Subgroups — #13507290

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 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×C₄×C₇×C₁₆	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	#11726802	C₂	1	4	yes	maximal, complemented
7	C₁₆×Dic₁₄	C₂	1	4	yes	maximal, complemented
8	#90161	C₄	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#89874	C₄	1	2	yes	—
13	#89873	C₂²	1	2	yes	—
14	C₂×#6386	C₂²	1	2	yes	—
15	#89696	C₄	1	1	yes	characteristic
16	#89451	C₄	1	4	yes	—
17	C₈×Dic₁₄	C₂²	1	4	yes	—
18	C₄×C₇×C₁₆	C₂²	1	2	yes	—
19	C₄×C₇×C₁₆	C₂²	1	2	yes	—
20	#89297	C₂²	1	4	yes	—
21	Dic₇⋊C₁₆	C₂²	1	8	yes	—
22	C₁₆×Dic₇	C₂²	1	4	yes	—
23	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
24	#33474	—	7	1	no	maximal, Sylow
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	#6489	C₂×C₄	1	2	yes	—
29	#6487	C₈	1	4	yes	—
30	C₂×#1136	C₂×C₄	1	2	yes	—
31	#6433	C₈	1	2	yes	—
32	#6432	C₈	1	2	yes	—
33	C₄×Dic₁₄	C₂×C₄	1	4	yes	—
34	C₂×C₇×C₁₆	—	2	2	no	—
35	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
36	C₂×C₇×C₁₆	C₂³	1	2	yes	—
37	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
39	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₇×C₈	C₂³	1	2	yes	—
41	#6389	C₂×C₄	1	4	yes	—
42	Dic₇⋊C₈	C₂×C₄	1	8	yes	—
43	C₈×Dic₇	C₂³	1	4	yes	—
44	#6386	—	2	4	no	—
45	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
46	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
47	#2960	—	7	1	no	—
48	#2183	—	7	4	no	—
49	Q₈×C₁₆	—	7	4	no	—
50	#2151	—	7	2	no	—
51	#2150	—	7	1	no	—
52	#2108	—	7	2	no	—
53	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₁₄	C₂×C₈	1	4	yes	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
61	C₂×C₇×C₈	#41	1	2	yes	—
62	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
63	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₇	C₂²×C₄	1	2	yes	—
66	C₂²⋊Dic₇	C₂×C₈	1	4	yes	—
67	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
68	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
69	Dic₇⋊C₄	C₂×C₈	1	8	yes	—
70	C₄×Dic₇	C₂²×C₄	1	4	yes	—
71	#1136	—	2	4	no	—
72	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
73	C₇×C₁₆	—	2	2	no	—
74	C₇×C₁₆	C₂×Q₈	1	2	yes	—
75	C₇×C₁₆	C₂×Q₈	1	2	yes	—
76	C₇⋊C₁₆	—	2	4	no	—
77	#533	—	7	1	no	—
78	C₂×M₃₂	—	7	2	no	—
79	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
80	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
81	#446	—	7	4	no	—
82	C₈×Q₈	—	7	4	no	—
83	#424	—	7	2	no	—
84	#423	—	7	1	no	—
85	#405	—	7	2	no	—
86	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	7	8	no	—
88	C₄⋊C₁₆	—	7	4	no	—
89	#346	—	7	4	no	—
90	C₄×C₁₆	—	7	4	no	—
91	C₄×C₁₆	C₂×D₇	1	2	yes	—
92	C₄×C₁₆	C₂×D₇	1	2	yes	—
93	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₇	#131	1	2	yes	—
100	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₇	C₂²×C₈	1	8	yes	—
102	Dic₁₄	—	2	8	no	—
103	C₇×C₈	—	2	2	no	—
104	C₇×C₈	C₄×Q₈	1	2	yes	—
105	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
106	C₇×C₈	C₂×#41	1	1	yes	characteristic
107	C₇⋊C₈	—	2	4	no	—
108	C₂×M₁₆	—	7	2	no	—
109	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
111	C₄⋊Q₈	—	7	2	no	—
112	#125	—	7	2	no	—
113	C₂²⋊Q₈	—	7	4	no	—
114	C₄×Q₈	—	7	4	no	—
115	#117	—	7	2	no	—
116	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
117	M₃₂	—	14	4	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	7	4	no	—
120	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
121	C₂×C₁₆	C₂²×D₇	1	2	yes	—
122	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
123	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
124	C₄⋊C₈	—	7	8	no	—
125	C₄⋊C₈	—	7	4	no	—
126	#97	—	7	4	no	—
127	C₄×C₈	—	7	4	no	—
128	C₄×C₈	C₄×D₇	1	2	yes	—
129	C₄×C₈	C₂²×D₇	1	2	yes	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	C₈×Q₈	1	2	yes	—
132	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	C₈×Q₈	1	2	yes	—
135	C₄×C₇	#504	1	4	yes	—
136	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₇	C₂×#131	1	1	yes	characteristic
138	Dic₇	—	2	8	no	—
139	C₂×Q₈	—	7	4	no	—
140	C₂²×C₄	C₈×D₇	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
142	M₁₆	—	14	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	7	4	no	—
145	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
146	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
147	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
148	C₂×C₈	#1157	1	2	yes	—
149	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
150	C₄⋊C₄	—	7	8	no	—
151	C₄⋊C₄	—	7	4	no	—
152	C₄⋊C₄	—	7	4	no	—
153	C₂²⋊C₄	—	7	4	no	—
154	C₄²	—	7	4	no	—
155	C₄²	C₈×D₇	1	1	yes	characteristic
156	C₄²	C₈×D₇	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₇	1	2	yes	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	14	4	no	—
160	C₁₆	C₂×Dic₁₄	1	2	yes	—
161	C₁₆	C₂×Dic₁₄	1	2	yes	—
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
164	C₂×C₇	C₂×#504	1	2	yes	—
165	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
166	Q₈	—	14	8	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	7	8	no	—
169	C₂×C₄	C₄×Dic₁₄	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
173	C₂×C₄	#6464	1	2	yes	—
174	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	14	4	no	—
177	C₈	C₄×Dic₁₄	1	2	yes	—
178	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
179	C₈	C₂×#1157	1	1	yes	characteristic
180	C₇	#33474	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Dic₁₄	1	2	yes	—
183	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	14	8	no	—
186	C₄	C₈×Dic₁₄	1	2	yes	—
187	C₄	#89673	1	4	yes	—
188	C₄	C₂×C₄×Dic₁₄	1	1	yes	characteristic
189	C₄	C₂×#6464	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Dic₁₄	1	1	yes	characteristic
192	C₂	C₂×#89673	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#33474
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
6. #11726802	C₂	semidirect
7. C₁₆×Dic₁₄	C₂	semidirect
35. C₂×C₇×C₁₆	Q₈	semidirect
120. C₂×C₁₆	Dic₁₄	semidirect
180. C₇	#33474	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×C₄×C₇×C₁₆	2
#11729155	2
#11729153	2
#11727497	2
#11726802	2
C₁₆×Dic₁₄	2
#33474	7