Subgroups — C₇×#12834 #13190050

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#366	C₂	1	2	yes	maximal, complemented
3	C₇×#1808	C₂	1	2	yes	maximal
4	C₇×#1380	C₂	1	2	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₇×C₁₆	—	2	4	no	—
7	C₂²×C₇×C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇×#107	C₂²	1	4	yes	—
11	C₇×#380	—	2	2	no	—
12	C₇×#366	—	2	4	no	—
13	C₇×#340	—	2	2	no	—
14	#12834	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂²×C₇×C₈	—	2	2	no	—
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	Q₈	1	2	yes	—
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₄	—	2	4	no	—
23	C₂×C₇×C₂²⋊C₄	—	4	2	no	—
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₂×C₇×C₁₆	—	2	8	no	—
26	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₇×C₁₆	D₄	1	4	yes	—
29	C₇×#107	—	2	4	no	—
30	C₇×#107	—	2	4	no	—
31	C₇×#95	—	4	2	no	—
32	C₂³×C₁₆	C₂×C₇	1	1	yes	characteristic
33	C₂×#366	C₂×C₇	1	2	yes	complemented
34	#1808	C₂×C₇	1	2	yes	—
35	#1380	C₂×C₇	1	2	yes	complemented
36	C₂⁴×C₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₇	—	2	2	no	—
38	C₂²×C₄×C₇	—	2	2	no	—
39	C₂²×C₄×C₇	—	4	4	no	—
40	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	—	2	2	no	—
47	C₂×C₇×C₈	—	2	4	no	—
48	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
51	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₇×C₈	#41	1	2	yes	—
53	C₇×C₄⋊C₄	—	4	4	no	—
54	C₇×C₄⋊C₄	—	4	4	no	—
55	C₇×C₂²⋊C₄	—	8	4	no	—
56	C₇×C₁₆	—	2	2	no	—
57	C₇×C₁₆	C₂²⋊C₄	1	4	yes	complemented
58	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
59	C₂²×C₁₆	—	2	4	no	—
60	C₂²×C₁₆	C₄×C₇	1	4	yes	complemented
61	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
62	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
63	C₂×#107	C₂²×C₇	1	4	yes	—
64	#380	—	2	2	no	—
65	#366	—	2	4	no	—
66	#340	—	2	2	no	—
67	C₂³×C₇	—	2	2	no	—
68	C₂³×C₇	—	2	2	no	—
69	C₂³×C₇	#107	1	2	yes	—
70	C₂³×C₇	D₁₆	1	2	yes	—
71	C₂³×C₇	Q₃₂	1	2	yes	—
72	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	—	8	4	no	—
78	C₂×C₄×C₇	—	8	4	no	—
79	C₂×C₄×C₇	—	8	4	no	—
80	C₂×C₄×C₇	#107	1	2	yes	—
81	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	#121	1	2	yes	—
83	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇	#135	1	2	yes	—
85	C₇×C₈	—	2	2	no	—
86	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₇×C₈	#117	1	1	yes	characteristic
88	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
89	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
93	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₇×Q₈	1	2	yes	—
95	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₂²⋊C₄	—	4	2	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	8	no	—
100	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
101	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
102	C₂×C₁₆	C₇×D₄	1	4	yes	—
103	#107	—	2	4	no	—
104	#107	—	2	4	no	—
105	#95	—	4	2	no	—
106	C₂²×C₇	—	2	2	no	—
107	C₂²×C₇	—	2	2	no	—
108	C₂²×C₇	—	2	4	no	—
109	C₂²×C₇	—	2	4	no	—
110	C₂²×C₇	#366	1	4	yes	complemented
111	C₂²×C₇	C₂×#107	1	1	yes	characteristic
112	C₂²×C₇	#466	1	2	yes	—
113	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
114	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
115	C₂²×C₇	#508	1	2	yes	—
116	C₄×C₇	—	2	2	no	—
117	C₄×C₇	—	16	4	no	—
118	C₄×C₇	#380	1	1	yes	characteristic
119	C₄×C₇	#427	1	1	yes	characteristic
120	C₄×C₇	#467	1	2	yes	—
121	C₂⁴	C₇×D₈	1	1	yes	characteristic
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
126	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
127	C₂²×C₄	C₇×Q₁₆	1	2	yes	—
128	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
134	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
135	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
136	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
137	C₂×C₈	C₇×#41	1	2	yes	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₂²⋊C₄	—	8	4	no	—
141	C₁₆	—	2	2	no	—
142	C₁₆	C₇×C₂²⋊C₄	1	4	yes	complemented
143	C₂×C₇	—	2	4	no	—
144	C₂×C₇	#1808	1	1	yes	characteristic
145	C₂×C₇	C₂×#366	1	1	yes	characteristic
146	C₂×C₇	#2158	1	1	yes	characteristic
147	C₂×C₇	#2215	1	2	yes	—
148	C₂×C₇	#2216	1	2	yes	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	2	no	—
151	C₂³	C₇×#107	1	2	yes	—
152	C₂³	C₇×D₁₆	1	2	yes	—
153	C₂³	C₇×Q₃₂	1	2	yes	—
154	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	C₇×#107	1	2	yes	—
163	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄	C₇×#121	1	2	yes	—
165	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
166	C₂×C₄	C₇×#135	1	2	yes	—
167	C₈	—	2	2	no	—
168	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
169	C₈	C₇×#117	1	1	yes	characteristic
170	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
171	C₇	#12834	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	C₇×#366	1	4	yes	complemented
177	C₂²	C₂×C₇×#107	1	1	yes	characteristic
178	C₂²	C₇×#466	1	2	yes	—
179	C₂²	C₂×C₇×D₁₆	1	1	yes	characteristic
180	C₂²	C₂×C₇×Q₃₂	1	1	yes	characteristic
181	C₂²	C₇×#508	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	16	4	no	—
184	C₄	C₇×#380	1	1	yes	characteristic
185	C₄	C₇×#427	1	1	yes	characteristic
186	C₄	C₇×#467	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₇×#1808	1	1	yes	characteristic
189	C₂	C₂×C₇×#366	1	1	yes	characteristic
190	C₂	C₇×#2158	1	1	yes	characteristic
191	C₂	C₇×#2215	1	2	yes	—
192	C₂	C₇×#2216	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#366	C₂	semidirect
4. C₇×#1380	C₂	semidirect
7. C₂²×C₇×C₁₆	C₄	semidirect
14. #12834	C₇	direct (N × H)
33. C₂×#366	C₂×C₇	semidirect
35. #1380	C₂×C₇	semidirect
57. C₇×C₁₆	C₂²⋊C₄	semidirect
60. C₂²×C₁₆	C₄×C₇	semidirect
110. C₂²×C₇	#366	semidirect
142. C₁₆	C₇×C₂²⋊C₄	semidirect
171. C₇	#12834	direct (N × H)
176. C₂²	C₇×#366	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₇×C₁₆	2
C₂×C₇×#366	2
C₇×#1808	2
C₇×#1380	2
#12834	7

Subgroups — C7×#12834 #13190050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12834 #13190050