Subgroups — #13243159

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1744	C₂	1	1	yes	maximal, characteristic
2	C₂×#89250	C₂	1	2	yes	maximal
3	C₂×#89249	C₂	1	2	yes	maximal
4	C₄×#6377	C₂	1	1	yes	maximal, characteristic
5	C₄²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
7	C₄³×C₇	C₂²	1	1	yes	characteristic
8	C₂×#6377	C₄	1	4	yes	—
9	C₂×C₄×C₇⋊C₈	C₄	1	4	yes	—
10	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
13	C₇×#336	C₄	1	4	yes	—
14	#89250	C₄	1	8	yes	complemented
15	#89249	C₄	1	8	yes	complemented
16	#7842	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
18	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	C₂×C₄	1	2	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	2	yes	—
23	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
26	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
27	#6377	—	2	8	no	—
28	C₄×C₇⋊C₈	—	2	8	no	—
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	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
33	#1744	D₇	1	1	yes	characteristic
34	C₄×#97	—	7	1	no	—
35	C₄²×C₈	—	7	1	no	—
36	C₂×#322	—	7	2	no	—
37	C₂×#322	—	7	2	no	—
38	C₂²×C₄×C₇	C₄²	1	4	yes	—
39	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₇×C₈	—	2	8	no	—
43	C₂×C₇×C₈	C₄²	1	8	yes	—
44	C₄²×C₇	—	2	4	no	—
45	C₄²×C₇	—	2	4	no	—
46	C₄²×C₇	M₁₆	1	8	yes	—
47	C₄²×C₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₇	C₂²×C₄	1	2	yes	—
49	C₂×C₇⋊C₈	—	2	4	no	—
50	C₂×C₇⋊C₈	—	2	8	no	—
51	C₂×C₇⋊C₈	C₄²	1	4	yes	—
52	C₂×C₇⋊C₈	C₄²	1	8	yes	—
53	C₂×C₇⋊C₈	C₄²	1	4	yes	—
54	C₂×C₇⋊C₈	M₁₆	1	8	yes	—
55	C₂×#97	—	7	4	no	—
56	C₂×C₄×C₈	—	7	1	no	—
57	C₂×C₄×C₈	—	7	1	no	—
58	C₂×C₄×C₈	—	7	2	no	—
59	C₂×C₄×C₈	—	7	4	no	—
60	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
61	C₄³	C₂×D₇	1	1	yes	characteristic
62	#336	Dic₇	1	4	yes	—
63	#322	—	7	8	no	—
64	#322	—	7	8	no	—
65	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇	—	2	4	no	—
67	C₂×C₄×C₇	—	2	4	no	—
68	C₂×C₄×C₇	#97	1	8	yes	—
69	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
71	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
73	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
74	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
75	C₂×C₄×C₇	#131	1	4	yes	—
76	C₇×C₈	—	2	8	no	—
77	C₇⋊C₈	—	2	8	no	—
78	C₇⋊C₈	—	2	4	no	—
79	C₇⋊C₈	#97	1	8	yes	complemented
80	C₂²×C₈	—	7	2	no	—
81	C₂²×C₈	—	7	2	no	—
82	C₂²×C₈	—	7	4	no	—
83	C₂²×C₈	C₂×Dic₇	1	4	yes	—
84	C₂×C₄²	C₄×D₇	1	2	yes	—
85	C₂×C₄²	C₄×D₇	1	2	yes	—
86	C₂×C₄²	C₂×Dic₇	1	2	yes	—
87	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
88	#97	—	14	8	no	—
89	C₄×C₈	—	7	4	no	—
90	C₄×C₈	—	7	4	no	—
91	C₄×C₈	—	7	4	no	—
92	C₄×C₈	—	7	4	no	—
93	C₄×C₈	—	14	8	no	—
94	C₄×C₈	C₄×D₇	1	4	yes	—
95	C₄×C₈	C₄×D₇	1	4	yes	—
96	C₂²×C₇	C₄³	1	1	yes	characteristic
97	C₂²×C₇	C₂×#97	1	2	yes	—
98	C₂²×C₇	C₄×M₁₆	1	2	yes	—
99	C₂²×C₇	#405	1	2	yes	—
100	C₄×C₇	—	2	8	no	—
101	C₄×C₇	C₂×#97	1	2	yes	—
102	C₄×C₇	C₄×M₁₆	1	4	yes	—
103	C₄×C₇	C₄×M₁₆	1	2	yes	—
104	C₄×C₇	#405	1	4	yes	—
105	C₂²×C₄	C₄×Dic₇	1	4	yes	—
106	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
108	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
109	C₂×C₈	—	2	8	no	—
110	C₂×C₈	—	7	4	no	—
111	C₂×C₈	—	7	8	no	—
112	C₂×C₈	—	7	8	no	—
113	C₂×C₈	—	7	4	no	—
114	C₂×C₈	—	14	8	no	—
115	C₂×C₈	—	14	4	no	—
116	C₂×C₈	C₄×Dic₇	1	8	yes	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	#1131	1	8	yes	—
120	C₄²	C₂×C₄×D₇	1	2	yes	—
121	C₄²	C₂×C₄×D₇	1	2	yes	—
122	C₂×C₇	#1451	1	4	yes	—
123	C₂×C₇	C₄×#97	1	1	yes	characteristic
124	C₂×C₇	C₄×#97	1	1	yes	characteristic
125	C₂×C₇	#1726	1	1	yes	characteristic
126	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	#6389	1	8	yes	—
130	C₂×C₄	C₄²×D₇	1	2	yes	—
131	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
132	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
133	C₂×C₄	C₂×#1131	1	4	yes	—
134	C₂×C₄	D₇×M₁₆	1	4	yes	—
135	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
136	C₂×C₄	#6511	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	7	8	no	—
139	C₈	—	14	8	no	—
140	C₈	—	14	4	no	—
141	C₇	#7842	1	1	yes	characteristic, complemented, Sylow
142	C₂²	C₄²×Dic₇	1	1	yes	characteristic
143	C₂²	#89769	1	2	yes	—
144	C₂²	C₂×#6389	1	2	yes	—
145	C₂²	M₁₆×Dic₇	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₄×#1131	1	4	yes	—
148	C₄	D₇×#97	1	2	yes	—
149	C₄	#89479	1	2	yes	—
150	C₄	#89769	1	4	yes	—
151	C₂	#11727188	1	1	yes	characteristic
152	C₂	C₄×#6389	1	1	yes	characteristic
153	C₂	Dic₇×#97	1	1	yes	characteristic
154	C₂	#11728495	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇
commutator	C₂²×C₇	C₄³
Fitting	C₇×#1744	C₂
socle, prime radical	C₂³×C₇	C₂×C₄²
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1744	D₇
hypocenter	C₇	#7842
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
14. #89250	C₄	semidirect
15. #89249	C₄	semidirect
79. C₇⋊C₈	#97	semidirect
141. C₇	#7842	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1744	2
C₂×#89250	2
C₂×#89249	2
C₄×#6377	2
C₄²×C₇⋊C₈	2
#7842	7