Subgroups — C₂×#11727035 #13507280

Order 1792.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇⋊C₁₆	C₂	1	2	yes	maximal
4	C₂×#89297	C₂	1	2	yes	maximal
5	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#11727035	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×Dic₁₄	C₄	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₇⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₇	C₂²	1	2	yes	—
12	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
15	C₈×Dic₁₄	C₄	1	4	yes	—
16	C₈×Dic₁₄	C₂²	1	4	yes	—
17	C₇×C₄⋊C₁₆	C₂²	1	4	yes	—
18	#89297	C₂²	1	8	yes	—
19	Dic₇⋊C₁₆	C₂²	1	8	yes	—
20	C₄×C₇⋊C₁₆	C₂²	1	4	yes	—
21	C₂×#2183	—	7	1	no	maximal, Sylow
22	C₂²×Dic₁₄	C₈	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₇	C₈	1	1	yes	characteristic
27	C₂×Dic₇⋊C₄	C₈	1	2	yes	—
28	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
29	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₄	C₈	1	4	yes	—
31	C₄×Dic₁₄	C₂×C₄	1	4	yes	—
32	C₂×C₇×C₁₆	—	2	4	no	—
33	C₂×C₇×C₁₆	C₂³	1	4	yes	—
34	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇×C₈	C₂³	1	2	yes	—
36	Dic₇⋊C₈	C₂×C₄	1	8	yes	—
37	C₈×Dic₇	C₂×C₄	1	4	yes	—
38	C₈×Dic₇	C₂³	1	4	yes	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₂×C₇⋊C₁₆	Q₈	1	4	yes	complemented
41	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
43	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
44	C₂×C₈×Q₈	—	7	1	no	—
45	#2183	—	7	8	no	—
46	C₂×C₄⋊C₁₆	—	7	2	no	—
47	C₂×C₄⋊C₁₆	D₇	1	1	yes	characteristic
48	C₂×#346	—	7	2	no	—
49	C₂×C₄×C₁₆	—	7	1	no	—
50	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₇	C₂×C₈	1	2	yes	—
54	C₂²×Dic₇	C₂×C₈	1	2	yes	—
55	C₂×Dic₁₄	—	2	4	no	—
56	C₂×Dic₁₄	C₂×C₈	1	4	yes	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
61	C₂×C₇×C₈	#41	1	2	yes	—
62	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
63	C₄²×C₇	C₂×C₈	1	2	yes	—
64	C₄²×C₇	C₂²×C₄	1	2	yes	—
65	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
66	Dic₇⋊C₄	C₂×C₈	1	8	yes	—
67	C₄×Dic₇	C₂×C₈	1	4	yes	—
68	C₄×Dic₇	C₂²×C₄	1	4	yes	—
69	C₂×C₇⋊C₈	—	2	4	no	—
70	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
71	C₇×C₁₆	—	2	4	no	—
72	C₇⋊C₁₆	—	2	2	no	—
73	C₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
74	C₂×C₄×Q₈	—	7	1	no	—
75	C₂²×C₁₆	—	7	1	no	—
76	C₂²×C₁₆	—	7	1	no	—
77	C₂²×C₁₆	C₂×D₇	1	2	yes	—
78	C₈×Q₈	—	7	4	no	—
79	C₈×Q₈	—	7	4	no	—
80	C₂×C₄⋊C₈	—	7	2	no	—
81	C₂×C₄⋊C₈	—	7	1	no	—
82	C₂×C₄×C₈	—	7	2	no	—
83	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
84	C₄⋊C₁₆	—	7	8	no	—
85	C₄⋊C₁₆	C₂×D₇	1	4	yes	—
86	#346	—	7	8	no	—
87	C₄×C₁₆	—	7	4	no	—
88	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	M₃₂	1	4	yes	—
91	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇	#131	1	2	yes	—
96	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₇	—	2	4	no	—
98	C₂×Dic₇	—	2	4	no	—
99	C₂×Dic₇	C₂²×C₈	1	4	yes	—
100	C₂×Dic₇	C₂²×C₈	1	4	yes	—
101	Dic₁₄	—	2	8	no	—
102	C₇×C₈	—	2	2	no	—
103	C₇×C₈	C₄×Q₈	1	2	yes	—
104	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
105	C₇×C₈	C₂×#41	1	1	yes	characteristic
106	C₇⋊C₈	—	2	4	no	—
107	C₂²×Q₈	—	7	1	no	—
108	C₂²×C₈	—	7	2	no	—
109	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
111	C₄×Q₈	—	7	4	no	—
112	C₄×Q₈	—	7	4	no	—
