Subgroups — #11737409

Order 896.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#89796	C₂	1	1	yes	maximal, characteristic, complemented
4	#89607	C₂	1	1	yes	maximal, characteristic
5	#89605	C₂	1	2	yes	maximal
6	#89602	C₂	1	1	yes	maximal, characteristic, complemented
7	#89600	C₂	1	2	yes	maximal, complemented
8	#89477	C₂	1	2	yes	maximal
9	#89473	C₂	1	2	yes	maximal, complemented
10	#89457	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₇	C₄	1	1	yes	characteristic
13	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
14	Q₈×Dic₇	C₄	1	1	yes	characteristic
15	D₇×C₄⋊C₄	C₄	1	1	yes	characteristic
16	D₇×C₄⋊C₄	C₄	1	2	yes	—
17	#6450	C₄	1	2	yes	—
18	#6450	C₄	1	1	yes	characteristic
19	C₄²×D₇	C₂²	1	2	yes	—
20	C₄²×D₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₄	C₂²	1	2	yes	—
22	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₇×#97	C₂²	1	2	yes	—
26	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
28	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
30	#6389	C₂²	1	2	yes	—
31	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
32	Dic₇⋊C₈	C₂²	1	2	yes	—
33	Dic₇⋊C₈	C₂²	1	2	yes	—
34	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₇	C₂²	1	2	yes	—
36	C₈×Dic₇	C₂²	1	2	yes	—
37	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
38	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
39	#6377	C₂²	1	2	yes	—
40	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
41	#2984	—	7	1	no	maximal, Sylow
42	C₂×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
43	Q₈×D₇	—	2	4	no	—
44	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₄	C₂×C₄	1	2	yes	—
50	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂³	1	2	yes	—
54	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₇	C₂³	1	1	yes	characteristic
57	C₄²×C₇	C₂³	1	2	yes	—
58	C₄⋊Dic₇	C₂×C₄	1	2	yes	—
59	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
60	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₇	C₂×C₄	1	2	yes	—
65	C₄×Dic₇	C₂×C₄	1	2	yes	—
66	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₇	C₂³	1	2	yes	—
68	C₄×Dic₇	C₂³	1	1	yes	characteristic
69	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇⋊C₈	C₂³	1	2	yes	—
72	C₂×C₄×Q₈	—	7	1	no	—
73	#446	—	7	1	no	—
74	#446	—	7	2	no	—
75	#446	—	7	2	no	—
76	#446	D₇	1	1	yes	characteristic, complemented
77	C₈×Q₈	—	7	1	no	—
78	C₈×Q₈	—	7	1	no	—
79	#432	—	7	1	no	—
80	#432	—	7	2	no	—
81	#431	—	7	2	no	—
82	#431	—	7	1	no	—
83	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
84	C₇×Q₈	—	2	2	no	—
85	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₇	C₂²×C₄	1	2	yes	—
95	C₄×D₇	—	2	4	no	—
96	C₄×D₇	—	2	2	no	—
97	C₄×D₇	—	2	2	no	—
98	C₄×D₇	—	2	1	no	—
99	C₄×D₇	—	2	1	no	—
100	C₄×D₇	—	2	4	no	—
101	Dic₁₄	—	2	4	no	—
102	Dic₁₄	M₁₆	1	4	yes	—
103	C₇×C₈	—	2	1	no	—
104	C₇×C₈	—	2	1	no	—
105	C₇×C₈	—	2	2	no	—
106	C₇⋊C₈	—	2	1	no	—
107	C₇⋊C₈	—	2	2	no	—
108	C₇⋊C₈	—	2	1	no	—
109	C₂²×Q₈	—	7	1	no	—
110	C₄×Q₈	—	7	1	no	—
111	C₄×Q₈	—	7	1	no	—
112	C₄×Q₈	—	7	2	no	—
113	C₄×Q₈	—	7	1	no	—
114	C₄×Q₈	—	7	2	no	—
115	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	7	2	no	—
117	C₂×C₄⋊C₄	—	7	1	no	—
118	C₂×C₄²	—	7	2	no	—
119	C₂×C₄²	—	7	1	no	—
120	C₄⋊C₈	—	7	1	no	—
121	C₄⋊C₈	—	7	1	no	—
122	C₄⋊C₈	—	7	2	no	—
