Subgroups — #13272011

Order 1792.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#89357	C₂	1	4	yes	maximal, complemented
3	#11727519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×D₈	—	2	1	no	—
6	C₂²×C₇×D₈	C₂²	1	2	yes	—
7	C₂×C₇×#127	—	2	1	no	—
8	C₇×#493	—	2	2	no	—
9	C₇×#493	C₄	1	4	yes	complemented
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×#6392	—	2	1	no	—
12	C₂×#6392	C₂²	1	2	yes	—
13	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#89707	—	2	1	no	—
15	#89357	—	2	8	no	—
16	C₄×C₇⋊C₁₆	—	2	2	no	—
17	#13202	—	7	1	no	maximal, Sylow
18	C₂²×C₇×D₄	—	2	2	no	—
19	C₂²×C₇×D₄	—	4	2	no	—
20	C₂×C₇×D₈	—	2	4	no	—
21	C₂×C₇×D₈	—	2	4	no	—
22	C₂×C₇×D₈	—	2	2	no	—
23	C₂×C₇×D₈	—	2	4	no	—
24	C₂×C₇×D₈	—	4	2	no	—
25	C₂×C₇×D₈	—	4	2	no	—
26	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
27	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₇×#127	—	2	4	no	—
30	C₇×#127	—	4	2	no	—
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₇	—	2	2	no	—
33	C₂×C₄⋊Dic₇	—	4	2	no	—
34	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇×C₈	D₄	1	2	yes	—
36	#6392	—	2	4	no	—
37	#6392	—	4	2	no	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₂×C₇⋊C₁₆	D₄	1	4	yes	complemented
41	C₂×#493	D₇	1	1	yes	characteristic
42	C₂×#357	—	7	4	no	—
43	C₂×C₄×C₁₆	—	7	1	no	—
44	#1845	—	7	1	no	—
45	C₂⁴×C₇	—	4	2	no	—
46	C₂×C₇×D₄	—	2	4	no	—
47	C₂×C₇×D₄	—	4	8	no	—
48	C₂×C₇×D₄	—	4	2	no	—
49	C₂×C₇×D₄	—	4	2	no	—
50	C₂×C₇×D₄	—	4	2	no	—
51	C₂×C₇×D₄	—	4	2	no	—
52	C₂×C₇×D₄	—	4	2	no	—
53	C₂×C₇×D₄	—	8	4	no	—
54	C₂×C₇×D₄	—	8	4	no	—
55	C₂²×C₄×C₇	D₈	1	2	yes	—
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇	—	4	2	no	—
58	C₇×D₈	—	2	8	no	—
59	C₇×D₈	—	4	8	no	—
60	C₇×D₈	—	4	4	no	—
61	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₇×C₈	D₈	1	2	yes	—
63	C₂×C₇×C₈	SD₁₆	1	2	yes	—
64	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	#41	1	2	yes	—
68	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇	SD₁₆	1	2	yes	—
70	C₄⋊Dic₇	—	4	2	no	—
71	C₄⋊Dic₇	—	4	2	no	—
72	C₄⋊Dic₇	—	8	4	no	—
73	C₇⋊C₁₆	—	2	4	no	—
74	C₂²×D₈	—	2	1	no	—
75	C₂²×D₈	C₂×D₇	1	2	yes	—
76	C₂×#127	—	2	1	no	—
77	C₂²×C₁₆	—	7	2	no	—
78	#493	—	2	2	no	—
79	#493	Dic₇	1	4	yes	complemented
80	C₂×#107	—	7	2	no	—
81	C₂×#107	—	14	1	no	—
82	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
83	#384	—	14	1	no	—
84	#357	—	14	8	no	—
85	C₄×C₁₆	—	14	2	no	—
86	C₂³×C₇	—	4	4	no	—
87	C₂³×C₇	—	8	2	no	—
88	C₂³×C₇	—	8	2	no	—
89	C₂³×C₇	—	8	4	no	—
90	C₂³×C₇	—	8	4	no	—
91	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
92	C₇×D₄	—	4	4	no	—
93	C₇×D₄	—	4	4	no	—
94	C₇×D₄	—	8	16	no	—
95	C₇×D₄	—	8	2	no	—
96	C₇×D₄	—	8	2	no	—
97	C₂×C₄×C₇	#102	1	4	yes	—
98	C₂×C₄×C₇	D₁₆	1	4	yes	—
99	C₂×C₄×C₇	SD₃₂	1	4	yes	—
100	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×Dic₇	—	8	2	no	—
107	C₂×Dic₇	—	8	4	no	—
108	C₇×C₈	#102	1	4	yes	—
109	C₇×C₈	C₄×D₄	1	2	yes	—
110	C₇×C₈	#121	1	2	yes	—
111	C₂²×D₄	—	2	2	no	—
112	C₂²×D₄	—	4	2	no	—
113	C₂×D₈	—	2	4	no	—
114	C₂×D₈	—	2	4	no	—
115	C₂×D₈	—	2	4	no	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	—	4	2	no	—
118	C₂×D₈	—	4	2	no	—
119	C₂×D₈	C₂×Dic₇	1	4	yes	—
120	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
122	#127	—	2	4	no	—
123	#127	—	4	2	no	—
124	C₂×C₄⋊C₄	—	14	2	no	—
125	C₂×C₄⋊C₄	—	28	2	no	—
126	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
127	C₂×C₁₆	—	7	4	no	—
128	C₂×C₁₆	—	14	2	no	—
129	C₂×C₁₆	—	14	2	no	—
130	#107	—	14	4	no	—
