Subgroups — #13269158

Order 1792.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic
2	#11729173	C₂	1	2	yes	maximal
3	#11727533	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#11725621	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6393	C₄	1	1	yes	characteristic
9	#89878	C₄	1	1	yes	characteristic
10	#89873	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₇×#364	C₄	1	4	yes	complemented
14	#89318	C₄	1	2	yes	—
15	#89297	—	2	2	no	—
16	C₁₆×Dic₇	—	2	2	no	—
17	C₄×C₇⋊C₁₆	C₄	1	4	yes	complemented
18	#12664	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	#6489	C₂×C₄	1	2	yes	—
23	C₂×#1136	C₂×C₄	1	2	yes	—
24	C₇×M₃₂	—	2	2	no	—
25	C₇×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₇×C₁₆	—	2	2	no	—
27	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
29	#6393	C₈	1	4	yes	—
30	#6392	C₈	1	2	yes	—
31	#6391	C₈	1	2	yes	—
32	#6389	—	2	2	no	—
33	C₈×Dic₇	—	2	2	no	—
34	C₂×C₇⋊C₁₆	—	2	1	no	—
35	C₂×C₇⋊C₁₆	—	2	1	no	—
36	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₇⋊C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
41	C₂×#364	D₇	1	1	yes	characteristic
42	#2108	—	7	2	no	—
43	C₂×C₄×C₁₆	—	7	1	no	—
44	#1772	—	7	1	no	—
45	#1379	—	7	2	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₄²	1	2	yes	—
49	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₇×C₈	#41	1	1	yes	characteristic
57	C₂×C₇×C₈	#41	1	1	yes	characteristic
58	C₄²×C₇	—	2	1	no	—
59	C₄²×C₇	C₄²	1	2	yes	—
60	C₂²⋊Dic₇	—	2	2	no	—
61	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
62	C₄×Dic₇	—	2	2	no	—
63	#1136	—	2	2	no	—
64	#1136	C₂×C₈	1	4	yes	—
65	C₂×C₇⋊C₈	—	2	2	no	—
66	C₇×C₁₆	—	2	4	no	—
67	C₇⋊C₁₆	—	2	2	no	—
68	C₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
69	C₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
70	C₂×M₃₂	C₂×D₇	1	2	yes	—
71	C₂²×C₁₆	—	7	1	no	—
72	C₂²×C₁₆	—	7	1	no	—
73	C₂×#108	—	7	1	no	—
74	#427	—	7	1	no	—
75	#405	—	7	2	no	—
76	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
77	#364	Dic₇	1	4	yes	complemented
78	#346	—	14	2	no	—
79	C₄×C₁₆	—	7	4	no	—
80	C₄×C₁₆	—	14	2	no	—
81	#339	—	7	2	no	—
82	C₂³×C₇	#131	1	1	yes	characteristic
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇	#131	1	1	yes	characteristic
91	C₂×Dic₇	—	2	4	no	—
92	C₇×C₈	C₄×C₈	1	2	yes	—
93	C₇×C₈	C₄×C₈	1	2	yes	—
94	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₇×C₈	#117	1	1	yes	characteristic
96	C₇×C₈	C₄×D₄	1	1	yes	characteristic
97	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
98	C₇⋊C₈	—	2	4	no	—
99	C₂×M₁₆	—	7	2	no	—
100	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
102	#117	—	7	2	no	—
103	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	C₂×Dic₇	1	4	yes	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	7	2	no	—
108	C₂×C₁₆	—	7	1	no	—
109	C₂×C₁₆	—	7	1	no	—
110	C₂×C₁₆	—	7	2	no	—
111	C₂×C₁₆	—	7	2	no	—
112	C₂×C₁₆	—	14	1	no	—
113	C₂×C₁₆	—	14	1	no	—
114	#108	—	7	4	no	—
115	#107	—	7	2	no	—
116	#106	—	7	2	no	—
117	#97	—	14	2	no	—
118	C₄×C₈	—	14	2	no	—
119	C₄×C₈	C₄×D₇	1	2	yes	—
120	C₄×C₈	C₂×Dic₇	1	2	yes	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	#405	1	1	yes	characteristic
123	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₇	C₈×Q₈	1	1	yes	characteristic
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
128	C₄×C₇	C₈×D₄	1	1	yes	characteristic
129	C₄×C₇	C₈×Q₈	1	1	yes	characteristic
130	Dic₇	—	4	4	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
133	M₁₆	—	7	4	no	—
134	M₁₆	—	14	2	no	—
135	C₂×C₈	—	14	2	no	—
136	C₂×C₈	C₈×D₇	1	1	yes	characteristic
137	C₂×C₈	C₈×D₇	1	1	yes	characteristic
138	C₂×C₈	C₈×D₇	1	2	yes	—
139	C₂×C₈	C₄×Dic₇	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
141	C₂×C₈	D₄×D₇	1	1	yes	characteristic
142	C₂×C₈	#1159	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
144	C₂×C₈	#1161	1	1	yes	characteristic
145	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
146	C₄⋊C₄	—	7	4	no	—
147	C₂²⋊C₄	—	14	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	14	2	no	—
150	C₄²	C₄×Dic₇	1	2	yes	—
151	C₁₆	—	2	4	no	—
152	C₁₆	—	7	2	no	—
153	C₁₆	—	7	2	no	—
154	C₁₆	—	14	2	no	—
155	C₂×C₇	—	2	1	no	—
156	C₂×C₇	—	2	1	no	—
157	C₂×C₇	C₈×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₇	#2170	1	2	yes	—
159	C₂³	#6470	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	14	4	no	—
163	C₂×C₄	#6450	1	1	yes	characteristic
164	C₂×C₄	#6456	1	1	yes	characteristic
165	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
166	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
167	C₂×C₄	#6470	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
169	C₈	—	14	4	no	—
170	C₈	C₈×Dic₇	1	2	yes	—
171	C₈	C₈×Dic₇	1	2	yes	—
172	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
173	C₈	#6455	1	1	yes	characteristic
174	C₈	D₄×Dic₇	1	1	yes	characteristic
175	C₈	Q₈×Dic₇	1	1	yes	characteristic
176	C₇	#12664	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	2	no	—
178	C₂²	#89602	1	1	yes	characteristic
179	C₂²	#89606	1	1	yes	characteristic
180	C₂²	#89894	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	28	4	no	—
183	C₄	#89602	1	1	yes	characteristic
184	C₄	#89606	1	1	yes	characteristic
185	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
186	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#11727054	1	2	yes	—
190	C₂	#11728219	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#364	C₂
socle	C₂²×C₇	C₈×D₄
prime radical	C₂³×C₇	#131
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#364	D₇
hypocenter	C₇	#12664
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. C₇×#364	C₄	semidirect
17. C₄×C₇⋊C₁₆	C₄	semidirect
77. #364	Dic₇	semidirect
176. C₇	#12664	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#364	2
#11729173	2
#11727533	2
C₂×C₄×C₇⋊C₁₆	2
#11725621	2
#12664	7