Subgroups — #11739896

Order 896.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	#90002	C₂	1	2	yes	maximal
3	#89757	C₂	1	2	yes	maximal
4	#89753	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₇	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
7	#89704	C₂	1	2	yes	maximal
8	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₄×C₇×Q₈	C₄	1	8	yes	complemented
11	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
17	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₇	C₂²	1	2	yes	—
19	C₂×C₄×Dic₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
22	C₇⋊#95	C₂²	1	4	yes	—
23	C₇⋊#95	C₂²	1	4	yes	—
24	C₇⋊#95	C₂²	1	4	yes	—
25	C₇⋊#95	C₂²	1	2	yes	—
26	#2295	—	7	1	no	maximal, Sylow
27	C₂×C₇×Q₈	—	2	4	no	—
28	C₂×C₇×Q₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₇	C₂³	1	4	yes	—
35	C₂²×Dic₇	C₂³	1	2	yes	—
36	C₂²×Dic₇	C₂³	1	2	yes	—
37	C₇×C₄⋊C₄	C₂×C₄	1	8	yes	—
38	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₄²×C₇	C₂×C₄	1	8	yes	—
40	C₄²×C₇	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₇	—	2	4	no	—
42	C₄⋊Dic₇	—	2	8	no	—
43	C₄×Dic₇	—	2	4	no	—
44	C₄×Dic₇	—	2	2	no	—
45	C₄×Dic₇	—	2	4	no	—
46	C₄×Dic₇	—	2	4	no	—
47	C₄×Dic₇	Q₈	1	4	yes	complemented
48	C₂×C₄×Q₈	D₇	1	1	yes	characteristic
49	#391	—	7	2	no	—
50	#389	—	7	2	no	—
51	#387	—	7	4	no	—
52	C₄×C₄⋊C₄	—	7	1	no	—
53	C₄×C₄⋊C₄	—	7	2	no	—
54	#376	—	7	2	no	—
55	C₄³	—	7	1	no	—
56	C₂³×C₇	C₂⁴	1	1	yes	characteristic
57	C₇×Q₈	—	2	8	no	—
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	4	no	—
60	C₂×C₄×C₇	—	2	4	no	—
61	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₇	#41	1	4	yes	—
68	C₂×C₄×C₇	#41	1	2	yes	—
69	C₂×Dic₇	—	2	2	no	—
70	C₂×Dic₇	—	2	2	no	—
71	C₂×Dic₇	—	2	2	no	—
72	C₂×Dic₇	—	2	2	no	—
73	C₂×Dic₇	—	2	4	no	—
74	C₂×Dic₇	—	2	4	no	—
75	C₂×Dic₇	—	2	4	no	—
76	C₂×Dic₇	C₂×Q₈	1	4	yes	—
77	C₂×Dic₇	#41	1	4	yes	—
78	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
79	C₄×Q₈	Dic₇	1	8	yes	complemented
80	C₂×C₄⋊C₄	—	7	4	no	—
81	C₂×C₄⋊C₄	—	7	2	no	—
82	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
84	C₂×C₄²	—	7	1	no	—
85	C₂×C₄²	—	7	2	no	—
86	C₂×C₄²	—	7	2	no	—
87	C₂×C₄²	—	7	1	no	—
88	C₂×C₄²	—	7	2	no	—
89	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₇	1	2	yes	—
91	#95	—	7	4	no	—
92	#95	—	7	4	no	—
93	#95	—	7	2	no	—
94	#95	—	7	4	no	—
95	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×C₇	C₂×#41	1	2	yes	—
98	C₂²×C₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₇	C₂×#41	1	2	yes	—
100	C₄×C₇	—	2	4	no	—
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	—	2	4	no	—
103	C₄×C₇	#117	1	4	yes	—
104	C₄×C₇	C₄×Q₈	1	4	yes	—
105	Dic₇	—	2	4	no	—
106	Dic₇	—	2	4	no	—
107	Dic₇	—	4	4	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	C₂×Dic₇	1	4	yes	—
110	C₂²×C₄	—	7	2	no	—
111	C₂²×C₄	—	7	2	no	—
112	C₂²×C₄	—	7	4	no	—
113	C₂²×C₄	C₂²×D₇	1	2	yes	—
114	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₇	1	2	yes	—
116	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
118	C₄⋊C₄	—	14	8	no	—
119	C₄⋊C₄	—	14	4	no	—
120	C₄⋊C₄	C₂×Dic₇	1	8	yes	—
121	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
122	C₄²	—	7	4	no	—
123	C₄²	—	14	4	no	—
124	C₄²	—	14	4	no	—
125	C₄²	—	14	2	no	—
126	C₄²	—	14	4	no	—
127	C₄²	C₂×Dic₇	1	8	yes	—
128	C₄²	C₂×Dic₇	1	4	yes	—
129	C₂×C₇	C₂×#117	1	1	yes	characteristic
130	C₂×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂×C₇	C₄×#41	1	1	yes	characteristic
132	C₂×C₇	#529	1	2	yes	—
133	C₂×C₇	#533	1	2	yes	—
134	C₂³	C₂³×D₇	1	1	yes	characteristic
135	Q₈	—	2	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	7	4	no	—
140	C₂×C₄	—	7	4	no	—
141	C₂×C₄	—	14	2	no	—
142	C₂×C₄	—	14	2	no	—
143	C₂×C₄	—	14	2	no	—
144	C₂×C₄	—	14	2	no	—
145	C₂×C₄	—	14	4	no	—
146	C₂×C₄	—	14	4	no	—
147	C₂×C₄	—	14	4	no	—
148	C₂×C₄	#1157	1	4	yes	—
149	C₂×C₄	Q₈×D₇	1	2	yes	—
150	C₂×C₄	#1161	1	2	yes	—
151	C₂×C₄	C₂²×Dic₇	1	2	yes	—
152	C₂×C₄	C₂²×Dic₇	1	4	yes	—
153	C₂×C₄	C₂²×Dic₇	1	4	yes	—
154	C₂×C₄	C₂²×Dic₇	1	2	yes	—
155	C₂×C₄	C₂²×Dic₇	1	2	yes	—
156	C₇	#2295	1	1	yes	characteristic, complemented, Sylow
157	C₂²	C₂×#1157	1	2	yes	—
158	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
159	C₂²	C₂×#1161	1	1	yes	characteristic
160	C₂²	D₇×#41	1	2	yes	—
161	C₂²	C₂³×Dic₇	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	14	4	no	—
166	C₄	—	14	4	no	—
167	C₄	—	28	4	no	—
168	C₄	#6489	1	4	yes	—
169	C₄	Q₈×Dic₇	1	4	yes	—
170	C₂	#90269	1	2	yes	—
171	C₂	#90271	1	2	yes	—
172	C₂	C₂×#6489	1	1	yes	characteristic
173	C₂	C₂×Q₈×Dic₇	1	1	yes	characteristic
174	C₂	#41×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂³×C₄
Frattini	C₂³	C₂³×D₇
Fitting	C₂×C₄×C₇×Q₈	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₇
hypocenter	C₇	#2295
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. C₄×C₇×Q₈	C₄	semidirect
47. C₄×Dic₇	Q₈	semidirect
79. C₄×Q₈	Dic₇	semidirect
156. C₇	#2295	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×Q₈	2
#90002	2
#89757	2
#89753	2
C₄×C₄⋊Dic₇	2
#89704	2
C₄²×Dic₇	2
#2295	7