Subgroups — #13269157

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	C₇×#1772	C₂	1	1	yes	maximal, characteristic
2	#11727635	C₂	1	2	yes	maximal
3	C₂×#89258	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#11725477	C₂	1	2	yes	maximal
6	C₂×C₇×#108	C₄	1	1	yes	characteristic
7	C₇×#427	C₄	1	1	yes	characteristic
8	C₇×#405	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6386	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₇×#339	C₄	1	2	yes	—
14	#89258	C₄	1	4	yes	complemented
15	#89256	—	2	2	no	—
16	C₄×C₇⋊C₁₆	—	2	2	no	—
17	C₄×C₇⋊C₁₆	C₄	1	4	yes	complemented
18	#12664	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₇×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₇×#108	C₈	1	4	yes	—
25	C₇×#107	C₈	1	2	yes	—
26	C₇×#106	C₈	1	2	yes	—
27	C₇×#97	—	2	2	no	—
28	C₄×C₇×C₈	—	2	2	no	—
29	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
31	#6386	—	2	2	no	—
32	#6386	C₂×C₄	1	4	yes	—
33	C₂×C₇⋊C₁₆	—	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	—	7	1	no	—
42	#2108	—	7	2	no	—
43	C₂×C₄×C₁₆	—	7	1	no	—
44	#1772	D₇	1	1	yes	characteristic
45	#1379	—	7	2	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₇×M₁₆	—	2	2	no	—
49	C₇×M₁₆	C₂×C₈	1	4	yes	—
50	C₂×C₇×C₈	—	2	2	no	—
51	C₂×C₇×C₈	C₄²	1	2	yes	—
52	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	1	yes	characteristic
60	C₂×C₇×C₈	#41	1	1	yes	characteristic
61	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
62	C₇×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₇	—	2	2	no	—
64	C₄²×C₇	—	2	1	no	—
65	C₄²×C₇	C₄²	1	2	yes	—
66	C₇⋊C₁₆	—	2	2	no	—
67	C₇⋊C₁₆	—	2	4	no	—
68	C₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
69	C₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
70	C₂×M₃₂	—	7	2	no	—
71	C₂²×C₁₆	—	7	1	no	—
72	C₂²×C₁₆	—	7	1	no	—
73	C₂×#108	Dic₇	1	1	yes	characteristic
74	#427	Dic₇	1	1	yes	characteristic
75	#405	C₂×D₇	1	2	yes	—
76	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
77	#364	—	7	4	no	—
78	#346	—	14	2	no	—
79	C₄×C₁₆	—	7	4	no	—
80	C₄×C₁₆	—	14	2	no	—
81	#339	Dic₇	1	2	yes	—
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	—	2	4	no	—
86	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₇	#131	1	1	yes	characteristic
92	C₇×C₈	—	2	4	no	—
93	C₇×C₈	C₄×C₈	1	2	yes	—
94	C₇×C₈	C₄×C₈	1	2	yes	—
95	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₇×C₈	#117	1	1	yes	characteristic
97	C₇×C₈	C₄×D₄	1	1	yes	characteristic
98	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
99	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
100	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
102	#117	C₂×Dic₇	1	2	yes	—
103	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
104	M₃₂	—	7	4	no	—
105	M₃₂	—	14	2	no	—
106	C₂×C₁₆	—	7	2	no	—
107	C₂×C₁₆	—	7	1	no	—
108	C₂×C₁₆	—	7	1	no	—
109	C₂×C₁₆	—	7	2	no	—
110	C₂×C₁₆	—	7	2	no	—
111	C₂×C₁₆	—	14	2	no	—
112	C₂×C₁₆	—	14	1	no	—
113	C₂×C₁₆	—	14	1	no	—
114	#108	C₇⋊C₈	1	4	yes	—
115	#107	C₇⋊C₈	1	2	yes	—
116	#106	C₇⋊C₈	1	2	yes	—
117	#97	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	C₄×D₇	1	2	yes	—
120	C₄×C₈	C₄×D₇	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	—	4	4	no	—
127	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
129	C₄×C₇	C₈×D₄	1	1	yes	characteristic
130	C₄×C₇	C₈×Q₈	1	1	yes	characteristic
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	C₂×C₇⋊C₈	1	4	yes	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
137	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
138	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
139	C₂×C₈	C₄×Dic₇	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
142	C₂×C₈	D₄×D₇	1	1	yes	characteristic
143	C₂×C₈	#1159	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
145	C₂×C₈	#1161	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄×Dic₇	1	2	yes	—
151	C₁₆	—	7	2	no	—
152	C₁₆	—	7	2	no	—
153	C₁₆	—	14	2	no	—
154	C₁₆	—	14	4	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	#6511	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
166	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
167	C₂×C₄	Q₈×Dic₇	1	1	yes	characteristic
168	C₂×C₄	#6511	1	1	yes	characteristic
169	C₈	—	2	4	no	—
170	C₈	C₄×C₇⋊C₈	1	2	yes	—
171	C₈	C₄×C₇⋊C₈	1	2	yes	—
172	C₈	#6450	1	1	yes	characteristic
173	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
174	C₈	#6455	1	1	yes	characteristic
175	C₈	#6456	1	1	yes	characteristic
176	C₇	#12664	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	2	no	—
178	C₂²	#89769	1	1	yes	characteristic
179	C₂²	D₄×C₇⋊C₈	1	1	yes	characteristic
180	C₂²	Q₈×C₇⋊C₈	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	4	4	no	—
183	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
184	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
185	C₄	D₄×C₇⋊C₈	1	1	yes	characteristic
186	C₄	Q₈×C₇⋊C₈	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₄⋊C₄×C₇⋊C₈	1	1	yes	characteristic
190	C₂	#11728677	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1772	C₂
socle	C₂²×C₇	C₈×D₄
prime radical	C₂³×C₇	#131
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1772	D₇
hypocenter	C₇	#12664
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
14. #89258	C₄	semidirect
17. C₄×C₇⋊C₁₆	C₄	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
C₇×#1772	2
#11727635	2
C₂×#89258	2
C₂×C₄×C₇⋊C₁₆	2
#11725477	2
#12664	7