Subgroups — C₇×#13284 #13190500

Order 1792.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
3	C₇×#1845	C₂	1	1	yes	maximal, characteristic
4	C₇×#1380	C₂	1	4	yes	maximal
5	C₂×C₇×#125	—	2	1	no	—
6	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
7	C₇×#499	—	2	2	no	—
8	C₇×#499	C₄	1	4	yes	complemented
9	C₂×C₇×#107	—	2	1	no	—
10	C₂×C₇×#107	C₂²	1	2	yes	—
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₇×#106	—	2	1	no	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×#384	—	2	1	no	—
15	C₇×#346	—	2	2	no	—
16	#13284	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₇×#125	—	2	4	no	—
20	C₇×#125	—	4	2	no	—
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	2	2	no	—
23	C₂×C₇×C₄⋊C₄	—	4	2	no	—
24	C₂×C₇×C₄⋊C₄	—	4	2	no	—
25	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
26	C₂×C₇×C₁₆	—	2	2	no	—
27	C₂×C₇×C₁₆	—	2	2	no	—
28	C₂×C₇×C₁₆	Q₈	1	4	yes	—
29	C₇×#107	—	2	2	no	—
30	C₇×#107	—	2	4	no	—
31	C₇×#107	—	4	2	no	—
32	C₇×#107	C₂×C₄	1	4	yes	—
33	C₇×#106	—	2	4	no	—
34	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇×C₈	D₄	1	2	yes	—
36	C₂×#499	C₂×C₇	1	1	yes	characteristic
37	C₂×#346	C₂×C₇	1	1	yes	characteristic
38	#1845	C₂×C₇	1	1	yes	characteristic
39	#1380	C₂×C₇	1	4	yes	—
40	C₂²×C₄×C₇	—	4	2	no	—
41	C₂²×C₄×C₇	—	4	2	no	—
42	C₂²×C₄×C₇	D₈	1	2	yes	—
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
45	C₂×C₇×C₈	SD₁₆	1	2	yes	—
46	C₂×C₇×C₈	Q₁₆	1	2	yes	—
47	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
50	C₂×C₇×C₈	#41	1	2	yes	—
51	C₇×C₄⋊C₄	—	2	4	no	—
52	C₇×C₄⋊C₄	—	4	8	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₄⋊C₄	—	8	4	no	—
57	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₇	SD₁₆	1	2	yes	—
59	C₇×C₁₆	—	2	4	no	—
60	C₂×#125	—	2	1	no	—
61	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
62	#499	—	2	2	no	—
63	#499	C₄×C₇	1	4	yes	complemented
64	C₂×#107	—	2	1	no	—
65	C₂×#107	C₂²×C₇	1	2	yes	—
66	C₂×#107	C₂²×C₇	1	2	yes	—
67	C₂×#106	—	2	1	no	—
68	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
69	#384	—	2	1	no	—
70	#346	—	2	2	no	—
71	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	4	4	no	—
75	C₂×C₄×C₇	—	8	2	no	—
76	C₂×C₄×C₇	—	8	2	no	—
77	C₂×C₄×C₇	—	8	2	no	—
78	C₂×C₄×C₇	—	8	4	no	—
79	C₂×C₄×C₇	#102	1	4	yes	—
80	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇	#124	1	1	yes	characteristic
82	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₇×C₈	#103	1	4	yes	—
87	C₇×C₈	C₄×Q₈	1	2	yes	—
88	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
89	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
91	#125	—	2	4	no	—
92	#125	—	4	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	C₇×Q₈	1	4	yes	—
101	#107	—	2	2	no	—
102	#107	—	2	4	no	—
103	#107	—	4	2	no	—
104	#107	C₂×C₄×C₇	1	4	yes	—
105	#106	—	2	4	no	—
106	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
107	C₄×C₈	C₇×D₄	1	2	yes	—
108	C₂²×C₇	C₂×#102	1	1	yes	characteristic
109	C₂²×C₇	#486	1	1	yes	characteristic
110	C₂²×C₇	#500	1	1	yes	characteristic
111	C₂²×C₇	#509	1	2	yes	—
112	C₂²×C₇	#510	1	2	yes	—
113	C₄×C₇	—	2	4	no	—
114	C₄×C₇	—	8	4	no	—
115	C₄×C₇	—	16	2	no	—
116	C₄×C₇	#391	1	1	yes	characteristic
117	C₄×C₇	C₂×#103	1	1	yes	characteristic
118	C₄×C₇	#487	1	1	yes	characteristic
119	C₄×C₇	#498	1	1	yes	characteristic
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	C₇×D₈	1	2	yes	—
123	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
124	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
125	C₂×C₈	C₇×SD₁₆	1	2	yes	—
126	C₂×C₈	C₇×Q₁₆	1	2	yes	—
127	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
128	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
129	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
130	C₂×C₈	C₇×#41	1	2	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	8	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
138	C₄²	C₇×SD₁₆	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂×C₇	#1990	1	1	yes	characteristic
141	C₂×C₇	#2139	1	2	yes	—
142	C₂×C₇	#2244	1	1	yes	characteristic
143	C₂×C₇	#2245	1	1	yes	characteristic
144	C₂×C₇	#2255	1	2	yes	—
145	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₇×#102	1	4	yes	—
154	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₇×#124	1	1	yes	characteristic
156	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
158	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
159	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
160	C₈	C₇×#103	1	4	yes	—
161	C₈	C₄×C₇×Q₈	1	2	yes	—
162	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
163	C₇	#13284	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²	C₂×C₇×#102	1	1	yes	characteristic
165	C₂²	C₇×#486	1	1	yes	characteristic
166	C₂²	C₇×#500	1	1	yes	characteristic
167	C₂²	C₇×#509	1	2	yes	—
168	C₂²	C₇×#510	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	8	4	no	—
171	C₄	—	16	2	no	—
172	C₄	C₇×#391	1	1	yes	characteristic
173	C₄	C₂×C₇×#103	1	1	yes	characteristic
174	C₄	C₇×#487	1	1	yes	characteristic
175	C₄	C₇×#498	1	1	yes	characteristic
176	C₂	C₇×#1990	1	1	yes	characteristic
177	C₂	C₇×#2139	1	2	yes	—
178	C₂	C₇×#2244	1	1	yes	characteristic
179	C₂	C₇×#2245	1	1	yes	characteristic
180	C₂	C₇×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₇×#499	C₄	semidirect
16. #13284	C₇	direct (N × H)
63. #499	C₄×C₇	semidirect
163. C₇	#13284	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×#499	2
C₂×C₇×#346	2
C₇×#1845	2
C₇×#1380	2
#13284	7

Subgroups — C7×#13284 #13190500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13284 #13190500