Subgroups — #11740674

Order 896.

552 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₇	C₂	1	2	yes	maximal
3	#90002	C₂	1	2	yes	maximal
4	#89757	C₂	1	2	yes	maximal
5	#89753	C₂	1	4	yes	maximal
6	C₄⋊C₄×Dic₇	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₈	C₂²	1	2	yes	—
10	C₇×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
14	Q₈×Dic₇	—	2	8	no	—
15	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₇	C₂²	1	4	yes	—
23	C₇⋊#95	C₂²	1	2	yes	—
24	C₇⋊#95	C₂²	1	2	yes	—
25	C₇⋊#95	C₂²	1	4	yes	—
26	C₇⋊#95	C₂²	1	4	yes	—
27	#2365	—	7	1	no	maximal, Sylow
28	C₂×C₇×Q₈	—	2	4	no	—
29	C₂×C₇×Q₈	—	2	4	no	—
30	C₂×C₇×Q₈	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₇	C₂³	1	2	yes	—
36	C₂²×Dic₇	C₂³	1	2	yes	—
37	C₂²×Dic₇	C₂³	1	2	yes	—
38	C₂²×Dic₇	C₂³	1	2	yes	—
39	C₇×C₄⋊C₄	C₂×C₄	1	8	yes	—
40	C₇×C₄⋊C₄	C₂×C₄	1	8	yes	—
41	C₄²×C₇	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₇	—	2	2	no	—
43	C₄⋊Dic₇	—	2	2	no	—
44	C₄⋊Dic₇	—	2	8	no	—
45	C₄⋊Dic₇	D₄	1	4	yes	—
46	C₄⋊Dic₇	Q₈	1	4	yes	complemented
47	C₄×Dic₇	—	2	4	no	—
48	C₄×Dic₇	—	2	4	no	—
49	C₄×Dic₇	—	2	8	no	—
50	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
51	C₂×C₄×Q₈	—	7	2	no	—
52	#391	—	7	2	no	—
53	#389	—	7	2	no	—
54	#387	—	7	4	no	—
55	C₄×C₄⋊C₄	—	7	2	no	—
56	C₄×C₄⋊C₄	—	7	1	no	—
57	C₄×C₄⋊C₄	—	7	1	no	—
58	C₂³×C₇	C₂⁴	1	1	yes	characteristic
59	C₇×Q₈	—	2	16	no	—
60	C₂×C₄×C₇	—	2	4	no	—
61	C₂×C₄×C₇	—	2	4	no	—
62	C₂×C₄×C₇	—	2	1	no	—
63	C₂×C₄×C₇	—	2	1	no	—
64	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₇	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₇	#41	1	2	yes	—
71	C₂×C₄×C₇	#41	1	2	yes	—
72	C₂×Dic₇	—	2	2	no	—
73	C₂×Dic₇	—	2	2	no	—
74	C₂×Dic₇	—	2	2	no	—
75	C₂×Dic₇	—	2	2	no	—
76	C₂×Dic₇	—	2	2	no	—
77	C₂×Dic₇	—	2	2	no	—
78	C₂×Dic₇	—	2	2	no	—
79	C₂×Dic₇	—	2	2	no	—
80	C₂×Dic₇	—	2	2	no	—
81	C₂×Dic₇	—	2	2	no	—
82	C₂×Dic₇	C₂×D₄	1	4	yes	—
83	C₂×Dic₇	C₂×Q₈	1	4	yes	—
84	C₂²×Q₈	C₂×D₇	1	2	yes	—
85	C₄⋊Q₈	Dic₇	1	8	yes	complemented
86	C₄×Q₈	—	14	8	no	—
87	C₂×C₄⋊C₄	—	7	1	no	—
88	C₂×C₄⋊C₄	—	7	1	no	—
89	C₂×C₄⋊C₄	—	7	4	no	—
90	C₂×C₄⋊C₄	—	7	1	no	—
91	C₂×C₄⋊C₄	—	7	1	no	—
92	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
93	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
94	C₂×C₄²	—	7	4	no	—
95	C₂×C₄²	—	7	2	no	—
96	C₂×C₄²	—	7	2	no	—
97	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
98	#95	—	7	2	no	—
99	#95	—	7	2	no	—
100	#95	—	7	4	no	—
101	#95	—	7	4	no	—
102	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂²×C₇	C₂×#41	1	1	yes	characteristic
106	C₂²×C₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄×C₇	—	2	8	no	—
110	C₄×C₇	—	2	2	no	—
111	C₄×C₇	C₄×D₄	1	4	yes	—
112	C₄×C₇	C₄×Q₈	1	4	yes	complemented
113	Dic₇	—	2	4	no	—
114	Dic₇	—	2	4	no	—
115	Dic₇	—	4	2	no	—
