Subgroups — #11736710

Order 896.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#431	C₂	1	1	yes	maximal, characteristic
3	#89698	C₂	1	1	yes	maximal, characteristic
4	#89600	C₂	1	4	yes	maximal, complemented
5	#89491	C₂	1	2	yes	maximal
6	#89490	C₂	1	2	yes	maximal
7	#89451	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₁₄	C₂	1	2	yes	maximal, complemented
9	C₂²×Dic₁₄	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇	C₄	1	1	yes	characteristic
12	C₂×Dic₇⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×Dic₇	C₂²	1	2	yes	—
14	C₄×Dic₁₄	C₄	1	1	yes	characteristic
15	C₄×Dic₁₄	C₄	1	2	yes	—
16	C₄×Dic₁₄	C₄	1	1	yes	characteristic
17	C₄×Dic₁₄	C₂²	1	2	yes	—
18	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
19	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
20	C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂²	1	2	yes	—
24	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
25	#6389	C₂²	1	4	yes	—
26	Dic₇⋊C₈	C₂²	1	4	yes	—
27	Dic₇⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₇	C₂²	1	4	yes	—
29	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
30	#6377	C₂²	1	2	yes	—
31	#2984	—	7	1	no	maximal, Sylow
32	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₇	C₂×C₄	1	2	yes	—
36	C₂²×Dic₇	C₂×C₄	1	2	yes	—
37	C₂×Dic₁₄	—	2	4	no	—
38	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₁₄	C₂×C₄	1	2	yes	—
40	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂³	1	2	yes	—
42	C₂×C₇×C₈	C₂³	1	2	yes	—
43	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₇	C₂×C₄	1	2	yes	—
48	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
50	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
51	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₇	C₂×C₄	1	2	yes	—
55	C₄×Dic₇	C₂×C₄	1	2	yes	—
56	C₄×Dic₇	C₂³	1	2	yes	—
57	C₄×Dic₇	C₂³	1	2	yes	—
58	C₂×C₇⋊C₈	C₂³	1	4	yes	—
59	C₂×C₄×Q₈	—	7	1	no	—
60	#446	—	7	4	no	—
61	#446	—	7	2	no	—
62	C₈×Q₈	—	7	2	no	—
63	#432	—	7	1	no	—
64	#432	—	7	2	no	—
65	#431	—	7	2	no	—
66	#431	D₇	1	1	yes	characteristic
67	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₇	—	2	4	no	—
78	C₂×Dic₇	—	2	4	no	—
79	C₂×Dic₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₇	C₂²×C₄	1	2	yes	—
83	Dic₁₄	—	2	2	no	—
84	Dic₁₄	—	2	4	no	—
85	Dic₁₄	M₁₆	1	4	yes	—
86	C₇×C₈	—	2	2	no	—
87	C₇×C₈	—	2	2	no	—
88	C₇⋊C₈	—	2	4	no	—
89	C₂²×Q₈	—	7	1	no	—
90	C₄×Q₈	—	7	1	no	—
91	C₄×Q₈	—	7	2	no	—
92	C₄×Q₈	—	7	1	no	—
93	C₄×Q₈	—	7	1	no	—
94	C₄×Q₈	—	7	2	no	—
95	C₄×Q₈	—	7	1	no	—
96	C₂×C₄⋊C₄	—	7	2	no	—
97	C₂×C₄⋊C₄	—	7	1	no	—
98	C₂×C₄²	—	7	2	no	—
99	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
100	C₄⋊C₈	—	7	4	no	—
101	C₄⋊C₈	—	7	4	no	—
102	C₄⋊C₈	—	7	2	no	—
103	C₄⋊C₈	C₂×D₇	1	2	yes	—
104	C₂²⋊C₈	—	7	2	no	—
105	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
106	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
107	#97	—	7	4	no	—
108	#97	—	7	2	no	—
109	C₄×C₈	—	7	4	no	—
110	C₄×C₈	C₂×D₇	1	2	yes	—
111	C₂²×C₇	—	2	1	no	—
112	C₂²×C₇	—	2	1	no	—
113	C₂²×C₇	#131	1	2	yes	—
114	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₇	—	2	1	no	—
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	—	2	1	no	—
118	C₄×C₇	—	2	1	no	—
119	C₄×C₇	C₂×M₁₆	1	2	yes	—
120	C₄×C₇	2₋¹⁺⁴	1	2	yes	—
121	Dic₇	—	2	2	no	—
122	Dic₇	—	2	2	no	—
123	Dic₇	—	2	2	no	—
124	Dic₇	C₂×M₁₆	1	4	yes	—
125	C₂×Q₈	—	7	1	no	—
126	C₂×Q₈	—	7	2	no	—
127	C₂×Q₈	—	7	1	no	—
128	C₂×Q₈	—	14	4	no	—
129	C₂²×C₄	—	7	2	no	—
130	C₂²×C₄	—	7	2	no	—
131	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
132	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
134	C₂×C₈	—	7	4	no	—
135	C₂×C₈	C₂²×D₇	1	2	yes	—
136	C₂×C₈	C₂²×D₇	1	2	yes	—
137	C₄⋊C₄	—	7	2	no	—
138	C₄⋊C₄	—	7	2	no	—
139	C₄⋊C₄	—	7	2	no	—
140	C₄⋊C₄	—	7	2	no	—
141	C₄⋊C₄	—	7	2	no	—
142	C₄⋊C₄	—	7	1	no	—
143	C₄⋊C₄	—	7	1	no	—
144	C₄²	—	7	2	no	—
145	C₄²	—	7	2	no	—
146	C₄²	—	7	2	no	—
147	C₄²	—	7	2	no	—
148	C₄²	C₄×D₇	1	1	yes	characteristic
149	C₄²	C₄×D₇	1	1	yes	characteristic
150	C₄²	C₂²×D₇	1	1	yes	characteristic
151	C₄²	C₂²×D₇	1	1	yes	characteristic
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	#519	1	1	yes	characteristic
154	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
155	C₂×C₇	C₂×#131	1	1	yes	characteristic
156	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
157	Q₈	—	7	4	no	—
158	Q₈	—	14	4	no	—
159	Q₈	—	14	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	7	2	no	—
164	C₂×C₄	—	7	2	no	—
165	C₂×C₄	—	7	2	no	—
166	C₂×C₄	—	7	2	no	—
167	C₂×C₄	—	14	4	no	—
168	C₂×C₄	—	14	4	no	—
169	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	14	4	no	—
178	C₇	#2984	1	1	yes	characteristic, complemented, Sylow
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#6464	1	2	yes	—
182	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	7	4	no	—
188	C₄	—	14	2	no	—
189	C₄	—	14	2	no	—
190	C₄	—	14	2	no	—
191	C₄	D₇×M₁₆	1	2	yes	—
192	C₄	#6554	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#90207	1	1	yes	characteristic
195	C₂	C₂×#6464	1	1	yes	characteristic
196	C₂	C₂×D₇×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#431	C₂
prime radical	C₂³×C₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#431	D₇
hypocenter	C₇	#2984
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
4. #89600	C₂	semidirect
7. #89451	C₂	semidirect
8. C₈×Dic₁₄	C₂	semidirect
178. C₇	#2984	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×Dic₁₄	2
C₇×#431	2
#89698	2
#89600	2
#89491	2
#89490	2
#89451	2
C₈×Dic₁₄	2
#2984	7