Subgroups — C₂×#11725275 #13265373

Order 1792.

262 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#89253	C₂	1	1	yes	maximal, characteristic
4	#11725275	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×C₄⋊C₈	—	2	1	no	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₇⋊C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
10	C₇×#335	—	2	1	no	—
11	C₇×#335	C₂²	1	2	yes	—
12	C₄×C₇⋊C₁₆	C₄	1	2	yes	—
13	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#89253	—	2	1	no	—
15	#89253	C₂²	1	2	yes	—
16	C₂×#1371	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	—	4	1	no	—
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂²×C₇⋊C₈	—	4	1	no	—
22	C₇×C₄⋊C₈	—	2	2	no	—
23	C₇×C₄⋊C₈	—	4	1	no	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	D₄	1	1	yes	characteristic
26	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₇⋊C₁₆	—	2	2	no	—
29	C₂×C₇⋊C₁₆	C₈	1	4	yes	complemented
30	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
32	C₄⋊(C₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	4	1	no	—
34	C₂×C₄×C₁₆	—	7	1	no	—
35	C₂×#335	—	7	1	no	—
36	C₂×#335	D₇	1	1	yes	characteristic
37	#1371	—	7	4	no	—
38	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
41	C₂×C₇×C₈	—	2	2	no	—
42	C₂×C₇×C₈	—	4	2	no	—
43	C₂×C₇×C₈	—	8	2	no	—
44	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₇×C₈	M₁₆	1	2	yes	—
48	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₇	D₈	1	1	yes	characteristic
52	C₄²×C₇	Q₁₆	1	1	yes	characteristic
53	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₇⋊C₈	—	4	2	no	—
55	C₂×C₇⋊C₈	—	8	2	no	—
56	C₇⋊C₁₆	—	2	2	no	—
57	C₇⋊C₁₆	C₂×C₈	1	4	yes	complemented
58	C₂²×C₁₆	—	7	1	no	—
59	C₂²×C₁₆	—	7	1	no	—
60	C₂×C₄⋊C₈	—	2	1	no	—
61	C₂×C₄⋊C₈	—	14	1	no	—
62	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
63	C₄×C₁₆	—	7	2	no	—
64	C₄×C₁₆	—	7	2	no	—
65	#335	—	2	1	no	—
66	#335	—	7	2	no	—
67	#335	—	14	1	no	—
68	#335	C₂×D₇	1	2	yes	—
69	C₂³×C₇	#107	1	1	yes	characteristic
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
72	C₂×C₄×C₇	#107	1	1	yes	characteristic
73	C₂×C₄×C₇	#107	1	1	yes	characteristic
74	C₂×C₄×C₇	#107	1	1	yes	characteristic
75	C₂×C₄×C₇	#108	1	2	yes	—
76	C₂×C₄×C₇	D₁₆	1	2	yes	—
77	C₂×C₄×C₇	Q₃₂	1	2	yes	—
78	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₇×C₈	—	2	1	no	—
82	C₇×C₈	—	2	1	no	—
83	C₇×C₈	—	8	2	no	—
84	C₇×C₈	C₄⋊C₈	1	2	yes	—
85	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
86	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
87	C₇⋊C₈	—	8	2	no	—
88	C₂²×C₈	—	4	1	no	—
89	C₂²×C₈	—	28	1	no	—
90	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
91	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
92	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
93	C₂×C₁₆	—	7	4	no	—
94	C₂×C₁₆	—	7	2	no	—
95	C₂×C₁₆	—	7	2	no	—
96	C₂×C₁₆	—	14	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	14	2	no	—
100	C₄⋊C₈	—	28	1	no	—
101	C₄×C₈	Dic₁₄	1	1	yes	characteristic
102	C₄×C₈	C₄×D₇	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
104	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
105	C₂²×C₇	#335	1	2	yes	—
106	C₂²×C₇	#366	1	2	yes	—
107	C₂²×C₇	#368	1	2	yes	—
108	C₂²×C₇	C₂×#107	1	1	yes	characteristic
109	C₄×C₇	—	2	1	no	—
110	C₄×C₇	—	2	1	no	—
111	C₄×C₇	#335	1	2	yes	—
112	C₄×C₇	#366	1	2	yes	—
113	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₇	C₂×#108	1	1	yes	characteristic
115	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
116	C₄×C₇	C₂×Q₃₂	1	1	yes	characteristic
117	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
118	C₂²×C₄	#1141	1	1	yes	characteristic
119	C₂²×C₄	#1144	1	1	yes	characteristic
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	8	2	no	—
123	C₂×C₈	—	28	2	no	—
124	C₂×C₈	—	56	2	no	—
125	C₂×C₈	C₈×D₇	1	2	yes	—
126	C₂×C₈	#1131	1	2	yes	—
127	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
131	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
132	C₄²	#1141	1	1	yes	characteristic
133	C₄²	#1144	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
135	C₁₆	—	7	4	no	—
136	C₁₆	—	14	2	no	—
137	C₂×C₇	#1371	1	4	yes	complemented
138	C₂×C₇	C₂×#335	1	1	yes	characteristic
139	C₂×C₇	C₂×#366	1	1	yes	characteristic
140	C₂×C₇	C₂×#368	1	1	yes	characteristic
141	C₂³	#6381	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	#6381	1	1	yes	characteristic
144	C₂×C₄	#6381	1	1	yes	characteristic
145	C₂×C₄	#6381	1	1	yes	characteristic
146	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
147	C₂×C₄	#6396	1	2	yes	—
148	C₂×C₄	#6400	1	2	yes	—
149	C₂×C₄	#6403	1	2	yes	—
150	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₂×#1141	1	1	yes	characteristic
152	C₂×C₄	C₂×#1144	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	8	2	no	—
156	C₈	—	56	2	no	—
157	C₈	Dic₇⋊C₈	1	2	yes	—
158	C₈	C₂×C₈×D₇	1	1	yes	characteristic
159	C₈	C₂×#1131	1	1	yes	characteristic
160	C₇	C₂×#1371	1	1	yes	characteristic, complemented, Sylow
161	C₂²	#89274	1	2	yes	—
162	C₂²	#89284	1	2	yes	—
163	C₂²	#89288	1	2	yes	—
164	C₂²	C₂×#6381	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	#89274	1	2	yes	—
168	C₄	#89284	1	2	yes	—
169	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
170	C₄	C₂×#6396	1	1	yes	characteristic
171	C₄	C₂×#6400	1	1	yes	characteristic
172	C₄	C₂×#6403	1	1	yes	characteristic
173	C₂	#11725275	1	4	yes	direct product factor
174	C₂	C₂×#89274	1	1	yes	characteristic
175	C₂	C₂×#89284	1	1	yes	characteristic
176	C₂	C₂×#89288	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#1141
commutator	C₇×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#335	C₂
socle, prime radical	C₂³×C₇	#107
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#335	D₇
hypocenter	C₇	C₂×#1371
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #11725275	C₂	direct (N × H)
29. C₂×C₇⋊C₁₆	C₈	semidirect
57. C₇⋊C₁₆	C₂×C₈	semidirect
137. C₂×C₇	#1371	semidirect
160. C₇	C₂×#1371	semidirect
173. C₂	#11725275	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×#335	2
C₂×C₄×C₇⋊C₁₆	2
C₂×#89253	2
#11725275	2
C₂×#1371	7

Subgroups — C2×#11725275 #13265373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725275 #13265373