Subgroups — #13268676

Order 1792.

606 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	C₂³×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#89878	C₂	1	2	yes	maximal
3	#11725612	C₂	1	4	yes	maximal, complemented
4	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
5	C₂×#6489	—	2	2	no	—
6	C₂²×C₇×C₁₆	—	2	4	no	—
7	C₂²×C₇×C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	#89878	—	2	4	no	—
10	#89878	C₄	1	8	yes	complemented
11	C₂×#6392	C₂²	1	4	yes	—
12	C₂×#6391	—	2	2	no	—
13	#12556	—	7	1	no	maximal, Sylow
14	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	D₄	1	4	yes	—
19	C₂²×C₇×C₈	Q₈	1	2	yes	—
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₇	—	4	2	no	—
22	#6489	—	2	8	no	—
23	#6489	—	4	4	no	—
24	C₂×C₄⋊Dic₇	—	2	4	no	—
25	C₂×C₄×Dic₇	—	4	2	no	—
26	C₂×C₇×C₁₆	—	2	8	no	—
27	C₂×C₇×C₁₆	—	2	4	no	—
28	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
29	#6392	—	2	4	no	—
30	#6392	C₂×C₄	1	8	yes	—
31	#6391	—	2	8	no	—
32	C₂³×C₁₆	D₇	1	1	yes	characteristic
33	C₂×#427	—	7	2	no	—
34	#1380	—	7	4	no	—
35	C₂⁴×C₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₇	—	4	4	no	—
43	C₂×C₇×C₈	C₄²	1	4	yes	—
44	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
45	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
46	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₇×C₈	C₄⋊C₄	1	8	yes	—
48	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
51	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂²⋊Dic₇	—	4	8	no	—
53	C₄⋊Dic₇	—	2	8	no	—
54	C₄⋊Dic₇	—	4	4	no	—
55	C₄×Dic₇	—	4	8	no	—
56	C₇×C₁₆	—	2	4	no	—
57	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
58	C₂×#117	—	14	2	no	—
59	C₂²×C₁₆	—	2	4	no	—
60	C₂²×C₁₆	Dic₇	1	4	yes	complemented
61	C₂²×C₁₆	C₂×D₇	1	2	yes	—
62	#427	—	7	8	no	—
63	#427	—	14	4	no	—
64	C₂×#107	—	7	4	no	—
65	C₂×#106	—	14	2	no	—
66	C₂³×C₇	—	2	4	no	—
67	C₂³×C₇	#102	1	4	yes	—
68	C₂³×C₇	#107	1	2	yes	—
69	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₇	#95	1	4	yes	—
71	C₂×C₄×C₇	#102	1	4	yes	—
72	C₂×C₄×C₇	#103	1	8	yes	—
73	C₂×C₄×C₇	#106	1	4	yes	—
74	C₂×C₄×C₇	#107	1	2	yes	—
75	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₇	—	4	8	no	—
80	C₂×Dic₇	—	8	8	no	—
81	C₇×C₈	#95	1	4	yes	—
82	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
83	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄	D₂₈	1	1	yes	characteristic
86	C₂²×C₈	Dic₁₄	1	2	yes	—
87	C₂²×C₈	C₄×D₇	1	4	yes	—
88	C₂²×C₈	D₂₈	1	1	yes	characteristic
89	C₂²×C₈	C₂×Dic₇	1	2	yes	—
90	C₂²×C₈	C₂²⋊D₇	1	4	yes	—
91	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
92	#117	—	14	8	no	—
93	#117	—	28	4	no	—
94	C₂×C₄⋊C₄	—	14	4	no	—
95	C₂×C₂²⋊C₄	—	28	2	no	—
96	C₂×C₄²	—	28	2	no	—
97	C₂×C₁₆	—	2	8	no	—
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
100	#107	—	7	8	no	—
101	#107	—	14	4	no	—
102	#106	—	14	8	no	—
103	C₂²×C₇	—	2	8	no	—