131	#107	—	28	2	no	—
132	C₄×C₈	C₂×Dic₇	1	2	yes	—
133	C₄×C₈	C₂²⋊D₇	1	2	yes	—
134	C₂²×C₇	—	8	8	no	—
135	C₂²×C₇	—	8	4	no	—
136	C₂²×C₇	—	16	4	no	—
137	C₂²×C₇	—	16	2	no	—
138	C₂²×C₇	—	16	2	no	—
139	C₂²×C₇	C₂×#102	1	1	yes	characteristic
140	C₂²×C₇	#486	1	1	yes	characteristic
141	C₂²×C₇	#493	1	1	yes	characteristic
142	C₂²×C₇	C₂×D₁₆	1	2	yes	—
143	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
144	C₄×C₇	#357	1	8	yes	complemented
145	C₄×C₇	#390	1	1	yes	characteristic
146	C₄×C₇	C₂×#102	1	1	yes	characteristic
147	C₄×C₇	#486	1	1	yes	characteristic
148	C₄×C₇	#492	1	1	yes	characteristic
149	Dic₇	—	16	2	no	—
150	C₂⁴	—	4	2	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	2	no	—
158	C₂×D₄	—	8	4	no	—
159	C₂×D₄	—	8	4	no	—
160	C₂²×C₄	—	28	2	no	—
161	C₂²×C₄	#1141	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
163	D₈	—	2	8	no	—
164	D₈	—	4	8	no	—
165	D₈	—	4	4	no	—
166	C₂×C₈	#1141	1	2	yes	—
167	C₂×C₈	#1142	1	2	yes	—
168	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
169	C₂×C₈	D₄×D₇	1	2	yes	—
170	C₂×C₈	#1159	1	2	yes	—
171	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
173	C₄⋊C₄	—	28	2	no	—
174	C₄⋊C₄	—	28	2	no	—
175	C₄⋊C₄	—	56	4	no	—
176	C₄²	#1142	1	2	yes	—
177	C₄²	C₂²⋊Dic₇	1	2	yes	—
178	C₁₆	—	14	4	no	—
179	C₂×C₇	—	16	4	no	—
180	C₂×C₇	#1957	1	1	yes	characteristic
181	C₂×C₇	C₂×#357	1	2	yes	—
182	C₂×C₇	#2240	1	2	yes	—
183	C₂×C₇	#2246	1	1	yes	characteristic
184	C₂×C₇	#2248	1	1	yes	characteristic
185	C₂³	—	4	4	no	—
186	C₂³	—	8	4	no	—
187	C₂³	—	8	4	no	—
188	C₂³	—	8	2	no	—
189	C₂³	—	8	2	no	—
190	C₂³	C₂×#1141	1	1	yes	characteristic
191	D₄	—	4	4	no	—
192	D₄	—	4	4	no	—
193	D₄	—	8	16	no	—
194	D₄	—	8	2	no	—
195	D₄	—	8	2	no	—
196	C₂×C₄	—	56	4	no	—
197	C₂×C₄	—	56	2	no	—
198	C₂×C₄	#6400	1	4	yes	—
199	C₂×C₄	#6401	1	4	yes	—
200	C₂×C₄	#6406	1	4	yes	—
201	C₂×C₄	C₂×#1141	1	1	yes	characteristic
202	C₂×C₄	C₂×#1142	1	1	yes	characteristic
203	C₂×C₄	C₂×#1142	1	1	yes	characteristic
204	C₂×C₄	#6499	1	1	yes	characteristic
205	C₂×C₄	#6503	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
207	C₈	#6406	1	4	yes	—
208	C₈	D₄×Dic₇	1	2	yes	—
209	C₈	#6501	1	2	yes	—
210	C₇	#13202	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	8	4	no	—
212	C₂²	—	8	8	no	—
213	C₂²	—	16	4	no	—
214	C₂²	—	16	2	no	—
215	C₂²	—	16	2	no	—
216	C₂²	#89842	1	1	yes	characteristic
217	C₂²	#89845	1	1	yes	characteristic
218	C₂²	C₂×#6400	1	2	yes	—
219	C₂²	C₂×#6401	1	2	yes	—
220	C₂²	C₂×#6406	1	1	yes	characteristic
221	C₄	—	112	2	no	—
222	C₄	#89357	1	8	yes	complemented
223	C₄	#89842	1	1	yes	characteristic
224	C₄	#89848	1	1	yes	characteristic
225	C₄	C₂×#6406	1	1	yes	characteristic
226	C₄	#89993	1	1	yes	characteristic
227	C₂	—	16	4	no	—
228	C₂	#11729042	1	2	yes	—
229	C₂	#11729044	1	1	yes	characteristic
230	C₂	#11729045	1	1	yes	characteristic
231	C₂	#11729518	1	1	yes	characteristic
232	C₂	C₂×#89357	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting, prime radical	C₂×C₇×#493	C₂
socle	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#493	D₇
hypocenter	C₇	#13202
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
2. C₂×#89357	C₂	semidirect
3. #11727519	C₂	semidirect
4. C₂×C₄×C₇⋊C₁₆	C₂	semidirect
9. C₇×#493	C₄	semidirect
40. C₂×C₇⋊C₁₆	D₄	semidirect
79. #493	Dic₇	semidirect
144. C₄×C₇	#357	semidirect
210. C₇	#13202	semidirect
222. C₄	#89357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#493	2
C₂×#89357	2
#11727519	2
C₂×C₄×C₇⋊C₁₆	2
#13202	7