Subgroups — #14217498

Order 1792.

2564 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×#90510	C₂	1	2	yes	maximal
3	#11741234	C₂	1	16	yes	maximal, complemented
4	C₂×#89897	C₂	1	12	yes	maximal, complemented
5	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
6	C₇×#568	C₄	1	16	yes	—
7	C₂×C₇×#131	C₂²	1	8	yes	—
8	C₂²×C₇×M₁₆	C₂²	1	6	yes	—
9	#90510	—	2	8	no	—
10	#90510	C₂²	1	16	yes	—
11	C₂×#6511	—	2	8	no	—
12	C₂²×#1136	C₂²	1	12	yes	—
13	#89897	C₂²	1	16	yes	—
14	#89897	C₂²	1	48	yes	complemented
15	C₂×#6393	C₂²	1	48	yes	complemented
16	#60113	—	7	1	no	maximal, Sylow
17	C₂×C₇×#41	C₂×C₄	1	8	yes	—
18	C₂×C₇×#41	D₄	1	8	yes	—
19	C₂×C₇×#41	Q₈	1	4	yes	—
20	C₂×C₇×#41	C₂³	1	4	yes	—
21	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
22	C₂²×C₇×D₄	Q₈	1	3	yes	—
23	C₂³×C₄×C₇	C₂³	1	3	yes	—
24	C₇×#131	C₂×C₄	1	64	yes	—
25	C₂×C₇×M₁₆	C₂×C₄	1	48	yes	—
26	C₂×C₇×M₁₆	C₂³	1	4	yes	—
27	C₂×C₇×M₁₆	C₂³	1	12	yes	—
28	C₂²×C₇×C₈	C₂³	1	12	yes	—
29	#6511	—	2	64	no	—
30	C₂×#1136	—	2	12	no	—
31	C₂×#1136	—	2	24	no	—
32	C₂×#1136	C₂³	1	48	yes	—
33	C₂×#1136	C₂³	1	8	yes	—
34	C₂²×C₇⋊C₈	—	2	12	no	—
35	#6393	C₂³	1	64	yes	—
36	C₂×#568	—	7	2	no	—
37	C₂×#568	D₇	1	1	yes	characteristic
38	#2916	—	7	16	no	—
39	C₂×#430	—	7	12	no	—
40	C₂⁴×C₇	C₂×Q₈	1	3	yes	—
41	C₇×#41	C₄⋊C₄	1	32	yes	—
42	C₇×#41	C₂²×C₄	1	16	yes	—
43	C₇×#41	C₂×D₄	1	16	yes	—
44	C₂×C₇×Q₈	C₄⋊C₄	1	8	yes	—
45	C₂×C₇×Q₈	C₂×Q₈	1	4	yes	—
46	C₂×C₇×D₄	C₄⋊C₄	1	24	yes	—
47	C₂×C₇×D₄	C₂×Q₈	1	12	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
50	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
52	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₇×M₁₆	C₂²×C₄	1	16	yes	—
55	C₇×M₁₆	C₂²×C₄	1	48	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	48	yes	—
57	C₂×C₇×C₈	C₂⁴	1	8	yes	—
58	#1136	—	2	8	no	—
59	#1136	—	2	48	no	—
60	#1136	C₂⁴	1	16	yes	—
61	C₂×C₇⋊C₈	—	2	8	no	—
62	C₂×C₇⋊C₈	—	2	48	no	—
63	C₂²×#41	C₂×D₇	1	1	yes	characteristic
64	#568	—	7	16	no	—
65	#568	—	14	8	no	—
66	#568	Dic₇	1	16	yes	—
67	C₂×#131	—	14	8	no	—
68	C₂×#131	C₂×D₇	1	8	yes	—
69	C₂²×M₁₆	—	7	12	no	—
70	C₂²×M₁₆	C₂×D₇	1	6	yes	—
71	#430	—	7	16	no	—
72	#430	—	7	48	no	—
73	C₂×#108	—	7	48	no	—
74	C₂³×C₇	—	8	3	no	—
75	C₂³×C₇	C₂×C₄⋊C₄	1	12	yes	—
76	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂³×C₇	C₂²×Q₈	1	6	yes	—
78	C₇×Q₈	C₂×C₄⋊C₄	1	16	yes	—
79	C₇×D₄	C₂×C₄⋊C₄	1	48	yes	—
80	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
81	C₂×C₄×C₇	C₂×C₄⋊C₄	1	24	yes	—
82	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
83	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
86	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂²×Q₈	1	6	yes	—
88	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
89	C₇×C₈	C₂³×C₄	1	16	yes	—
90	C₇⋊C₈	—	2	16	no	—
91	C₂×#41	Dic₁₄	1	4	yes	—
92	C₂×#41	D₂₈	1	8	yes	—
93	C₂×#41	C₂×Dic₇	1	8	yes	—
94	C₂×#41	C₂²×D₇	1	4	yes	—
95	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
96	C₂²×D₄	Dic₁₄	1	3	yes	—
97	C₂³×C₄	C₂²×D₇	1	3	yes	—
98	#131	—	14	64	no	—