104	C₂²×C₇	—	2	4	no	—
105	C₂²×C₇	#340	1	4	yes	—
106	C₂²×C₇	C₂×#102	1	2	yes	—
107	C₂²×C₇	C₂×#107	1	1	yes	characteristic
108	C₂²×C₇	#508	1	4	yes	—
109	C₄×C₇	#340	1	4	yes	—
110	C₄×C₇	C₂×#95	1	1	yes	characteristic
111	C₄×C₇	C₂×#103	1	2	yes	—
112	C₄×C₇	C₂×#106	1	1	yes	characteristic
113	Dic₇	—	8	8	no	—
114	C₂⁴	D₅₆	1	1	yes	characteristic
115	C₂²×C₄	—	28	4	no	—
116	C₂²×C₄	#1132	1	4	yes	—
117	C₂²×C₄	D₅₆	1	1	yes	characteristic
118	C₂²×C₄	Dic₂₈	1	2	yes	—
119	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
120	C₂²×C₄	(C₂×D₇)⋊C₄	1	4	yes	—
121	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
122	C₂×C₈	C₄×Dic₇	1	4	yes	—
123	C₂×C₈	Dic₇⋊C₄	1	8	yes	—
124	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
125	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
126	C₂×C₈	C₂²⋊Dic₇	1	4	yes	—
127	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
129	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
130	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
131	C₄⋊C₄	—	14	8	no	—
132	C₄⋊C₄	—	28	4	no	—
133	C₂²⋊C₄	—	28	8	no	—
134	C₄²	—	28	8	no	—
135	C₁₆	—	2	4	no	—
136	C₂×C₇	—	2	4	no	—
137	C₂×C₇	C₂×#340	1	1	yes	characteristic
138	C₂×C₇	#2138	1	4	yes	—
139	C₂×C₇	#2158	1	2	yes	—
140	C₂³	—	2	4	no	—
141	C₂³	#6392	1	2	yes	—
142	C₂³	#6395	1	4	yes	—
143	C₂³	C₂×D₅₆	1	1	yes	characteristic
144	C₂×C₄	—	28	8	no	—
145	C₂×C₄	—	56	8	no	—
146	C₂×C₄	#6390	1	8	yes	—
147	C₂×C₄	#6391	1	4	yes	—
148	C₂×C₄	#6392	1	2	yes	—
149	C₂×C₄	#6395	1	4	yes	—
150	C₂×C₄	C₇⋊#95	1	4	yes	—
151	C₂×C₄	C₂×#1132	1	2	yes	—
152	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
154	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
155	C₈	C₇⋊#95	1	4	yes	—
156	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
157	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
158	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
159	C₇	#12556	1	1	yes	characteristic, complemented, Sylow
160	C₂²	—	2	4	no	—
161	C₂²	—	2	8	no	—
162	C₂²	#89346	1	4	yes	—
163	C₂²	#89676	1	4	yes	—
164	C₂²	C₂×#6392	1	1	yes	characteristic
165	C₂²	C₂×#6395	1	2	yes	—
166	C₄	—	56	8	no	—
167	C₄	#89346	1	4	yes	—
168	C₄	C₂×#6390	1	2	yes	—
169	C₄	C₂×#6391	1	1	yes	characteristic
170	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	#11729158	1	2	yes	—
173	C₂	#11729166	1	4	yes	—
174	C₂	C₂×#89346	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₅₆
commutator	C₇×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂³×C₇×C₁₆	C₂
socle	C₂³×C₇	C₂×D₈
prime radical	C₂⁴×C₇	D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂³×C₁₆	D₇
hypocenter	C₇	#12556
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #11725612	C₂	semidirect
7. C₂²×C₇×C₁₆	C₄	semidirect
10. #89878	C₄	semidirect
60. C₂²×C₁₆	Dic₇	semidirect
159. C₇	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₇×C₁₆	2
C₂×#89878	2
#11725612	2
#12556	7