Subgroups — #13509306

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736618	C₂	1	2	yes	maximal
2	C₇×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#89299	C₂	1	1	yes	maximal, characteristic
4	#11729153	C₂	1	1	yes	maximal, characteristic
5	#11728403	C₂	1	2	yes	maximal
6	#11727095	C₂	1	4	yes	maximal, complemented
7	#11727075	C₂	1	2	yes	maximal, complemented
8	#11727074	C₂	1	2	yes	maximal, complemented
9	#90210	—	2	2	no	—
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₇×#427	C₂²	1	2	yes	—
12	C₂×#6392	C₂²	1	2	yes	—
13	#89873	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	1	yes	characteristic
15	#89767	—	2	2	no	—
16	#89650	C₂²	1	4	yes	—
17	#89646	C₂²	1	4	yes	—
18	#89628	—	2	4	no	—
19	C₇×#367	C₂²	1	2	yes	—
20	C₇×#366	C₂²	1	1	yes	characteristic
21	C₇×#366	C₂²	1	1	yes	characteristic
22	#89299	C₂²	1	2	yes	—
23	#89299	C₂²	1	2	yes	—
24	#89297	C₂²	1	2	yes	—
25	C₁₆×Dic₇	C₂²	1	2	yes	—
26	#89285	C₂²	1	4	yes	—
27	#89284	C₂²	1	4	yes	—
28	#33729	—	7	1	no	maximal, Sylow
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×#117	—	2	2	no	—
31	#6489	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₇	—	2	2	no	—
33	C₂×#1136	D₄	1	1	yes	characteristic
34	#6453	—	2	4	no	—
35	#6451	—	2	4	no	—
36	#6441	—	4	4	no	—
37	#6433	—	4	2	no	—
38	#6432	—	4	2	no	—
39	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
40	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
42	C₇×#107	C₂³	1	2	yes	—
43	C₇×#107	C₂³	1	2	yes	—
44	C₇×#106	—	2	2	no	—
45	#6392	—	2	2	no	—
46	#6392	C₂³	1	4	yes	—
47	#6389	D₄	1	2	yes	—
48	C₈×Dic₇	C₂³	1	2	yes	—
49	#6386	Q₈	1	4	yes	complemented
50	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
51	#6382	—	2	4	no	—
52	#6381	—	2	4	no	—
53	#3164	—	7	2	no	—
54	#2255	—	7	4	no	—
55	#2253	—	7	2	no	—
56	#2252	—	7	2	no	—
57	#2159	—	7	2	no	—
58	#2158	D₇	1	1	yes	characteristic
59	C₂×#366	—	7	1	no	—
60	#2108	—	7	1	no	—
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₇	—	4	2	no	—
63	C₂×Dic₁₄	—	4	4	no	—
64	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
68	C₇×C₄⋊C₄	—	2	2	no	—
69	C₇×C₄⋊C₄	—	2	2	no	—
70	C₇×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₇	—	4	2	no	—
72	C₂²⋊Dic₇	D₈	1	2	yes	—
73	C₄⋊Dic₇	—	2	4	no	—
74	C₄⋊Dic₇	—	4	2	no	—
75	C₄⋊Dic₇	—	4	4	no	—
76	C₄⋊Dic₇	D₈	1	1	yes	characteristic
77	C₄⋊Dic₇	D₈	1	1	yes	characteristic
78	Dic₇⋊C₄	—	4	4	no	—
79	Dic₇⋊C₄	—	4	4	no	—
80	C₄×Dic₇	C₂×D₄	1	2	yes	—
81	#1136	Q₁₆	1	4	yes	—
82	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
83	C₇×C₁₆	C₂×Q₈	1	2	yes	—
84	C₇×C₁₆	C₂×Q₈	1	2	yes	—
85	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
86	#544	—	14	2	no	—
87	C₂×M₃₂	—	7	1	no	—
88	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
89	#501	—	14	4	no	—
90	#500	—	7	4	no	—
91	#499	—	7	4	no	—
92	#428	—	14	2	no	—
93	#427	C₂×D₇	1	2	yes	—
94	C₂×#107	—	7	2	no	—
95	#405	—	7	1	no	—
96	#367	—	7	4	no	—
97	#367	C₂×D₇	1	2	yes	—
98	#366	—	7	2	no	—
99	#366	—	7	2	no	—
100	#366	—	7	4	no	—
101	#366	C₂×D₇	1	1	yes	characteristic
102	#366	C₂×D₇	1	1	yes	characteristic
103	#346	—	7	2	no	—
104	C₄×C₁₆	—	7	2	no	—
105	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇	—	4	4	no	—
113	C₂×Dic₇	—	16	2	no	—
114	C₂×Dic₇	C₂×D₈	1	2	yes	—
115	C₂×Dic₇	C₂×D₈	1	2	yes	—
116	Dic₁₄	—	16	4	no	—
117	C₇×C₈	C₄⋊Q₈	1	2	yes	—
118	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