116	Dic₇	—	4	2	no	—
117	C₂×Q₈	—	2	4	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	C₂×Dic₇	1	8	yes	—
120	C₂²×C₄	—	7	2	no	—
121	C₂²×C₄	—	7	2	no	—
122	C₂²×C₄	—	7	2	no	—
123	C₂²×C₄	—	7	2	no	—
124	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₇	1	4	yes	—
126	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
128	C₄⋊C₄	—	7	4	no	—
129	C₄⋊C₄	—	7	4	no	—
130	C₄⋊C₄	—	14	8	no	—
131	C₄⋊C₄	—	14	2	no	—
132	C₄⋊C₄	—	14	2	no	—
133	C₄⋊C₄	C₂×Dic₇	1	8	yes	—
134	C₄⋊C₄	C₂×Dic₇	1	8	yes	—
135	C₄²	—	14	8	no	—
136	C₄²	—	14	4	no	—
137	C₄²	—	14	4	no	—
138	C₄²	C₂×Dic₇	1	4	yes	—
139	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂×C₇	#519	1	1	yes	characteristic
142	C₂×C₇	#547	1	1	yes	characteristic
143	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
144	C₂×C₇	#556	1	1	yes	characteristic
145	C₂×C₇	#557	1	1	yes	characteristic
146	C₂³	C₂³×D₇	1	1	yes	characteristic
147	Q₈	—	2	16	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	7	4	no	—
153	C₂×C₄	—	7	4	no	—
154	C₂×C₄	—	14	2	no	—
155	C₂×C₄	—	14	2	no	—
156	C₂×C₄	—	14	2	no	—
157	C₂×C₄	—	14	2	no	—
158	C₂×C₄	—	14	2	no	—
159	C₂×C₄	—	14	2	no	—
160	C₂×C₄	—	14	2	no	—
161	C₂×C₄	—	14	2	no	—
162	C₂×C₄	—	14	2	no	—
163	C₂×C₄	—	14	2	no	—
164	C₂×C₄	D₄×D₇	1	2	yes	—
165	C₂×C₄	#1159	1	2	yes	—
166	C₂×C₄	Q₈×D₇	1	2	yes	—
167	C₂×C₄	#1161	1	2	yes	—
168	C₂×C₄	C₂²×Dic₇	1	2	yes	—
169	C₂×C₄	C₂²×Dic₇	1	8	yes	—
170	C₂×C₄	C₂²×Dic₇	1	2	yes	—
171	C₂×C₄	C₂²×Dic₇	1	2	yes	—
172	C₇	#2365	1	1	yes	characteristic, complemented, Sylow
173	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
174	C₂²	C₂×#1159	1	1	yes	characteristic
175	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
176	C₂²	C₂×#1161	1	1	yes	characteristic
177	C₂²	#6551	1	1	yes	characteristic
178	C₂²	#6551	1	1	yes	characteristic
179	C₂²	C₂³×Dic₇	1	1	yes	characteristic
180	C₄	—	2	8	no	—
181	C₄	—	2	2	no	—
182	C₄	—	14	4	no	—
183	C₄	—	14	4	no	—
184	C₄	—	28	2	no	—
185	C₄	—	28	2	no	—
186	C₄	D₄×Dic₇	1	4	yes	—
187	C₄	Q₈×Dic₇	1	4	yes	complemented
188	C₂	#90412	1	1	yes	characteristic
189	C₂	#90417	1	1	yes	characteristic
190	C₂	#90418	1	1	yes	characteristic
191	C₂	#90423	1	1	yes	characteristic
192	C₂	C₂×D₄×Dic₇	1	1	yes	characteristic
193	C₂	C₂×Q₈×Dic₇	1	1	yes	characteristic
194	C₂	#90503	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator	C₂²×C₇	C₂³×C₄
Fitting	C₂×C₇×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#2365
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
10. C₇×C₄⋊Q₈	C₄	semidirect
46. C₄⋊Dic₇	Q₈	semidirect
85. C₄⋊Q₈	Dic₇	semidirect
112. C₄×C₇	C₄×Q₈	semidirect
172. C₇	#2365	semidirect
187. C₄	Q₈×Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×Q₈×Dic₇	2
#90002	2
#89757	2
#89753	2
C₄⋊C₄×Dic₇	2
C₄×C₄⋊Dic₇	2
#2365	7