Subgroups — #13269146

Order 1792.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#89297	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#11725610	C₂	1	2	yes	maximal
6	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇	C₂²	1	2	yes	—
9	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₇	C₄	1	1	yes	characteristic
12	C₂×C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
13	C₇×C₄⋊C₁₆	C₄	1	4	yes	complemented
14	#89345	C₄	1	2	yes	—
15	#89297	—	2	4	no	—
16	C₄×C₇⋊C₁₆	C₄	1	4	yes	complemented
17	#12662	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇	C₈	1	2	yes	—
22	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
23	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₂×C₇×C₁₆	—	2	2	no	—
26	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
27	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₇⋊#95	C₈	1	2	yes	—
30	C₈×Dic₇	—	2	2	no	—
31	C₈×Dic₇	—	2	2	no	—
32	C₂×C₇⋊C₁₆	—	2	1	no	—
33	C₂×C₇⋊C₁₆	—	2	1	no	—
34	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
37	C₂×C₇⋊C₁₆	Q₈	1	2	yes	—
38	C₄⋊(C₇⋊C₈)	C₈	1	4	yes	—
39	C₂×C₄⋊C₁₆	D₇	1	1	yes	characteristic
40	C₂×#346	—	7	2	no	—
41	C₂×C₄×C₁₆	—	7	1	no	—
42	C₈×C₄⋊C₄	—	7	1	no	—
43	#1374	—	7	2	no	—
44	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₇	C₂×C₈	1	4	yes	—
48	C₂×C₇×C₈	—	2	1	no	—
49	C₂×C₇×C₈	—	2	1	no	—
50	C₂×C₇×C₈	C₄²	1	2	yes	—
51	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₇×C₈	#41	1	1	yes	characteristic
56	C₂×C₇×C₈	#41	1	1	yes	characteristic
57	C₄²×C₇	C₄²	1	2	yes	—
58	C₄²×C₇	C₂×C₈	1	2	yes	—
59	C₄⋊Dic₇	—	2	4	no	—
60	C₄×Dic₇	—	2	2	no	—
61	C₄×Dic₇	—	2	2	no	—
62	C₂×C₇⋊C₈	—	2	2	no	—
63	C₂×C₇⋊C₈	—	2	2	no	—
64	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
65	C₇×C₁₆	—	2	4	no	—
66	C₇⋊C₁₆	—	2	2	no	—
67	C₇⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
68	C₂²×C₁₆	—	7	1	no	—
69	C₂²×C₁₆	—	7	1	no	—
70	C₂²×C₁₆	C₂×D₇	1	2	yes	—
71	C₂×C₄⋊C₈	—	7	1	no	—
72	C₂×C₄×C₈	—	7	2	no	—
73	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
74	C₄×C₄⋊C₄	—	7	1	no	—
75	C₄⋊C₁₆	Dic₇	1	4	yes	complemented
76	#346	—	14	4	no	—
77	C₄×C₁₆	—	7	4	no	—
78	#336	—	7	2	no	—
79	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₇	M₃₂	1	4	yes	—
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	#131	1	1	yes	characteristic
89	C₂×C₄×C₇	#131	1	1	yes	characteristic
90	C₂×Dic₇	—	2	4	no	—
91	C₂×Dic₇	—	2	4	no	—
92	C₂×Dic₇	—	2	4	no	—
93	C₇×C₈	—	2	2	no	—
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	C₂²×C₈	—	7	2	no	—
100	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	7	2	no	—
103	C₂×C₄²	—	7	2	no	—
104	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	7	2	no	—
108	C₂×C₁₆	—	7	2	no	—
109	C₂×C₁₆	—	7	2	no	—
110	C₂×C₁₆	—	7	2	no	—
111	C₂×C₁₆	—	14	1	no	—
112	C₂×C₁₆	—	14	1	no	—
113	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
114	C₄⋊C₈	—	7	4	no	—
115	C₄×C₈	—	14	2	no	—
116	C₄×C₈	—	14	2	no	—
117	C₄×C₈	C₄×D₇	1	2	yes	—
118	C₄×C₈	C₂×Dic₇	1	2	yes	—
119	#95	—	7	2	no	—
