Subgroups — C₂×C₇×#2862 #13213848

Order 1792.

2814 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₇×#2862	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₇×#391	C₂	1	8	yes	maximal
4	C₂×C₇×#382	C₂	1	4	yes	maximal
5	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇×Q₈	C₂²	1	4	yes	—
7	C₂×C₇×C₄⋊Q₈	—	2	16	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	16	yes	—
9	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₄²×C₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₇	C₂²	1	2	yes	—
14	C₇×#391	C₂²	1	64	yes	—
15	C₇×#382	C₂²	1	32	yes	—
16	C₂×C₇×#95	C₂²	1	16	yes	—
17	C₄³×C₇	C₂²	1	8	yes	—
18	C₂×#2862	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×Q₈	—	2	16	no	—
20	C₂²×C₇×Q₈	—	2	32	no	—
21	C₂²×C₇×Q₈	C₂³	1	16	yes	—
22	C₂³×C₄×C₇	C₂³	1	8	yes	—
23	C₂³×C₄×C₇	C₂³	1	4	yes	—
24	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₇	C₂³	1	2	yes	—
26	C₇×C₄⋊Q₈	—	2	64	no	—
27	C₂×C₇×C₄⋊C₄	—	2	16	no	—
28	C₂×C₇×C₄⋊C₄	—	2	16	no	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
30	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
31	C₂×C₄²×C₇	—	2	16	no	—
32	C₂×C₄²×C₇	D₄	1	8	yes	—
33	C₂×C₄²×C₇	Q₈	1	16	yes	complemented
34	C₂×C₄²×C₇	C₂³	1	16	yes	—
35	C₂×C₄²×C₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₇	C₂³	1	8	yes	—
37	C₇×#95	C₂³	1	64	yes	—
38	C₂²×C₄⋊Q₈	C₂×C₇	1	2	yes	—
39	#2862	C₂×C₇	1	16	yes	direct product factor
40	C₂×#391	C₂×C₇	1	8	yes	—
41	C₂×#382	C₂×C₇	1	4	yes	—
42	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
43	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
44	C₂×C₇×Q₈	—	2	32	no	—
45	C₂×C₇×Q₈	—	2	64	no	—
46	C₂×C₇×Q₈	—	4	64	no	—
47	C₂²×C₄×C₇	—	2	16	no	—
48	C₂²×C₄×C₇	—	2	16	no	—
49	C₂²×C₄×C₇	—	2	32	no	—
50	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
51	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂×Q₈	1	16	yes	—
53	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₇	#41	1	16	yes	—
55	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
58	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
59	C₇×C₄⋊C₄	—	2	32	no	—
60	C₇×C₄⋊C₄	—	2	32	no	—
61	C₄²×C₇	—	2	32	no	—
62	C₄²×C₇	C₂×D₄	1	16	yes	—
63	C₄²×C₇	C₂×Q₈	1	32	yes	complemented
64	C₂³×Q₈	C₂²×C₇	1	4	yes	—
65	C₂×C₄⋊Q₈	—	2	16	no	—
66	C₂×C₄⋊Q₈	C₂²×C₇	1	16	yes	—
67	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
68	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
69	C₂²×C₄²	C₂²×C₇	1	4	yes	—
70	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
71	C₂²×C₄²	C₂²×C₇	1	2	yes	—
72	#391	C₂²×C₇	1	64	yes	—
73	#382	C₂²×C₇	1	32	yes	—
74	C₂×#95	C₂²×C₇	1	16	yes	—
75	C₄³	C₂²×C₇	1	8	yes	—
76	C₂³×C₇	C₂²×D₄	1	2	yes	—
77	C₂³×C₇	C₂²×Q₈	1	4	yes	—
78	C₂³×C₇	C₂×#41	1	8	yes	—
79	C₂³×C₇	C₂⁵	1	1	yes	characteristic
80	C₇×Q₈	—	4	64	no	—
81	C₂×C₄×C₇	—	2	32	no	—
82	C₂×C₄×C₇	—	2	16	no	—
83	C₂×C₄×C₇	—	2	16	no	—
84	C₂×C₄×C₇	—	2	32	no	—
85	C₂×C₄×C₇	—	4	32	no	—
86	C₂×C₄×C₇	#124	1	32	yes	—
87	C₂×C₄×C₇	C₄⋊Q₈	1	16	yes	—
88	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
89	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
90	C₂×C₄×C₇	C₂²×Q₈	1	16	yes	—
91	C₂×C₄×C₇	C₂²×Q₈	1	8	yes	—
92	C₂×C₄×C₇	C₂×#41	1	16	yes	—
93	C₂²×Q₈	—	2	16	no	—
94	C₂²×Q₈	—	2	32	no	—
95	C₂²×Q₈	C₂³×C₇	1	16	yes	—
96	C₂³×C₄	C₂³×C₇	1	8	yes	—
97	C₂³×C₄	C₂³×C₇	1	4	yes	—
98	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
