Subgroups — #13269107

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₄×C₈×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	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
13	C₄²×Dic₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₇×C₄⋊C₁₆	C₄	1	4	yes	complemented
16	#89297	—	2	4	no	—
17	C₄⋊(C₇⋊C₁₆)	C₄	1	4	yes	complemented
18	#12653	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇	C₈	1	2	yes	—
23	C₂×C₄×Dic₇	C₈	1	2	yes	—
24	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×C₁₆	—	2	2	no	—
29	C₂×C₇×C₁₆	—	2	2	no	—
30	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
31	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
33	C₈×Dic₇	—	2	1	no	—
34	C₈×Dic₇	—	2	1	no	—
35	C₈×Dic₇	—	2	2	no	—
36	C₈×Dic₇	—	2	2	no	—
37	C₈×Dic₇	D₄	1	2	yes	—
38	C₈×Dic₇	Q₈	1	2	yes	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₂×C₇⋊C₁₆	—	2	2	no	—
41	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
42	C₄×C₇⋊C₈	C₈	1	4	yes	—
43	C₂×C₄⋊C₁₆	—	7	1	no	—
44	C₂×C₄⋊C₁₆	D₇	1	1	yes	characteristic
45	C₂×#346	—	7	2	no	—
46	C₄²×C₈	—	7	1	no	—
47	#1374	—	7	2	no	—
48	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₇	C₂×C₈	1	2	yes	—
52	C₂²×Dic₇	C₂×C₈	1	2	yes	—
53	C₂×C₇×C₈	—	2	1	no	—
54	C₂×C₇×C₈	—	2	1	no	—
55	C₂×C₇×C₈	C₄²	1	2	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₇×C₈	#41	1	1	yes	characteristic
61	C₂×C₇×C₈	#41	1	1	yes	characteristic
62	C₄²×C₇	C₄²	1	2	yes	—
63	C₄²×C₇	C₂×C₈	1	2	yes	—
64	C₄×Dic₇	—	2	4	no	—
65	C₄×Dic₇	—	2	1	no	—
66	C₄×Dic₇	—	2	1	no	—
67	C₄×Dic₇	—	2	4	no	—
68	C₄×Dic₇	C₄⋊C₄	1	2	yes	—
69	C₄×Dic₇	M₁₆	1	2	yes	—
70	C₂×C₇⋊C₈	—	2	1	no	—
71	C₂×C₇⋊C₈	—	2	1	no	—
72	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
74	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
75	C₂×C₇⋊C₈	M₁₆	1	2	yes	—
76	C₇×C₁₆	—	2	4	no	—
77	C₇⋊C₁₆	—	2	4	no	—
78	C₂²×C₁₆	—	7	2	no	—
79	C₂²×C₁₆	C₂×D₇	1	2	yes	—
80	C₂×C₄×C₈	—	7	1	no	—
81	C₂×C₄×C₈	—	7	1	no	—
82	C₂×C₄×C₈	—	7	1	no	—
83	C₂×C₄×C₈	—	7	1	no	—
84	C₂×C₄×C₈	—	7	1	no	—
85	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
86	C₄³	—	7	1	no	—
87	C₄⋊C₁₆	—	7	4	no	—
88	C₄⋊C₁₆	Dic₇	1	4	yes	complemented
89	#346	—	14	4	no	—
90	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
93	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₇	M₃₂	1	4	yes	—
95	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₇	—	2	2	no	—
102	C₂×Dic₇	—	2	2	no	—
103	C₂×Dic₇	—	2	2	no	—
104	C₂×Dic₇	—	2	2	no	—
105	C₂×Dic₇	C₄⋊C₈	1	4	yes	—
106	C₂×Dic₇	M₃₂	1	4	yes	—
107	C₇×C₈	—	2	2	no	—
108	C₇×C₈	C₄×D₄	1	1	yes	characteristic
109	C₇×C₈	C₄×D₄	1	1	yes	characteristic
110	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
111	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₇⋊C₈	—	2	2	no	—
113	C₇⋊C₈	C₄⋊C₈	1	4	yes	—
114	C₂²×C₈	—	7	1	no	—
115	C₂²×C₈	—	7	1	no	—
116	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
118	C₂×C₄²	—	7	1	no	—
119	C₂×C₄²	—	7	2	no	—
120	C₂×C₄²	—	7	2	no	—
121	C₂×C₄²	—	7	1	no	—
122	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	7	4	no	—
126	C₂×C₁₆	—	14	2	no	—
127	C₂×C₁₆	—	14	2	no	—
128	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
129	C₄×C₈	—	7	2	no	—
130	C₄×C₈	—	7	2	no	—
131	C₄×C₈	—	7	4	no	—
132	C₄×C₈	—	14	2	no	—
133	C₄×C₈	—	14	2	no	—
134	C₄×C₈	—	14	1	no	—
