Subgroups — C₂×#11727074 #13509272

Order 1792.

558 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	C₂×#89646	C₂	1	2	yes	maximal
2	C₂×C₇×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×#89299	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#89284	C₂	1	2	yes	maximal
6	#11727074	C₂	1	8	yes	maximal, direct product factor
7	C₂×#6451	—	2	2	no	—
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₇×#107	C₂²	1	2	yes	—
10	C₂×#6392	C₂²	1	2	yes	—
11	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#6381	—	2	2	no	—
14	#89646	—	2	4	no	—
15	#89646	C₂²	1	8	yes	—
16	C₇×#366	C₂²	1	4	yes	—
17	#89299	C₂²	1	4	yes	—
18	C₁₆×Dic₇	C₂²	1	4	yes	—
19	#89284	C₂²	1	8	yes	—
20	C₂×#2252	—	7	1	no	maximal, Sylow
21	C₂²×Dic₁₄	—	4	2	no	—
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊Dic₇	—	2	2	no	—
25	C₂×Dic₇⋊C₄	—	4	2	no	—
26	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
28	#6451	—	2	8	no	—
29	#6451	—	4	4	no	—
30	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
31	C₂×C₇×C₁₆	C₂³	1	2	yes	—
32	C₇×#107	—	2	2	no	—
33	C₇×#107	C₂³	1	4	yes	—
34	#6392	—	2	2	no	—
35	#6392	C₂³	1	4	yes	—
36	C₈×Dic₇	D₄	1	2	yes	—
37	C₈×Dic₇	C₂³	1	2	yes	—
38	C₂×C₇⋊C₁₆	Q₈	1	4	yes	complemented
39	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
40	#6381	—	2	8	no	—
41	C₂×#500	—	7	2	no	—
42	#2252	—	7	8	no	—
43	C₂×#366	—	7	1	no	—
44	C₂×#366	—	7	2	no	—
45	C₂×#366	D₇	1	1	yes	characteristic
46	C₂×C₄×C₁₆	—	7	1	no	—
47	C₂²×C₄×C₇	—	4	2	no	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₇	—	4	2	no	—
50	C₂²×Dic₇	D₈	1	1	yes	characteristic
51	C₂²×Dic₇	D₈	1	1	yes	characteristic
52	C₂×Dic₁₄	—	4	8	no	—
53	C₂×Dic₁₄	—	8	8	no	—
54	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
56	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	2	4	no	—
59	C₇×C₄⋊C₄	—	4	2	no	—
60	C₄⋊Dic₇	—	2	4	no	—
61	C₄⋊Dic₇	—	4	2	no	—
62	Dic₇⋊C₄	—	4	8	no	—
63	C₄×Dic₇	D₈	1	2	yes	—
64	C₄×Dic₇	C₂×D₄	1	2	yes	—
65	C₂×C₇⋊C₈	Q₁₆	1	4	yes	—
66	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₇×C₁₆	C₂×Q₈	1	4	yes	complemented
68	C₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
69	C₂×C₄⋊Q₈	—	14	2	no	—
70	C₂²×C₁₆	—	7	1	no	—
71	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
72	#500	—	7	8	no	—
73	#500	—	14	4	no	—
74	C₂×#107	—	7	2	no	—
75	C₂×#107	—	14	2	no	—
76	C₂×#107	C₂×D₇	1	2	yes	—
77	C₂×C₄×C₈	—	7	1	no	—
78	#366	—	7	4	no	—
79	#366	—	7	8	no	—
80	#366	C₂×D₇	1	4	yes	—
81	C₄×C₁₆	—	7	4	no	—
82	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₇	—	4	4	no	—
84	C₂×C₄×C₇	—	8	4	no	—
85	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×Dic₇	—	4	4	no	—
90	C₂×Dic₇	—	8	4	no	—
91	C₂×Dic₇	D₁₆	1	4	yes	—
92	C₂×Dic₇	Q₃₂	1	4	yes	—
93	C₂×Dic₇	C₂×D₈	1	2	yes	—
94	C₂×Dic₇	C₂×D₈	1	2	yes	—
95	Dic₁₄	—	8	16	no	—
96	C₇×C₈	C₄⋊Q₈	1	2	yes	—
97	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
98	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
99	C₇⋊C₈	C₂×Q₁₆	1	4	yes	—
100	C₂²×Q₈	—	28	2	no	—
101	C₂²×C₈	—	7	1	no	—
102	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
103	C₄⋊Q₈	—	14	8	no	—
104	C₄⋊Q₈	—	28	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	14	2	no	—
107	C₂×C₄⋊C₄	—	28	2	no	—
108	C₂×C₄²	—	7	1	no	—
109	C₂×C₁₆	—	7	4	no	—
