteristic
165	C₂×C₇	C₂²×#108	1	1	yes	characteristic
166	C₂³	C₂×C₇×C₄⋊C₄	1	3	yes	—
167	Q₈	C₇×#108	1	4	yes	—
168	D₄	—	2	6	no	—
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	3	no	—
174	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
175	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
176	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
177	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
180	C₈	—	2	3	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	3	no	—
183	C₈	—	4	6	no	—
184	C₈	—	4	2	no	—
185	C₈	C₇×#131	1	2	yes	—
186	C₇	#17837	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²	—	2	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
191	C₄	—	2	3	no	—
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×C₇×#108	1	6	yes	—
195	C₄	C₂×C₇×#131	1	1	yes	characteristic
196	C₄	C₇×#568	1	1	yes	characteristic
197	C₄	C₇×#576	1	1	yes	characteristic
198	C₄	C₇×#578	1	1	yes	characteristic
199	C₂	—	4	3	no	—
200	C₂	C₇×#2906	1	1	yes	characteristic
201	C₂	C₇×#2913	1	1	yes	characteristic
202	C₂	C₂²×C₇×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1574	C₂	semidirect
4. C₇×#1567	C₂	semidirect
5. C₇×#1564	C₂	semidirect
21. #17837	C₇	direct (N × H)
46. #1574	C₂×C₇	semidirect
47. #1567	C₂×C₇	semidirect
48. #1564	C₂×C₇	semidirect
186. C₇	#17837	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×#2991	2
C₇×C₈×#41	2
C₇×#1574	2
C₇×#1567	2
C₇×#1564	2
#17837	7

Subgroups — C7×#17837 #13195053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17837 #13195053