113	C₂×C₄⋊C₄	—	7	2	no	—
114	C₂×C₄⋊C₄	—	7	1	no	—
115	C₂×C₄²	—	7	2	no	—
116	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	7	4	no	—
119	C₂×C₁₆	—	7	2	no	—
120	C₂×C₁₆	—	7	2	no	—
121	C₂×C₁₆	—	14	2	no	—
122	C₂×C₁₆	C₂²×D₇	1	4	yes	—
123	C₄⋊C₈	—	7	8	no	—
124	C₄⋊C₈	—	7	4	no	—
125	C₄×C₈	—	7	4	no	—
126	C₄×C₈	—	7	4	no	—
127	C₄×C₈	C₄×D₇	1	2	yes	—
128	C₄×C₈	C₂²×D₇	1	2	yes	—
129	C₂²×C₇	C₈×Q₈	1	2	yes	—
130	C₂²×C₇	C₂×M₃₂	1	2	yes	—
131	C₂²×C₇	#504	1	2	yes	—
132	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	C₈×Q₈	1	2	yes	—
135	C₄×C₇	C₂×M₃₂	1	4	yes	—
136	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₇	C₂×#131	1	1	yes	characteristic
138	Dic₇	—	2	4	no	—
139	Dic₇	—	2	4	no	—
140	C₂×Q₈	—	7	4	no	—
141	C₂×Q₈	—	14	4	no	—
142	C₂²×C₄	—	7	2	no	—
143	C₂²×C₄	—	7	2	no	—
144	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
145	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	7	4	no	—
149	C₂×C₈	—	14	4	no	—
150	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
152	C₂×C₈	#1159	1	2	yes	—
153	C₂×C₈	Q₈×D₇	1	2	yes	—
154	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
155	C₄⋊C₄	—	7	8	no	—
156	C₄⋊C₄	—	7	4	no	—
157	C₄²	—	7	4	no	—
158	C₄²	—	7	4	no	—
159	C₄²	C₈×D₇	1	2	yes	—
160	C₄²	C₂×C₄×D₇	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₁₆	—	7	4	no	—
163	C₁₆	—	14	2	no	—
164	C₂×C₇	#2183	1	4	yes	complemented
165	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
166	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
167	C₂×C₇	C₂×#504	1	1	yes	characteristic
168	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
169	Q₈	—	14	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	7	4	no	—
172	C₂×C₄	—	7	4	no	—
173	C₂×C₄	—	14	4	no	—
174	C₂×C₄	—	14	4	no	—
175	C₂×C₄	#6372	1	4	yes	—
176	C₂×C₄	#6450	1	2	yes	—
177	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
178	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
180	C₂×C₄	#6470	1	2	yes	—
181	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	14	4	no	—
184	C₈	#6450	1	2	yes	—
185	C₈	C₂×#1159	1	1	yes	characteristic
186	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
187	C₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
188	C₂²	#89602	1	2	yes	—
189	C₂²	C₂×#6372	1	2	yes	—
190	C₂²	#89679	1	2	yes	—
191	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	14	4	no	—
194	C₄	—	14	4	no	—
195	C₄	#89602	1	2	yes	—
196	C₄	C₂×#6372	1	4	yes	—
197	C₄	C₂×#6450	1	1	yes	characteristic
198	C₄	C₂×#6470	1	1	yes	characteristic
199	C₂	#11727035	1	4	yes	direct product factor
200	C₂	C₂×#89602	1	1	yes	characteristic
201	C₂	C₂²×#6372	1	1	yes	characteristic
202	C₂	C₂×#89679	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₇
hypocenter	C₇	C₂×#2183
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
6. #11727035	C₂	direct (N × H)
40. C₂×C₇⋊C₁₆	Q₈	semidirect
73. C₇⋊C₁₆	C₂×Q₈	semidirect
164. C₂×C₇	#2183	semidirect
187. C₇	C₂×#2183	semidirect
199. C₂	#11727035	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×C₈×Dic₁₄	2
C₂×C₇×C₄⋊C₁₆	2
C₂×Dic₇⋊C₁₆	2
C₂×#89297	2
C₂×C₄×C₇⋊C₁₆	2
#11727035	2
C₂×#2183	7

Subgroups — C2×#11727035 #13507280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727035 #13507280