120	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₇	C₈×Q₈	1	1	yes	characteristic
123	C₂²×C₇	C₂×M₃₂	1	2	yes	—
124	C₂²×C₇	#504	1	2	yes	—
125	C₄×C₇	—	2	2	no	—
126	C₄×C₇	#346	1	4	yes	complemented
127	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₇	#405	1	1	yes	characteristic
129	C₄×C₇	C₈×D₄	1	1	yes	characteristic
130	C₄×C₇	C₈×Q₈	1	1	yes	characteristic
131	Dic₇	—	4	4	no	—
132	C₂²×C₄	—	7	4	no	—
133	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
134	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	7	4	no	—
139	C₂×C₈	—	14	2	no	—
140	C₂×C₈	—	14	2	no	—
141	C₂×C₈	C₄×Dic₇	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
143	C₂×C₈	D₄×D₇	1	1	yes	characteristic
144	C₂×C₈	#1159	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
146	C₂×C₈	#1161	1	1	yes	characteristic
147	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
148	C₄⋊C₄	—	14	4	no	—
149	C₄²	—	14	2	no	—
150	C₄²	—	14	2	no	—
151	C₄²	C₈×D₇	1	2	yes	—
152	C₄²	C₄×Dic₇	1	2	yes	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	7	4	no	—
155	C₁₆	—	14	2	no	—
156	C₂×C₇	C₈×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₇	C₂×#346	1	1	yes	characteristic
158	C₂×C₇	#2108	1	1	yes	characteristic
159	C₂×C₇	#2169	1	2	yes	—
160	C₂×C₇	#2183	1	2	yes	—
161	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	14	4	no	—
164	C₂×C₄	—	14	4	no	—
165	C₂×C₄	—	14	4	no	—
166	C₂×C₄	#6372	1	4	yes	—
167	C₂×C₄	C₈×Dic₇	1	2	yes	—
168	C₂×C₄	C₈×Dic₇	1	2	yes	—
169	C₂×C₄	#6450	1	1	yes	characteristic
170	C₂×C₄	#6456	1	1	yes	characteristic
171	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
172	C₂×C₄	#6470	1	1	yes	characteristic
173	C₂×C₄	#6470	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	14	4	no	—
177	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
178	C₈	#6455	1	1	yes	characteristic
179	C₈	D₄×Dic₇	1	1	yes	characteristic
180	C₈	Q₈×Dic₇	1	1	yes	characteristic
181	C₇	#12662	1	1	yes	characteristic, complemented, Sylow
182	C₂²	#89602	1	1	yes	characteristic
183	C₂²	#89606	1	1	yes	characteristic
184	C₂²	C₂×#6372	1	2	yes	—
185	C₂²	#89679	1	2	yes	—
186	C₂²	C₂×C₈×Dic₇	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	28	4	no	—
189	C₄	#89297	1	4	yes	complemented
190	C₄	#89602	1	1	yes	characteristic
191	C₄	#89606	1	1	yes	characteristic
192	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
193	C₄	#89894	1	1	yes	characteristic
194	C₂	#11727033	1	2	yes	—
195	C₂	#11727035	1	2	yes	—
196	C₂	#11728219	1	1	yes	characteristic
197	C₂	C₂×#89297	1	1	yes	characteristic
198	C₂	#11729173	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₂×C₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇
hypocenter	C₇	#12662
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. C₇×C₄⋊C₁₆	C₄	semidirect
16. C₄×C₇⋊C₁₆	C₄	semidirect
67. C₇⋊C₁₆	C₄⋊C₄	semidirect
75. C₄⋊C₁₆	Dic₇	semidirect
126. C₄×C₇	#346	semidirect
181. C₇	#12662	semidirect
189. C₄	#89297	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₁₆	2
C₂×#89297	2
C₈×C₄⋊Dic₇	2
C₂×C₄×C₇⋊C₁₆	2
#11725610	2
#12662	7