Subgroups — C₂×#11728703 #13507284

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×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#11728703	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#89256	C₂	1	2	yes	maximal
6	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
8	C₇×C₈×Q₈	C₄	1	4	yes	—
9	C₇×C₈×Q₈	C₂²	1	4	yes	—
10	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
14	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
18	C₄⋊(C₇⋊C₁₆)	C₂²	1	8	yes	—
19	#89256	C₂²	1	8	yes	—
20	C₄×C₇⋊C₁₆	C₂²	1	4	yes	—
21	C₂×#2183	—	7	1	no	maximal, Sylow
22	C₂²×C₇×Q₈	C₈	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₄×C₇×Q₈	C₈	1	4	yes	—
27	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
28	C₂×C₇×C₄⋊C₄	C₈	1	2	yes	—
29	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
33	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
36	C₄×C₇×C₈	C₂³	1	2	yes	—
37	C₄×C₇×C₈	C₂³	1	4	yes	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×C₇⋊C₁₆	—	2	4	no	—
40	C₂×C₇⋊C₁₆	Q₈	1	4	yes	complemented
41	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
43	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
45	#2183	—	7	8	no	—
46	C₂×C₄⋊C₁₆	—	7	1	no	—
47	C₂×C₄⋊C₁₆	—	7	2	no	—
48	C₂×#346	—	7	2	no	—
49	C₂×C₄×C₁₆	—	7	1	no	—
50	C₂×C₇×Q₈	—	2	4	no	—
51	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	4	no	—
59	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
63	C₂×C₇×C₈	#41	1	2	yes	—
64	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
65	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
66	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
67	C₄²×C₇	C₂×C₈	1	4	yes	—
68	C₄²×C₇	C₂×C₈	1	2	yes	—
69	C₄²×C₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₇	C₂²×C₄	1	2	yes	—
71	C₇⋊C₁₆	—	2	2	no	—
72	C₇⋊C₁₆	—	2	4	no	—
73	C₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
74	C₂×C₄×Q₈	Dic₇	1	1	yes	characteristic
75	C₂²×C₁₆	—	7	1	no	—
76	C₂²×C₁₆	—	7	1	no	—
77	C₂²×C₁₆	—	7	2	no	—
78	C₈×Q₈	Dic₇	1	4	yes	—
79	C₈×Q₈	C₂×D₇	1	4	yes	—
80	C₂×C₄⋊C₈	Dic₇	1	2	yes	—
81	C₂×C₄⋊C₈	Dic₇	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
84	C₄⋊C₁₆	—	7	4	no	—
85	C₄⋊C₁₆	—	7	8	no	—
86	#346	—	7	8	no	—
87	C₄×C₁₆	—	7	4	no	—
88	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
89	C₇×Q₈	—	2	8	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	M₃₂	1	4	yes	—
94	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
96	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
98	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₇	#131	1	2	yes	—
101	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
102	C₇×C₈	—	2	2	no	—
103	C₇×C₈	—	2	4	no	—
104	C₇×C₈	C₄×Q₈	1	2	yes	—
105	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
106	C₇×C₈	C₂×#41	1	1	yes	characteristic
107	C₂²×Q₈	C₇⋊C₈	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₇	1	2	yes	—
110	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
111	C₄×Q₈	C₇⋊C₈	1	4	yes	—
112	C₄×Q₈	C₂×Dic₇	1	4	yes	—
113	C₂×C₄⋊C₄	C₇⋊C₈	1	2	yes	—
114	C₂×C₄⋊C₄	C₇⋊C₈	1	1	yes	characteristic
115	C₂×C₄²	C₂×Dic₇	1	2	yes	—
116	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
117	C₂×C₁₆	—	7	4	no	—
118	C₂×C₁₆	—	7	2	no	—
119	C₂×C₁₆	—	7	2	no	—
120	C₂×C₁₆	—	7	4	no	—
121	C₂×C₁₆	—	14	2	no	—
122	C₂×C₁₆	—	14	4	no	—
123	C₄⋊C₈	C₂×Dic₇	1	8	yes	—
124	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
125	C₄×C₈	C₂×Dic₇	1	2	yes	—
126	C₄×C₈	C₂×Dic₇	1	4	yes	—
127	C₄×C₈	C₂²×D₇	1	2	yes	—
128	C₄×C₈	C₂²×D₇	1	4	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	4	no	—
134	C₄×C₇	—	2	4	no	—
135	C₄×C₇	—	2	2	no	—
136	C₄×C₇	C₈×Q₈	1	2	yes	—
137	C₄×C₇	C₂×M₃₂	1	4	yes	—
138	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₇	C₂×#131	1	1	yes	characteristic
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	C₂×C₇⋊C₈	1	4	yes	—
142	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
143	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
145	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	Q₈×D₇	1	2	yes	—
150	C₂×C₈	#1161	1	2	yes	—
151	C₂×C₈	C₂²×Dic₇	1	2	yes	—
152	C₂×C₈	C₂²×Dic₇	1	4	yes	—
153	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
154	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
155	C₄⋊C₄	C₂×C₇⋊C₈	1	8	yes	—
156	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
157	C₄²	C₂×C₇⋊C₈	1	4	yes	—
158	C₄²	C₂×C₇⋊C₈	1	2	yes	—
159	C₄²	C₂²×Dic₇	1	4	yes	—
160	C₄²	C₂²×Dic₇	1	2	yes	—
161	C₁₆	—	7	4	no	—
162	C₁₆	—	14	2	no	—
163	C₁₆	—	14	4	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	C₂²×C₇⋊C₈	1	1	yes	characteristic
169	Q₈	—	2	8	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	#6386	1	4	yes	—
174	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
175	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
176	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
177	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
178	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
179	C₂×C₄	Q₈×Dic₇	1	2	yes	—
180	C₂×C₄	#6511	1	2	yes	—
181	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	4	no	—
184	C₈	Q₈×Dic₇	1	2	yes	—
185	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
186	C₈	C₂×#1161	1	1	yes	characteristic
187	C₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
188	C₂²	Q₈×C₇⋊C₈	1	2	yes	—
189	C₂²	C₂×#6386	1	2	yes	—
190	C₂²	#89912	1	2	yes	—
191	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	Q₈×C₇⋊C₈	1	2	yes	—
196	C₄	C₂×#6386	1	4	yes	—
197	C₄	C₂×Q₈×Dic₇	1	1	yes	characteristic
198	C₄	C₂×#6511	1	1	yes	characteristic
199	C₂	#11728703	1	4	yes	direct product factor
200	C₂	C₂×Q₈×C₇⋊C₈	1	1	yes	characteristic
201	C₂	C₂²×#6386	1	1	yes	characteristic
202	C₂	C₂×#89912	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×C₈×Q₈	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	C₂×#2183
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #11728703	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	#11728703	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×C₈×Q₈	2
#11728703	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₂×#89256	2
C₂×C₄×C₇⋊C₁₆	2
C₂×#2183	7

Subgroups — C2×#11728703 #13507284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11728703 #13507284