Subgroups — C₂×#11739880 #14218735

Order 1792.

2234 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	#11739880	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×C₇⋊C₈	C₂	1	8	yes	maximal, complemented
4	C₂×#89697	C₂	1	6	yes	maximal
5	C₂³×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×Q₈	C₄	1	4	yes	—
7	C₂×C₄×C₇×Q₈	C₄	1	8	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
9	C₂×C₄×C₇×Q₈	C₂²	1	8	yes	—
10	C₂²×C₇×C₄⋊C₄	C₄	1	3	yes	—
11	C₂²×C₄²×C₇	C₂²	1	3	yes	—
12	C₂×C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	—
13	Q₈×C₇⋊C₈	C₂²	1	64	yes	complemented
14	#89697	C₂²	1	48	yes	—
15	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	12	yes	—
16	C₂×C₄×C₇⋊C₈	C₂²	1	12	yes	—
17	C₂×#2982	—	7	1	no	maximal, Sylow
18	C₂²×C₇×Q₈	C₈	1	16	yes	complemented
19	C₂²×C₇×Q₈	C₂×C₄	1	4	yes	—
20	C₂²×C₇×Q₈	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₄×C₇×Q₈	C₂×C₄	1	16	yes	—
25	C₄×C₇×Q₈	C₂×C₄	1	32	yes	—
26	C₄×C₇×Q₈	C₂³	1	16	yes	—
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	12	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	24	yes	—
29	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₇	C₂³	1	6	yes	—
32	C₂×C₄²×C₇	C₂³	1	12	yes	—
33	C₂²×C₇⋊C₈	C₂³	1	8	yes	—
34	C₂²⋊(C₇⋊C₈)	C₂³	1	16	yes	—
35	C₄⋊(C₇⋊C₈)	C₂³	1	48	yes	—
36	C₄×C₇⋊C₈	C₂³	1	48	yes	—
37	C₂²×C₄×Q₈	D₇	1	1	yes	characteristic
38	#2982	—	7	16	no	—
39	C₂×C₈×Q₈	—	7	8	no	—
40	C₂×#431	—	7	6	no	—
41	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₇×Q₈	C₂×C₈	1	32	yes	—
43	C₂×C₇×Q₈	C₂×C₈	1	64	yes	complemented
44	C₂×C₇×Q₈	C₂²×C₄	1	16	yes	—
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	4	6	no	—
47	C₂²×C₄×C₇	C₂×C₈	1	24	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
55	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
56	C₇×C₄⋊C₄	C₂²×C₄	1	48	yes	—
57	C₄²×C₇	C₂²×C₄	1	24	yes	—
58	C₄²×C₇	C₂²×C₄	1	12	yes	—
59	C₄²×C₇	C₂⁴	1	12	yes	—
60	C₂×C₇⋊C₈	—	2	16	no	—
61	C₂×C₇⋊C₈	C₂⁴	1	16	yes	—
62	C₂³×Q₈	Dic₇	1	1	yes	characteristic
63	C₂×C₄×Q₈	Dic₇	1	4	yes	—
64	C₂×C₄×Q₈	Dic₇	1	8	yes	—
65	C₂×C₄×Q₈	C₂×D₇	1	4	yes	—
66	C₂×C₄×Q₈	C₂×D₇	1	8	yes	—
67	C₂²×C₄⋊C₄	Dic₇	1	3	yes	—
68	C₂²×C₄²	C₂×D₇	1	3	yes	—
69	C₈×Q₈	—	7	64	no	—
70	#431	—	7	48	no	—
71	C₂×C₄⋊C₈	—	7	12	no	—
72	C₂×C₂²⋊C₈	—	7	4	no	—
73	C₂×C₄×C₈	—	7	12	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	C₂²×C₈	1	4	yes	—
77	C₂³×C₇	C₂³×C₄	1	2	yes	—
78	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
79	C₇×Q₈	C₂²×C₈	1	64	yes	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	12	no	—
82	C₂×C₄×C₇	—	2	4	no	—
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	4	12	no	—
85	C₂×C₄×C₇	C₂²×C₈	1	24	yes	—
86	C₂×C₄×C₇	C₂²×C₈	1	48	yes	—
87	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
91	C₂×C₄×C₇	2₋¹⁺⁴	1	4	yes	—
92	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
93	C₇⋊C₈	—	2	16	no	—
94	C₂²×Q₈	C₇⋊C₈	1	16	yes	complemented
95	C₂²×Q₈	C₂×Dic₇	1	4	yes	—
96	C₂²×Q₈	C₂×Dic₇	1	8	yes	—
97	C₂³×C₄	C₂×Dic₇	1	3	yes	—
98	C₂³×C₄	C₂×Dic₇	1	3	yes	—
99	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
100	C₂²×C₈	—	7	8	no	—
101	C₄×Q₈	C₂×Dic₇	1	16	yes	—
102	C₄×Q₈	C₂×Dic₇	1	32	yes	—
103	C₄×Q₈	C₂²×D₇	1	16	yes	—
104	C₂×C₄⋊C₄	C₂×Dic₇	1	12	yes	—
105	C₂×C₄⋊C₄	C₂×Dic₇	1	24	yes	—
106	C₂×C₄²	C₂×Dic₇	1	6	yes	—
107	C₂×C₄²	C₂×Dic₇	1	12	yes	—
