Subgroups — #13246728

Order 1792.

398 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#11729391	C₂	1	2	yes	maximal
3	#11727485	C₂	1	2	yes	maximal
4	#11725511	C₂	1	2	yes	maximal
5	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
6	C₇×C₈×Q₈	—	2	2	no	—
7	C₇×C₈×Q₈	—	2	2	no	—
8	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6410	—	2	2	no	—
13	#89696	C₂²	1	4	yes	—
14	#89325	—	2	2	no	—
15	#89253	—	2	2	no	—
16	#89253	—	2	2	no	—
17	#8617	—	7	1	no	maximal, Sylow
18	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×Q₈	—	2	2	no	—
23	C₄×C₇×Q₈	—	2	2	no	—
24	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
28	C₂×#1136	—	2	4	no	—
29	C₇×C₄⋊C₈	—	2	4	no	—
30	C₇×C₄⋊C₈	—	2	2	no	—
31	C₄×C₇×C₈	—	2	2	no	—
32	C₄×C₇×C₈	—	2	2	no	—
33	C₄×C₇×C₈	D₄	1	2	yes	—
34	C₄×C₇×C₈	D₄	1	2	yes	—
35	#6410	—	4	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₇⋊C₈)	—	2	4	no	—
38	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
39	#1801	—	7	2	no	—
40	#1567	—	7	2	no	—
41	#1292	—	7	2	no	—
42	C₂×C₇×Q₈	—	2	2	no	—
43	C₂×C₇×Q₈	—	2	2	no	—
44	C₂×C₇×Q₈	D₈	1	2	yes	—
45	C₂×C₇×Q₈	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₇×C₈	—	2	2	no	—
52	C₂×C₇×C₈	—	2	2	no	—
53	C₂×C₇×C₈	—	4	1	no	—
54	C₂×C₇×C₈	—	4	2	no	—
55	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	2	2	no	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₄²×C₇	—	2	2	no	—
61	C₄²×C₇	—	2	2	no	—
62	C₄²×C₇	C₂×D₄	1	2	yes	—
63	C₄²×C₇	#41	1	2	yes	—
64	#1136	—	4	4	no	—
65	#1136	—	4	4	no	—
66	C₂×C₇⋊C₈	—	4	4	no	—
67	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
68	C₈×Q₈	—	2	2	no	—
69	C₈×Q₈	—	2	2	no	—
70	#423	—	7	4	no	—
71	C₂×C₄⋊C₈	Dic₇	1	2	yes	—
72	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
73	C₂×#101	—	14	2	no	—
74	C₂×C₄×C₈	Dic₇	1	2	yes	—
75	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
76	#343	—	14	2	no	—
77	#335	—	14	2	no	—
78	#335	—	14	2	no	—
79	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₇×Q₈	—	2	4	no	—
81	C₇×Q₈	—	2	4	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	4	1	no	—
88	C₂×C₄×C₇	—	4	2	no	—
89	C₂×C₄×C₇	#107	1	4	yes	—
90	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₇	#121	1	2	yes	—
93	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	#135	1	2	yes	—
97	C₇×C₈	—	2	1	no	—
98	C₇×C₈	—	2	2	no	—
99	C₇×C₈	—	4	2	no	—
100	C₇×C₈	#101	1	2	yes	complemented
101	C₇⋊C₈	—	8	4	no	—
102	C₂²×Q₈	D₂₈	1	1	yes	characteristic
103	C₂×M₁₆	—	14	4	no	—
104	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₇	1	2	yes	—
106	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
107	C₄×Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	C₂×C₄⋊C₄	Dic₁₄	1	2	yes	—
110	C₂×C₄⋊C₄	D₂₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₇	1	2	yes	—
112	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	14	4	no	—
116	C₄⋊C₈	—	14	4	no	—
117	#101	—	28	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₂²⋊D₇	1	2	yes	—
121	C₄×C₈	C₂²⋊D₇	1	2	yes	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	2	1	no	—
124	C₂²×C₇	#380	1	1	yes	characteristic
125	C₂²×C₇	C₂×#107	1	1	yes	characteristic
126	C₂²×C₇	#427	1	1	yes	characteristic
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	4	2	no	—
131	C₄×C₇	C₂×#101	1	1	yes	characteristic
132	C₄×C₇	#413	1	1	yes	characteristic
133	C₄×C₇	#466	1	2	yes	—
134	C₄×C₇	#467	1	2	yes	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	D₅₆	1	2	yes	—
138	C₂×Q₈	Dic₂₈	1	2	yes	—
139	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
140	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
141	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
143	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
144	M₁₆	—	28	4	no	—
145	M₁₆	—	28	4	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	28	4	no	—
151	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
153	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	#1157	1	2	yes	—
159	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
160	C₂×C₇	—	2	1	no	—
161	C₂×C₇	—	2	1	no	—
162	C₂×C₇	#1801	1	1	yes	characteristic
163	C₂×C₇	#1808	1	1	yes	characteristic
164	C₂×C₇	#1813	1	1	yes	characteristic
165	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
166	Q₈	—	2	4	no	—
167	Q₈	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	#6392	1	4	yes	—
176	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
177	C₂×C₄	#6467	1	2	yes	—
178	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
179	C₂×C₄	#6487	1	2	yes	—
180	C₂×C₄	#6489	1	1	yes	characteristic
181	C₂×C₄	#6493	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	2	no	—
185	C₈	—	4	2	no	—
186	C₈	—	56	4	no	—
187	C₈	#6410	1	2	yes	complemented
188	C₇	#8617	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂×#6392	1	1	yes	characteristic
192	C₂²	#89878	1	1	yes	characteristic
193	C₂²	#89984	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	4	2	no	—
198	C₄	#89887	1	2	yes	—
199	C₄	#89888	1	2	yes	—
200	C₄	C₂×#6410	1	1	yes	characteristic
201	C₄	#90009	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	#11729391	1	1	yes	characteristic
205	C₂	#11729725	1	1	yes	characteristic
206	C₂	#11729768	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89984
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×C₈×Q₈	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	#8617
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
100. C₇×C₈	#101	semidirect
187. C₈	#6410	semidirect
188. C₇	#8617	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
#11729391	2
#11727485	2
#11725511	2
#8617	7