135	C₄×C₈	—	14	1	no	—
136	C₄×C₈	C₄×D₇	1	2	yes	—
137	C₄×C₈	C₂×Dic₇	1	2	yes	—
138	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
139	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂²×C₇	#431	1	1	yes	characteristic
141	C₂²×C₇	C₂×M₃₂	1	2	yes	—
142	C₂²×C₇	C₂×M₃₂	1	2	yes	—
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	#346	1	4	yes	complemented
145	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₇	#431	1	1	yes	characteristic
149	Dic₇	—	2	4	no	—
150	Dic₇	—	2	4	no	—
151	C₂²×C₄	—	7	2	no	—
152	C₂²×C₄	—	7	2	no	—
153	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
154	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	7	2	no	—
159	C₂×C₈	—	7	2	no	—
160	C₂×C₈	—	7	2	no	—
161	C₂×C₈	—	7	2	no	—
162	C₂×C₈	—	14	1	no	—
163	C₂×C₈	—	14	1	no	—
164	C₂×C₈	C₄×Dic₇	1	2	yes	—
165	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
166	C₂×C₈	D₄×D₇	1	1	yes	characteristic
167	C₂×C₈	#1159	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
169	C₂×C₈	#1161	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
171	C₄²	—	7	2	no	—
172	C₄²	—	7	2	no	—
173	C₄²	—	14	4	no	—
174	C₄²	—	14	1	no	—
175	C₄²	—	14	1	no	—
176	C₄²	—	14	4	no	—
177	C₄²	C₈×D₇	1	2	yes	—
178	C₄²	C₄×Dic₇	1	2	yes	—
179	C₁₆	—	2	4	no	—
180	C₁₆	—	14	4	no	—
181	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
182	C₂×C₇	C₂×#346	1	1	yes	characteristic
183	C₂×C₇	C₄×M₃₂	1	1	yes	characteristic
184	C₂×C₇	#2150	1	2	yes	—
185	C₂×C₇	#2163	1	2	yes	—
186	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	7	4	no	—
189	C₂×C₄	—	7	4	no	—
190	C₂×C₄	—	14	2	no	—
191	C₂×C₄	—	14	2	no	—
192	C₂×C₄	—	14	2	no	—
193	C₂×C₄	—	14	2	no	—
194	C₂×C₄	#6372	1	4	yes	—
195	C₂×C₄	C₈×Dic₇	1	2	yes	—
196	C₂×C₄	C₈×Dic₇	1	2	yes	—
197	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#6455	1	1	yes	characteristic
199	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
200	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
201	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	7	4	no	—
205	C₈	—	14	2	no	—
206	C₈	#6450	1	1	yes	characteristic
207	C₈	#6456	1	1	yes	characteristic
208	C₈	D₄×Dic₇	1	1	yes	characteristic
209	C₈	Q₈×Dic₇	1	1	yes	characteristic
210	C₇	#12653	1	1	yes	characteristic, complemented, Sylow
211	C₂²	D₇×C₄⋊C₈	1	1	yes	characteristic
212	C₂²	#89605	1	1	yes	characteristic
213	C₂²	C₂×#6372	1	2	yes	—
214	C₂²	D₇×M₃₂	1	2	yes	—
215	C₂²	C₂×C₈×Dic₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	14	4	no	—
218	C₄	—	14	4	no	—
219	C₄	#89297	1	4	yes	complemented
220	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
221	C₄	#89605	1	1	yes	characteristic
222	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
223	C₄	M₁₆×Dic₇	1	1	yes	characteristic
224	C₂	#11727029	1	2	yes	—
225	C₂	#11727038	1	2	yes	—
226	C₂	Dic₇×C₄⋊C₈	1	1	yes	characteristic
227	C₂	C₂×#89297	1	1	yes	characteristic
228	C₂	Dic₇×M₃₂	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	#12653
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
15. C₇×C₄⋊C₁₆	C₄	semidirect
17. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
88. C₄⋊C₁₆	Dic₇	semidirect
144. C₄×C₇	#346	semidirect
210. C₇	#12653	semidirect
219. 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
C₄×C₈×Dic₇	2
C₂×C₄⋊(C₇⋊C₁₆)	2
#11725610	2
#12653	7