Subgroups — #13269162

Order 1792.

326 conjugacy classes of subgroups.

The subgroup lattice has 188 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×#89256	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	#89256	C₄	1	4	yes	complemented
17	C₄×C₇⋊C₁₆	—	2	2	no	—
18	#12665	—	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	2	yes	—
39	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
40	C₂×#364	—	7	1	no	—
41	#2108	—	7	2	no	—
42	C₂×#346	—	7	1	no	—
43	#1772	D₇	1	1	yes	characteristic
44	#1379	—	7	2	no	—
45	C₂²×C₄×C₇	—	2	1	no	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₇×M₁₆	—	2	2	no	—
48	C₇×M₁₆	C₂×C₈	1	4	yes	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	C₄²	1	2	yes	—
51	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₇×C₈	#41	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
61	C₇×C₂²⋊C₄	—	2	2	no	—
62	C₄²×C₇	—	2	2	no	—
63	C₄²×C₇	—	2	1	no	—
64	C₄²×C₇	C₄²	1	2	yes	—
65	C₇⋊C₁₆	—	2	2	no	—
66	C₇⋊C₁₆	—	2	4	no	—
67	C₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
68	C₂×M₃₂	—	7	2	no	—
69	C₂²×C₁₆	—	7	1	no	—
70	C₂²×C₁₆	—	7	1	no	—
71	C₂×#108	Dic₇	1	1	yes	characteristic
72	#427	Dic₇	1	1	yes	characteristic
73	#405	C₂×D₇	1	2	yes	—
74	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
75	#364	—	7	4	no	—
76	#346	—	7	4	no	—
77	#346	—	14	2	no	—
78	C₄×C₁₆	—	14	2	no	—
79	#339	Dic₇	1	2	yes	—
80	C₂³×C₇	#131	1	1	yes	characteristic
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	4	no	—
83	C₂×C₄×C₇	—	4	2	no	—
84	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇	#131	1	1	yes	characteristic
90	C₇×C₈	—	2	4	no	—
91	C₇×C₈	C₄×C₈	1	2	yes	—
92	C₇×C₈	C₄×C₈	1	2	yes	—
93	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₇×C₈	#117	1	1	yes	characteristic
95	C₇×C₈	C₄×D₄	1	1	yes	characteristic
96	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
97	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
98	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
100	#117	C₂×Dic₇	1	2	yes	—
101	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
102	M₃₂	—	7	4	no	—
103	M₃₂	—	14	2	no	—
104	C₂×C₁₆	—	7	2	no	—
105	C₂×C₁₆	—	7	2	no	—
106	C₂×C₁₆	—	7	2	no	—
107	C₂×C₁₆	—	7	2	no	—
108	C₂×C₁₆	—	14	2	no	—
109	C₂×C₁₆	—	14	1	no	—
110	C₂×C₁₆	—	14	1	no	—
111	#108	C₇⋊C₈	1	4	yes	—
112	#107	C₇⋊C₈	1	2	yes	—
113	#106	C₇⋊C₈	1	2	yes	—
114	#97	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	C₄×D₇	1	2	yes	—
117	C₄×C₈	C₄×D₇	1	2	yes	—
118	C₂²×C₇	—	2	2	no	—
119	C₂²×C₇	#405	1	1	yes	characteristic
120	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₇	C₈×Q₈	1	1	yes	characteristic
122	C₄×C₇	—	4	2	no	—
123	C₄×C₇	—	4	4	no	—
124	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
126	C₄×C₇	C₈×D₄	1	1	yes	characteristic
127	C₄×C₇	C₈×Q₈	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
130	M₁₆	—	2	2	no	—
131	M₁₆	C₂×C₇⋊C₈	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
134	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
135	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
136	C₂×C₈	C₄×Dic₇	1	2	yes	—
137	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
139	C₂×C₈	D₄×D₇	1	1	yes	characteristic
140	C₂×C₈	#1159	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
142	C₂×C₈	#1161	1	1	yes	characteristic
143	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	1	no	—
147	C₄²	C₄×Dic₇	1	2	yes	—
148	C₁₆	—	7	4	no	—
149	C₁₆	—	14	2	no	—
150	C₁₆	—	14	4	no	—
151	C₂×C₇	—	2	1	no	—
152	C₂×C₇	—	2	1	no	—
153	C₂×C₇	C₈×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₇	#2171	1	2	yes	—
155	C₂³	#6511	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
163	C₂×C₄	Q₈×Dic₇	1	1	yes	characteristic
164	C₂×C₄	#6511	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₈	C₄×C₇⋊C₈	1	2	yes	—
167	C₈	C₄×C₇⋊C₈	1	2	yes	—
168	C₈	#6450	1	1	yes	characteristic
169	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
170	C₈	#6455	1	1	yes	characteristic
171	C₈	#6456	1	1	yes	characteristic
172	C₇	#12665	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	#89769	1	1	yes	characteristic
175	C₂²	D₄×C₇⋊C₈	1	1	yes	characteristic
176	C₂²	Q₈×C₇⋊C₈	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	4	4	no	—
179	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
180	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
181	C₄	D₄×C₇⋊C₈	1	1	yes	characteristic
182	C₄	Q₈×C₇⋊C₈	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₄⋊C₄×C₇⋊C₈	1	1	yes	characteristic
186	C₂	#11728678	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	#12665
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
16. #89256	C₄	semidirect
172. C₇	#12665	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×#89256	2
#11725477	2
#12665	7