Subgroups — #13243166

Order 1792.

518 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#89250	C₂	1	1	yes	maximal, characteristic
3	C₂×#89249	C₂	1	2	yes	maximal
4	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#11727142	C₂	1	2	yes	maximal
6	C₂×C₇×#97	C₄	1	4	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
12	#89319	C₄	1	8	yes	—
13	#89250	C₄	1	4	yes	complemented
14	#89249	C₄	1	8	yes	complemented
15	C₈×C₇⋊C₈	C₄	1	4	yes	complemented
16	#7844	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
22	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₇×#97	—	2	8	no	—
25	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
27	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
32	#1744	—	7	2	no	—
33	C₄×#97	D₇	1	1	yes	characteristic
34	C₂×#322	—	7	2	no	—
35	C₂×#322	—	7	1	no	—
36	C₂×C₈²	—	7	1	no	—
37	C₂²×C₄×C₇	C₄²	1	4	yes	—
38	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
40	C₂×C₇×C₈	—	2	4	no	—
41	C₂×C₇×C₈	C₄²	1	4	yes	—
42	C₂×C₇×C₈	C₄²	1	4	yes	—
43	C₂×C₇×C₈	M₁₆	1	8	yes	—
44	C₄²×C₇	—	2	4	no	—
45	C₄²×C₇	—	2	8	no	—
46	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇	C₂²×C₄	1	2	yes	—
49	C₂×C₇⋊C₈	—	2	8	no	—
50	C₂×C₇⋊C₈	—	2	8	no	—
51	C₂×C₇⋊C₈	C₄²	1	8	yes	—
52	C₂×C₇⋊C₈	C₄²	1	8	yes	—
53	C₂×#97	Dic₇	1	4	yes	—
54	C₂×C₄×C₈	—	7	2	no	—
55	C₂×C₄×C₈	—	7	2	no	—
56	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
58	C₄³	C₂×D₇	1	1	yes	characteristic
59	#336	—	7	8	no	—
60	#322	—	7	8	no	—
61	#322	—	7	4	no	—
62	C₈²	—	7	4	no	—
63	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₇	—	2	4	no	—
65	C₂×C₄×C₇	—	2	8	no	—
66	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
67	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
71	C₂×C₄×C₇	#131	1	8	yes	—
72	C₇×C₈	—	2	4	no	—
73	C₇×C₈	#97	1	8	yes	—
74	C₇⋊C₈	—	2	8	no	—
75	C₇⋊C₈	—	2	8	no	—
76	C₂²×C₈	—	7	4	no	—
77	C₂²×C₈	—	7	4	no	—
78	C₂²×C₈	C₂×Dic₇	1	2	yes	—
79	C₂²×C₈	C₂×Dic₇	1	2	yes	—
80	C₂×C₄²	C₄×D₇	1	4	yes	—
81	C₂×C₄²	C₂×Dic₇	1	2	yes	—
82	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
83	#97	—	2	8	no	—
84	C₄×C₈	—	7	4	no	—
85	C₄×C₈	—	7	4	no	—
86	C₄×C₈	—	7	4	no	—
87	C₄×C₈	—	7	4	no	—
88	C₄×C₈	C₄×D₇	1	2	yes	—
89	C₄×C₈	C₄×D₇	1	4	yes	—
90	C₄×C₈	C₄×D₇	1	2	yes	—
91	C₂²×C₇	C₄³	1	1	yes	characteristic
92	C₂²×C₇	C₄×M₁₆	1	2	yes	—
93	C₂²×C₇	#405	1	4	yes	—
94	C₄×C₇	—	4	4	no	—
95	C₄×C₇	C₂×#97	1	2	yes	—
96	C₄×C₇	C₄×M₁₆	1	2	yes	—
97	C₄×C₇	#405	1	4	yes	—
98	C₄×C₇	#405	1	4	yes	—
99	C₂²×C₄	C₄×Dic₇	1	4	yes	—
100	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
101	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	7	8	no	—
104	C₂×C₈	—	7	8	no	—
105	C₂×C₈	—	14	8	no	—
106	C₂×C₈	—	14	8	no	—
107	C₂×C₈	#1136	1	8	yes	—
108	C₂×C₈	C₄×Dic₇	1	4	yes	—
109	C₂×C₈	C₄×Dic₇	1	4	yes	—
110	C₄²	—	2	8	no	—
111	C₄²	—	2	4	no	—
112	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
113	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
114	C₄²	C₂×C₄×D₇	1	2	yes	—
115	C₂×C₇	#1453	1	4	yes	—
116	C₂×C₇	C₄×#97	1	1	yes	characteristic
117	C₂×C₇	#1726	1	2	yes	—
118	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
119	C₂×C₄	—	2	8	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄²×D₇	1	2	yes	—
122	C₂×C₄	C₄²×D₇	1	2	yes	—
123	C₂×C₄	#6470	1	8	yes	—
124	C₂×C₄	C₂×#1136	1	4	yes	—
125	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	14	8	no	—
129	C₈	—	14	8	no	—
130	C₈	#6377	1	8	yes	—
131	C₇	#7844	1	1	yes	characteristic, complemented, Sylow
132	C₂²	C₄×#1136	1	2	yes	—
133	C₂²	C₄²×Dic₇	1	1	yes	characteristic
134	C₂²	#89894	1	4	yes	—
135	C₄	—	4	4	no	—
136	C₄	#89480	1	4	yes	—
137	C₄	#89480	1	4	yes	—
138	C₄	C₂×#6377	1	2	yes	—
139	C₄	C₄×#1136	1	2	yes	—
140	C₂	C₄×#6377	1	1	yes	characteristic
141	C₂	#11727558	1	2	yes	—
142	C₂	#11727602	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇
commutator	C₂²×C₇	C₄³
Fitting	C₄×C₇×#97	C₂
socle, prime radical	C₂³×C₇	C₂×C₄²
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×#97	D₇
hypocenter	C₇	#7844
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
13. #89250	C₄	semidirect
14. #89249	C₄	semidirect
15. C₈×C₇⋊C₈	C₄	semidirect
131. C₇	#7844	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#97	2
C₂×#89250	2
C₂×#89249	2
C₂×C₈×C₇⋊C₈	2
#11727142	2
#7844	7