99	C₂³×C₄	C₂³×C₇	1	2	yes	—
100	C₄⋊Q₈	—	2	64	no	—
101	C₂×C₄⋊C₄	—	2	16	no	—
102	C₂×C₄⋊C₄	—	2	16	no	—
103	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
104	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
105	C₂×C₄²	—	2	16	no	—
106	C₂×C₄²	C₇×D₄	1	8	yes	—
107	C₂×C₄²	C₇×Q₈	1	16	yes	complemented
108	C₂×C₄²	C₂³×C₇	1	16	yes	—
109	C₂×C₄²	C₂³×C₇	1	4	yes	—
110	C₂×C₄²	C₂³×C₇	1	8	yes	—
111	#95	C₂³×C₇	1	64	yes	—
112	C₂²×C₇	C₂×#124	1	8	yes	—
113	C₂²×C₇	C₂×C₄⋊Q₈	1	4	yes	—
114	C₂²×C₇	#533	1	16	yes	—
115	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
116	C₂²×C₇	C₂³×Q₈	1	2	yes	—
117	C₂²×C₇	C₂²×#41	1	4	yes	—
118	C₄×C₇	—	2	16	no	—
119	C₄×C₇	—	4	16	no	—
120	C₄×C₇	C₂×#124	1	16	yes	—
121	C₄×C₇	C₂×C₄⋊Q₈	1	8	yes	—
122	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
123	C₂×Q₈	—	2	32	no	—
124	C₂×Q₈	—	2	64	no	—
125	C₂×Q₈	—	4	64	no	—
126	C₂²×C₄	—	2	16	no	—
127	C₂²×C₄	—	2	16	no	—
128	C₂²×C₄	—	2	32	no	—
129	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
130	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
131	C₂²×C₄	C₂×C₇×Q₈	1	16	yes	—
132	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
133	C₂²×C₄	C₇×#41	1	16	yes	—
134	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
135	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
136	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
137	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
138	C₄⋊C₄	—	2	32	no	—
139	C₄⋊C₄	—	2	32	no	—
140	C₄²	—	2	32	no	—
141	C₄²	C₂×C₇×D₄	1	16	yes	—
142	C₄²	C₂×C₇×Q₈	1	32	yes	complemented
143	C₂×C₇	#2862	1	8	yes	direct product factor
144	C₂×C₇	C₂²×#124	1	2	yes	—
145	C₂×C₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
146	C₂×C₇	C₂×#533	1	4	yes	—
147	C₂³	C₂²×C₇×D₄	1	2	yes	—
148	C₂³	C₂²×C₇×Q₈	1	4	yes	—
149	C₂³	C₂×C₇×#41	1	8	yes	—
150	C₂³	C₂⁵×C₇	1	1	yes	characteristic
151	Q₈	—	4	64	no	—
152	C₂×C₄	—	2	16	no	—
153	C₂×C₄	—	2	32	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	2	32	no	—
156	C₂×C₄	—	4	32	no	—
157	C₂×C₄	C₇×#124	1	32	yes	—
158	C₂×C₄	C₇×C₄⋊Q₈	1	16	yes	—
159	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
160	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
161	C₂×C₄	C₂²×C₇×Q₈	1	16	yes	—
162	C₂×C₄	C₂²×C₇×Q₈	1	8	yes	—
163	C₂×C₄	C₂×C₇×#41	1	16	yes	—
164	C₇	C₂×#2862	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²	C₂×C₇×#124	1	8	yes	—
166	C₂²	C₂×C₇×C₄⋊Q₈	1	4	yes	—
167	C₂²	C₇×#533	1	16	yes	—
168	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
169	C₂²	C₂³×C₇×Q₈	1	2	yes	—
170	C₂²	C₂²×C₇×#41	1	4	yes	—
171	C₄	—	2	16	no	—
172	C₄	—	4	16	no	—
173	C₄	C₂×C₇×#124	1	16	yes	—
174	C₄	C₂×C₇×C₄⋊Q₈	1	8	yes	—
175	C₂	C₇×#2862	1	8	yes	direct product factor
176	C₂	C₂²×C₇×#124	1	2	yes	—
177	C₂	C₂²×C₇×C₄⋊Q₈	1	1	yes	characteristic
178	C₂	C₂×C₇×#533	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2862	C₂	direct (N × H)
18. C₂×#2862	C₇	direct (N × H)
33. C₂×C₄²×C₇	Q₈	semidirect
39. #2862	C₂×C₇	direct (N × H)
63. C₄²×C₇	C₂×Q₈	semidirect
107. C₂×C₄²	C₇×Q₈	semidirect
142. C₄²	C₂×C₇×Q₈	semidirect
143. C₂×C₇	#2862	direct (N × H)
164. C₇	C₂×#2862	direct (N × H)
175. C₂	C₇×#2862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₄⋊Q₈	2
C₇×#2862	2
C₂×C₇×#391	2
C₂×C₇×#382	2
C₂×C₄³×C₇	2
C₂×#2862	7

Subgroups — C2×C7×#2862 #13213848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2862 #13213848