119	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
120	C₇⋊C₈	C₂×Q₁₆	1	4	yes	—
121	C₂×M₁₆	—	7	1	no	—
122	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	14	4	no	—
124	C₄⋊Q₈	—	28	2	no	—
125	#125	—	14	4	no	—
126	#125	—	28	2	no	—
127	C₂²⋊Q₈	—	28	4	no	—
128	#117	—	2	2	no	—
129	#117	—	7	1	no	—
130	C₂×C₄⋊C₄	—	14	2	no	—
131	M₃₂	—	7	4	no	—
132	C₂×C₁₆	—	7	2	no	—
133	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
134	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
135	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
136	#107	—	7	4	no	—
137	#107	—	14	4	no	—
138	#107	—	14	2	no	—
139	#107	C₂²×D₇	1	2	yes	—
140	#107	C₂²×D₇	1	2	yes	—
141	#106	—	2	2	no	—
142	#106	—	14	4	no	—
143	#97	—	7	2	no	—
144	C₄×C₈	—	7	2	no	—
145	C₂²×C₇	—	2	1	no	—
146	C₂²×C₇	—	2	1	no	—
147	C₂²×C₇	#500	1	2	yes	—
148	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₇	—	8	2	no	—
150	C₄×C₇	—	8	2	no	—
151	C₄×C₇	#500	1	2	yes	—
152	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
154	Dic₇	—	2	2	no	—
155	Dic₇	—	2	2	no	—
156	Dic₇	—	8	4	no	—
157	C₂×Q₈	—	28	4	no	—
158	C₂²×C₄	—	28	2	no	—
159	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
160	M₁₆	—	7	4	no	—
161	C₂×C₈	—	7	2	no	—
162	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
163	C₂×C₈	D₄×D₇	1	1	yes	characteristic
164	C₂×C₈	Q₈×D₇	1	2	yes	—
165	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	7	1	no	—
169	C₄⋊C₄	—	7	1	no	—
170	C₄⋊C₄	—	14	4	no	—
171	C₄⋊C₄	—	28	4	no	—
172	C₄⋊C₄	—	28	4	no	—
173	C₄⋊C₄	—	28	4	no	—
174	C₄⋊C₄	—	28	2	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	7	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	—	7	2	no	—
179	C₁₆	—	7	4	no	—
180	C₁₆	C₂×Dic₁₄	1	2	yes	—
181	C₁₆	C₂×Dic₁₄	1	2	yes	—
182	C₂×C₇	—	2	2	no	—
183	C₂×C₇	C₂×#500	1	1	yes	characteristic
184	C₂×C₇	#3415	1	1	yes	characteristic
185	C₂×C₇	#3417	1	1	yes	characteristic
186	C₂³	D₇×D₈	1	1	yes	characteristic
187	Q₈	—	112	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	7	2	no	—
191	C₂×C₄	—	7	2	no	—
192	C₂×C₄	—	28	4	no	—
193	C₂×C₄	—	112	2	no	—
194	C₂×C₄	#6451	1	2	yes	—
195	C₂×C₄	D₇×D₈	1	1	yes	characteristic
196	C₂×C₄	D₇×Q₁₆	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
198	C₈	—	7	4	no	—
199	C₈	#6451	1	2	yes	—
200	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
201	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
202	C₇	#33729	1	1	yes	characteristic, complemented, Sylow
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#89646	1	2	yes	—
206	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	—	14	2	no	—
210	C₄	—	14	2	no	—
211	C₄	—	56	4	no	—
212	C₄	#89646	1	2	yes	—
213	C₄	C₂×#6451	1	1	yes	characteristic
214	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#89646	1	1	yes	characteristic
217	C₂	#11739476	1	1	yes	characteristic
218	C₂	#11739478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2158	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#2158	D₇
hypocenter	C₇	#33729
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
6. #11727095	C₂	semidirect
7. #11727075	C₂	semidirect
8. #11727074	C₂	semidirect
39. C₂×C₇×C₁₆	Q₈	semidirect
49. #6386	Q₈	semidirect
133. C₂×C₁₆	Dic₁₄	semidirect
202. C₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#11736618	2
C₇×#2158	2
C₂×#89299	2
#11729153	2
#11728403	2
#11727095	2
#11727075	2
#11727074	2
#33729	7