110	C₂×C₁₆	—	7	2	no	—
111	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
112	C₂×C₁₆	C₂²×D₇	1	2	yes	—
113	#107	—	2	2	no	—
114	#107	—	7	4	no	—
115	#107	—	14	8	no	—
116	#107	—	14	2	no	—
117	#107	C₂²×D₇	1	4	yes	—
118	C₄×C₈	—	7	2	no	—
119	C₄×C₈	—	7	2	no	—
120	C₂²×C₇	#500	1	2	yes	—
121	C₂²×C₇	C₂×D₁₆	1	2	yes	—
122	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
123	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₇	—	8	4	no	—
125	C₄×C₇	#500	1	2	yes	—
126	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
128	Dic₇	—	8	4	no	—
129	Dic₇	C₂×D₁₆	1	4	yes	—
130	Dic₇	C₂×Q₃₂	1	4	yes	—
131	C₂×Q₈	—	28	8	no	—
132	C₂×Q₈	—	56	8	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	7	1	no	—
135	C₂²×C₄	—	7	1	no	—
136	C₂²×C₄	—	28	2	no	—
137	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
138	C₂×C₈	—	7	4	no	—
139	C₂×C₈	—	7	2	no	—
140	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
141	C₂×C₈	D₄×D₇	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₇	1	2	yes	—
143	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	14	4	no	—
147	C₄⋊C₄	—	28	8	no	—
148	C₄⋊C₄	—	28	2	no	—
149	C₄²	—	7	2	no	—
150	C₄²	—	7	2	no	—
151	C₁₆	—	7	4	no	—
152	C₁₆	C₂×Dic₁₄	1	4	yes	complemented
153	C₂×C₇	#2252	1	4	yes	complemented
154	C₂×C₇	C₂×#500	1	1	yes	characteristic
155	C₂×C₇	C₂²×D₁₆	1	1	yes	characteristic
156	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
157	C₂³	D₇×D₈	1	1	yes	characteristic
158	Q₈	—	56	16	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	7	2	no	—
161	C₂×C₄	—	7	2	no	—
162	C₂×C₄	—	7	4	no	—
163	C₂×C₄	—	7	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	—	28	4	no	—
166	C₂×C₄	—	56	4	no	—
167	C₂×C₄	#6451	1	2	yes	—
168	C₂×C₄	D₇×D₈	1	1	yes	characteristic
169	C₂×C₄	D₇×Q₁₆	1	2	yes	—
170	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
171	C₈	—	7	4	no	—
172	C₈	#6451	1	2	yes	—
173	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
175	C₇	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
176	C₂²	#89646	1	2	yes	—
177	C₂²	D₇×D₁₆	1	2	yes	—
178	C₂²	D₇×Q₃₂	1	2	yes	—
179	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
180	C₄	—	7	4	no	—
181	C₄	—	7	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	56	4	no	—
184	C₄	#89646	1	2	yes	—
185	C₄	C₂×#6451	1	1	yes	characteristic
186	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
187	C₂	#11727074	1	4	yes	direct product factor
188	C₂	C₂×#89646	1	1	yes	characteristic
189	C₂	C₂×D₇×D₁₆	1	1	yes	characteristic
190	C₂	C₂×D₇×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#366	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×#366	D₇
hypocenter	C₇	C₂×#2252
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
6. #11727074	C₂	direct (N × H)
30. C₂×C₇×C₁₆	Q₈	semidirect
38. C₂×C₇⋊C₁₆	Q₈	semidirect
67. C₇×C₁₆	C₂×Q₈	semidirect
68. C₇⋊C₁₆	C₂×Q₈	semidirect
111. C₂×C₁₆	Dic₁₄	semidirect
152. C₁₆	C₂×Dic₁₄	semidirect
153. C₂×C₇	#2252	semidirect
175. C₇	C₂×#2252	semidirect
187. C₂	#11727074	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×#89646	2
C₂×C₇×#366	2
C₂×#89299	2
C₂×C₁₆×Dic₇	2
C₂×#89284	2
#11727074	2
C₂×#2252	7

Subgroups — C2×#11727074 #13509272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727074 #13509272