123	C₄⋊C₈	—	7	2	no	—
124	C₄⋊C₈	—	7	2	no	—
125	C₄⋊C₈	—	7	1	no	—
126	C₄⋊C₈	C₂×D₇	1	2	yes	—
127	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
128	C₂²⋊C₈	—	7	1	no	—
129	C₂²⋊C₈	—	7	2	no	—
130	C₂²⋊C₈	—	7	1	no	—
131	#97	—	7	2	no	—
132	#97	—	7	2	no	—
133	#97	C₂×D₇	1	2	yes	—
134	C₄×C₈	—	7	2	no	—
135	C₄×C₈	—	7	2	no	—
136	C₄×C₈	—	7	1	no	—
137	C₄×C₈	C₂×D₇	1	1	yes	characteristic
138	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₇	—	2	1	no	—
140	C₂×D₇	—	2	1	no	—
141	C₂×D₇	#131	1	2	yes	—
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	C₂×M₁₆	1	2	yes	—
146	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₇	—	2	1	no	—
149	Dic₇	—	2	1	no	—
150	Dic₇	—	2	2	no	—
151	Dic₇	—	2	1	no	—
152	Dic₇	C₂×M₁₆	1	4	yes	—
153	C₂×Q₈	—	7	1	no	—
154	C₂×Q₈	—	7	2	no	—
155	C₂×Q₈	—	14	4	no	—
156	C₂×Q₈	C₄×D₇	1	1	yes	characteristic
157	C₂²×C₄	—	7	2	no	—
158	C₂²×C₄	—	7	1	no	—
159	C₂²×C₄	—	7	2	no	—
160	C₂²×C₄	—	7	1	no	—
161	C₂²×C₄	—	7	1	no	—
162	C₂×C₈	—	7	1	no	—
163	C₂×C₈	—	7	2	no	—
164	C₂×C₈	—	7	1	no	—
165	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₇	1	2	yes	—
167	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
168	C₄⋊C₄	—	7	2	no	—
169	C₄⋊C₄	—	7	2	no	—
170	C₄⋊C₄	—	7	2	no	—
171	C₄⋊C₄	—	7	2	no	—
172	C₄⋊C₄	—	7	1	no	—
173	C₄⋊C₄	C₄×D₇	1	2	yes	—
174	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
175	C₄²	—	7	2	no	—
176	C₄²	—	7	2	no	—
177	C₄²	—	7	1	no	—
178	C₄²	—	7	1	no	—
179	C₄²	—	7	2	no	—
180	C₄²	—	7	1	no	—
181	C₄²	C₂²×D₇	1	1	yes	characteristic
182	C₄²	C₂²×D₇	1	2	yes	—
183	C₂×C₇	#519	1	1	yes	characteristic
184	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
185	C₂×C₇	C₂×#131	1	1	yes	characteristic
186	D₇	—	2	2	no	—
187	C₂³	—	7	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	7	4	no	—
190	Q₈	—	14	4	no	—
191	C₂×C₄	—	7	2	no	—
192	C₂×C₄	—	7	1	no	—
193	C₂×C₄	—	7	2	no	—
194	C₂×C₄	—	7	1	no	—
195	C₂×C₄	—	7	1	no	—
196	C₂×C₄	—	14	4	no	—
197	C₂×C₄	—	14	2	no	—
198	C₂×C₄	—	14	4	no	—
199	C₂×C₄	—	14	2	no	—
200	C₂×C₄	—	14	1	no	—
201	C₂×C₄	—	14	1	no	—
202	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
206	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	14	1	no	—
211	C₈	—	14	2	no	—
212	C₈	—	14	1	no	—
213	C₇	#2984	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	7	2	no	—
215	C₂²	—	14	1	no	—
216	C₂²	—	14	1	no	—
217	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	7	4	no	—
222	C₄	—	14	1	no	—
223	C₄	—	14	2	no	—
224	C₄	—	14	1	no	—
225	C₄	—	14	1	no	—
226	C₄	D₇×M₁₆	1	2	yes	—
227	C₄	#6551	1	1	yes	characteristic
228	C₄	#6554	1	1	yes	characteristic
229	C₂	—	14	2	no	—
230	C₂	#90263	1	1	yes	characteristic
231	C₂	C₂×D₇×M₁₆	1	1	yes	characteristic
232	C₂	D₇×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#446	C₂
prime radical	C₂²×D₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#446	D₇
hypocenter	C₇	#2984
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₇×#446	C₂	semidirect
3. #89796	C₂	semidirect
6. #89602	C₂	semidirect
7. #89600	C₂	semidirect
9. #89473	C₂	semidirect
11. C₈×Dic₁₄	C₂	semidirect
76. #446	D₇	semidirect
213. C₇	#2984	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₇	2
C₇×#446	2
#89796	2
#89607	2
#89605	2
#89602	2
#89600	2
#89477	2
#89473	2
#89457	2
C₈×Dic₁₄	2
#2984	7