108	C₂×C₄²	C₂²×D₇	1	12	yes	—
109	C₂×C₄²	C₂²×D₇	1	6	yes	—
110	C₄⋊C₈	—	7	48	no	—
111	C₂²⋊C₈	—	7	16	no	—
112	C₄×C₈	—	7	48	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	—	2	8	no	—
115	C₂²×C₇	—	2	2	no	—
116	C₂²×C₇	#519	1	2	yes	—
117	C₂²×C₇	C₂³×C₈	1	4	yes	—
118	C₂²×C₇	C₂³×C₈	1	2	yes	—
119	C₂²×C₇	#568	1	2	yes	—
120	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
121	C₄×C₇	—	2	2	no	—
122	C₄×C₇	—	2	12	no	—
123	C₄×C₇	#519	1	2	yes	—
124	C₄×C₇	C₂³×C₈	1	24	yes	—
125	C₄×C₇	C₂×2₋¹⁺⁴	1	2	yes	—
126	C₂⁴	C₂²×Dic₇	1	1	yes	characteristic
127	C₂×Q₈	C₂×C₇⋊C₈	1	32	yes	—
128	C₂×Q₈	C₂×C₇⋊C₈	1	64	yes	complemented
129	C₂×Q₈	C₂²×Dic₇	1	16	yes	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	4	6	no	—
132	C₂²×C₄	C₂×C₇⋊C₈	1	24	yes	—
133	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
136	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
137	C₂²×C₄	C₂²×Dic₇	1	6	yes	—
138	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
139	C₂²×C₄	C₂³×D₇	1	2	yes	—
140	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
141	C₂×C₈	—	7	16	no	—
142	C₂×C₈	—	14	16	no	—
143	C₄⋊C₄	C₂²×Dic₇	1	48	yes	—
144	C₄²	C₂²×Dic₇	1	24	yes	—
145	C₄²	C₂²×Dic₇	1	12	yes	—
146	C₄²	C₂³×D₇	1	12	yes	—
147	C₂×C₇	—	2	4	no	—
148	C₂×C₇	#2982	1	4	yes	complemented
149	C₂×C₇	C₂×#519	1	1	yes	characteristic
150	C₂×C₇	C₂⁴×C₈	1	1	yes	characteristic
151	C₂×C₇	C₂×#568	1	1	yes	characteristic
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	C₂²×C₇⋊C₈	1	4	yes	—
155	C₂³	C₂³×Dic₇	1	2	yes	—
156	C₂³	C₂³×Dic₇	1	1	yes	characteristic
157	Q₈	C₂²×C₇⋊C₈	1	64	yes	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	12	no	—
163	C₂×C₄	C₂²×C₇⋊C₈	1	24	yes	—
164	C₂×C₄	C₂²×C₇⋊C₈	1	48	yes	—
165	C₂×C₄	#6551	1	4	yes	—
166	C₂×C₄	C₂³×Dic₇	1	2	yes	—
167	C₂×C₄	C₂³×Dic₇	1	12	yes	—
168	C₂×C₄	C₂³×Dic₇	1	12	yes	—
169	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
170	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
171	C₈	—	14	16	no	—
172	C₇	C₂×#2982	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	8	no	—
176	C₂²	C₂³×C₇⋊C₈	1	4	yes	—
177	C₂²	C₂³×C₇⋊C₈	1	2	yes	—
178	C₂²	#90503	1	2	yes	—
179	C₂²	#90510	1	2	yes	—
180	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	12	no	—
183	C₄	C₂³×C₇⋊C₈	1	24	yes	—
184	C₄	#90503	1	2	yes	—
185	C₄	C₂×#6551	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	#11739880	1	4	yes	direct product factor
188	C₂	C₂⁴×C₇⋊C₈	1	1	yes	characteristic
189	C₂	C₂×#90503	1	1	yes	characteristic
190	C₂	C₂×#90510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₇
commutator	C₂×C₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂²×C₄×C₇×Q₈	C₂
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₇
hypocenter	C₇	C₂×#2982
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #11739880	C₂	direct (N × H)
3. C₂×Q₈×C₇⋊C₈	C₂	semidirect
13. Q₈×C₇⋊C₈	C₂²	semidirect
18. C₂²×C₇×Q₈	C₈	semidirect
43. C₂×C₇×Q₈	C₂×C₈	semidirect
94. C₂²×Q₈	C₇⋊C₈	semidirect
128. C₂×Q₈	C₂×C₇⋊C₈	semidirect
148. C₂×C₇	#2982	semidirect
172. C₇	C₂×#2982	semidirect
187. C₂	#11739880	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×C₄×C₇×Q₈	2
#11739880	2
C₂×Q₈×C₇⋊C₈	2
C₂×#89697	2
C₂×#2982	7

Subgroups — C2×#11739880 #14218735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11739880 #14218735