99	#131	C₂×Dic₇	1	64	yes	—
100	C₂×M₁₆	—	7	8	no	—
101	C₂×M₁₆	—	7	48	no	—
102	C₂×M₁₆	—	14	12	no	—
103	C₂×M₁₆	—	14	24	no	—
104	C₂×M₁₆	C₂×Dic₇	1	48	yes	—
105	C₂×M₁₆	C₂²×D₇	1	4	yes	—
106	C₂×M₁₆	C₂²×D₇	1	12	yes	—
107	C₂²×C₈	—	14	12	no	—
108	C₂²×C₈	C₂²×D₇	1	12	yes	—
109	#108	—	7	64	no	—
110	C₂²×C₇	—	4	1	no	—
111	C₂²×C₇	—	4	12	no	—
112	C₂²×C₇	—	4	6	no	—
113	C₂²×C₇	C₂²×C₄⋊C₄	1	2	yes	—
114	C₂²×C₇	C₂²×C₄⋊C₄	1	12	yes	—
115	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
116	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
117	C₄×C₇	C₂²×C₄⋊C₄	1	12	yes	—
118	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
120	C₂⁴	C₂×Dic₁₄	1	3	yes	—
121	#41	C₄⋊Dic₇	1	32	yes	—
122	#41	C₂×D₂₈	1	16	yes	—
123	#41	C₂²×Dic₇	1	16	yes	—
124	C₂×Q₈	C₄⋊Dic₇	1	8	yes	—
125	C₂×Q₈	C₂×Dic₁₄	1	4	yes	—
126	C₂×D₄	C₄⋊Dic₇	1	24	yes	—
127	C₂×D₄	C₂×Dic₁₄	1	12	yes	—
128	C₂²×C₄	C₂×Dic₁₄	1	3	yes	—
129	C₂²×C₄	C₂×Dic₁₄	1	6	yes	—
130	C₂²×C₄	C₂×D₂₈	1	12	yes	—
131	C₂²×C₄	C₂²×Dic₇	1	12	yes	—
132	C₂²×C₄	C₂³×D₇	1	6	yes	—
133	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
134	M₁₆	—	7	16	no	—
135	M₁₆	—	14	8	no	—
136	M₁₆	—	14	48	no	—
137	M₁₆	C₂²×Dic₇	1	16	yes	—
138	M₁₆	C₂²×Dic₇	1	48	yes	—
139	C₂×C₈	—	14	8	no	—
140	C₂×C₈	—	14	48	no	—
141	C₂×C₈	C₂²×Dic₇	1	48	yes	—
142	C₂×C₈	C₂³×D₇	1	8	yes	—
143	C₂×C₇	—	2	2	no	—
144	C₂×C₇	—	2	1	no	—
145	C₂×C₇	—	2	12	no	—
146	C₂×C₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
147	C₂³	—	8	3	no	—
148	C₂³	C₂×C₄⋊Dic₇	1	12	yes	—
149	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
150	C₂³	C₂²×Dic₁₄	1	6	yes	—
151	Q₈	C₂×C₄⋊Dic₇	1	16	yes	—
152	D₄	C₂×C₄⋊Dic₇	1	48	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₇	1	12	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₇	1	24	yes	—
155	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×Dic₁₄	1	6	yes	—
157	C₂×C₄	C₂²×D₂₈	1	2	yes	—
158	C₂×C₄	C₂²×D₂₈	1	12	yes	—
159	C₂×C₄	C₂³×Dic₇	1	12	yes	—
160	C₂×C₄	C₂³×Dic₇	1	2	yes	—
161	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
162	C₈	—	14	16	no	—
163	C₈	C₂³×Dic₇	1	16	yes	—
164	C₇	#60113	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	4	1	no	—
166	C₂²	—	4	12	no	—
167	C₂²	—	4	6	no	—
168	C₂²	C₂²×C₄⋊Dic₇	1	2	yes	—
169	C₂²	C₂²×C₄⋊Dic₇	1	12	yes	—
170	C₂²	C₂³×Dic₁₄	1	1	yes	characteristic
171	C₄	C₂²×C₄⋊Dic₇	1	2	yes	—
172	C₄	C₂²×C₄⋊Dic₇	1	12	yes	—
173	C₄	C₂³×D₂₈	1	1	yes	characteristic
174	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	2	12	no	—
178	C₂	C₂³×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×D₂₈
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂×C₇×#568	C₂
socle	C₂×C₇	C₂³×C₄⋊C₄
prime radical	C₂²×C₇×#41	C₂²
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂×#568	D₇
hypocenter	C₇	#60113
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #11741234	C₂	semidirect
4. C₂×#89897	C₂	semidirect
14. #89897	C₂²	semidirect
15. C₂×#6393	C₂²	semidirect
164. C₇	#60113	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#568	2
C₂×#90510	2
#11741234	2
C₂×#89897	2
#60113	7