Subgroups — #13507296

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735890	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
3	#11729155	C₂	1	2	yes	maximal
4	#11729153	C₂	1	2	yes	maximal
5	#11727497	C₂	1	1	yes	maximal, characteristic
6	#11726822	C₂	1	8	yes	maximal, complemented
7	#90161	C₄	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	#89874	C₄	1	2	yes	—
12	#89873	C₂²	1	2	yes	—
13	C₂×#6386	C₂²	1	2	yes	—
14	#89696	C₄	1	1	yes	characteristic
15	#89451	C₄	1	4	yes	—
16	C₈×Dic₁₄	C₂²	1	4	yes	—
17	C₇×#346	C₂²	1	4	yes	—
18	#89297	C₂²	1	4	yes	—
19	Dic₇⋊C₁₆	C₂²	1	8	yes	—
20	C₁₆×Dic₇	C₂²	1	4	yes	—
21	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
22	#33475	—	7	1	no	maximal, Sylow
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	#6489	C₂×C₄	1	2	yes	—
27	#6487	C₈	1	4	yes	—
28	C₂×#1136	C₂×C₄	1	2	yes	—
29	#6433	C₈	1	2	yes	—
30	#6432	C₈	1	2	yes	—
31	C₄×Dic₁₄	C₂×C₄	1	4	yes	—
32	C₂×C₇×C₁₆	—	2	2	no	—
33	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
34	C₂×C₇×C₁₆	C₂³	1	2	yes	—
35	C₂×C₇×C₁₆	C₂³	1	2	yes	—
36	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₇×C₈	C₂³	1	2	yes	—
38	#6389	C₂×C₄	1	4	yes	—
39	Dic₇⋊C₈	C₂×C₄	1	8	yes	—
40	C₈×Dic₇	C₂³	1	4	yes	—
41	#6386	—	2	4	no	—
42	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
43	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
44	#2960	—	7	1	no	—
45	#2183	—	7	8	no	—
46	#2151	—	7	2	no	—
47	#2150	—	7	1	no	—
48	#2108	—	7	2	no	—
49	C₂×#346	D₇	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₁₄	C₂×C₈	1	4	yes	—
53	C₂×C₇×C₈	—	2	2	no	—
54	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₇×C₈	#41	1	2	yes	—
58	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
59	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₇	C₂²×C₄	1	2	yes	—
62	C₂²⋊Dic₇	C₂×C₈	1	4	yes	—
63	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
64	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
65	Dic₇⋊C₄	C₂×C₈	1	8	yes	—
66	C₄×Dic₇	C₂²×C₄	1	4	yes	—
67	#1136	—	2	4	no	—
68	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
69	C₇×C₁₆	—	2	2	no	—
70	C₇×C₁₆	C₂×Q₈	1	4	yes	—
71	C₇⋊C₁₆	—	2	4	no	—
72	#533	—	7	1	no	—
73	C₂×M₃₂	—	7	2	no	—
74	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
75	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
76	#446	—	7	4	no	—
77	C₈×Q₈	—	7	4	no	—
78	#424	—	7	2	no	—
79	#423	—	7	1	no	—
80	#405	—	7	2	no	—
81	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
82	C₄⋊C₁₆	—	7	8	no	—
83	C₄⋊C₁₆	—	7	4	no	—
84	#346	—	7	4	no	—
85	#346	C₂×D₇	1	4	yes	—
86	C₄×C₁₆	—	7	4	no	—
87	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇	—	4	2	no	—
89	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
90	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₇	#131	1	2	yes	—
94	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×Dic₇	C₂²×C₈	1	8	yes	—
96	Dic₁₄	—	4	4	no	—
97	C₇×C₈	—	2	2	no	—
98	C₇×C₈	C₄×Q₈	1	2	yes	—
99	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
100	C₇×C₈	C₂×#41	1	1	yes	characteristic
101	C₇⋊C₈	—	2	4	no	—
102	C₂×M₁₆	—	7	2	no	—
103	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
105	C₄⋊Q₈	—	7	2	no	—
106	#125	—	7	2	no	—
107	C₂²⋊Q₈	—	7	4	no	—
108	C₄×Q₈	—	7	4	no	—
109	#117	—	7	2	no	—
110	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
111	M₃₂	—	14	4	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	7	4	no	—
114	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
115	C₂×C₁₆	C₂²×D₇	1	2	yes	—
116	C₂×C₁₆	C₂²×D₇	1	2	yes	—
117	C₄⋊C₈	—	7	8	no	—
118	C₄⋊C₈	—	7	4	no	—
119	#97	—	7	4	no	—
120	C₄×C₈	—	7	4	no	—
121	C₄×C₈	C₄×D₇	1	2	yes	—
122	C₄×C₈	C₂²×D₇	1	2	yes	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	C₈×Q₈	1	2	yes	—
125	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₇	—	2	2	no	—
127	C₄×C₇	—	2	2	no	—
128	C₄×C₇	C₈×Q₈	1	2	yes	—
129	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₇	C₂×#131	1	1	yes	characteristic
131	Dic₇	—	2	8	no	—
132	C₂×Q₈	—	7	4	no	—
133	C₂²×C₄	C₈×D₇	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
135	M₁₆	—	14	4	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	7	4	no	—
138	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
139	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
141	C₂×C₈	#1157	1	2	yes	—
142	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
143	C₄⋊C₄	—	7	8	no	—
144	C₄⋊C₄	—	7	4	no	—
145	C₄⋊C₄	—	7	4	no	—
146	C₂²⋊C₄	—	7	4	no	—
147	C₄²	—	7	4	no	—
148	C₄²	C₈×D₇	1	1	yes	characteristic
149	C₄²	C₈×D₇	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₇	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	14	4	no	—
153	C₁₆	C₂×Dic₁₄	1	4	yes	—
154	C₂×C₇	—	2	2	no	—
155	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
156	C₂×C₇	#3407	1	2	yes	—
157	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
158	Q₈	—	28	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	7	8	no	—
161	C₂×C₄	C₄×Dic₁₄	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
163	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
165	C₂×C₄	#6464	1	2	yes	—
166	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	14	4	no	—
169	C₈	C₄×Dic₁₄	1	2	yes	—
170	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
171	C₈	C₂×#1157	1	1	yes	characteristic
172	C₇	#33475	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	C₈×Dic₁₄	1	2	yes	—
175	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	14	8	no	—
179	C₄	C₈×Dic₁₄	1	2	yes	—
180	C₄	C₂×C₄×Dic₁₄	1	1	yes	characteristic
181	C₄	C₂×#6464	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₈×Dic₁₄	1	1	yes	characteristic
184	C₂	#11739452	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₂×C₇×#346	C₂
prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#346	D₇
hypocenter	C₇	#33475
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
6. #11726822	C₂	semidirect
33. C₂×C₇×C₁₆	Q₈	semidirect
114. C₂×C₁₆	Dic₁₄	semidirect
172. C₇	#33475	semidirect

Maximal subgroups

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

subgroup	index
#11735890	2
C₂×C₇×#346	2
#11729155	2
#11729153	2
#11727497	2
#11726822	2
#33475	7