Subgroups — #13509327

Order 1792.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×#89455	C₂	1	1	yes	maximal, characteristic
3	#11729140	C₂	1	4	yes	maximal, complemented
4	#11729108	C₂	1	4	yes	maximal, complemented
5	#11728403	C₂	1	4	yes	maximal
6	C₄×#6386	C₂	1	1	yes	maximal, characteristic
7	C₇×#533	—	2	1	no	—
8	C₂×#6433	—	2	1	no	—
9	C₇×#500	C₂²	1	2	yes	—
10	C₇×#499	—	2	1	no	—
11	C₇×#499	C₂²	1	2	yes	—
12	C₇×#498	—	2	1	no	—
13	C₇×#427	—	2	1	no	—
14	C₇×#427	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×#6392	C₂²	1	2	yes	—
17	C₂×#6391	—	2	1	no	—
18	C₂×#6386	C₂²	1	2	yes	—
19	#89455	—	2	1	no	—
20	#89455	—	2	1	no	—
21	#89455	C₂²	1	2	yes	—
22	#89455	C₂²	1	2	yes	—
23	#89285	C₂²	1	8	yes	—
24	#89284	C₂²	1	8	yes	—
25	#89256	C₂²	1	2	yes	—
26	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
27	#33732	—	7	1	no	maximal, Sylow
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×#125	—	2	2	no	—
32	C₇×C₂²⋊Q₈	—	4	2	no	—
33	C₄×C₇×Q₈	—	4	2	no	—
34	C₇×#117	—	2	2	no	—
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇	—	2	2	no	—
37	C₂×Dic₇⋊C₄	—	4	2	no	—
38	#6433	—	2	2	no	—
39	#6433	—	2	2	no	—
40	#6433	—	4	2	no	—
41	C₇×#107	—	2	2	no	—
42	C₇×#107	C₂³	1	4	yes	—
43	C₇×#106	—	2	2	no	—
44	C₇×#106	—	2	2	no	—
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	1	yes	characteristic
47	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
49	#6392	—	2	2	no	—
50	#6392	C₂³	1	4	yes	—
51	#6391	—	2	2	no	—
52	#6391	—	2	2	no	—
53	#6386	Q₈	1	8	yes	complemented
54	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
55	#3160	D₇	1	1	yes	characteristic
56	C₂×#499	—	7	1	no	—
57	#2255	—	7	4	no	—
58	#2253	—	7	4	no	—
59	#2159	—	7	4	no	—
60	C₄×M₃₂	—	7	1	no	—
61	C₂×C₇×Q₈	—	4	2	no	—
62	C₂²×C₄×C₇	D₈	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₇	—	4	2	no	—
65	C₂×C₇×C₈	Q₁₆	1	4	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
70	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	4	4	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₇	—	4	2	no	—
76	C₄²×C₇	D₈	1	2	yes	—
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
79	C₄⋊Dic₇	—	2	4	no	—
80	C₄⋊Dic₇	—	4	2	no	—
81	Dic₇⋊C₄	—	4	4	no	—
82	Dic₇⋊C₄	—	4	4	no	—
83	C₇⋊C₁₆	C₂×Q₈	1	8	yes	—
84	#533	—	2	1	no	—
85	C₂×#125	—	14	1	no	—
86	C₂×M₃₂	—	7	2	no	—
87	#500	C₂×D₇	1	2	yes	—
88	#499	—	2	1	no	—
89	#499	—	7	2	no	—
90	#499	—	7	2	no	—
91	#499	—	14	1	no	—
92	#499	—	14	1	no	—
93	#499	C₂×D₇	1	2	yes	—
94	#498	—	2	1	no	—
95	#427	—	2	1	no	—
96	#427	C₂×D₇	1	2	yes	—
97	C₂×#107	—	7	2	no	—
98	C₂×#106	—	14	1	no	—
99	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
100	#367	—	7	8	no	—
101	#366	—	7	8	no	—
102	#346	—	7	2	no	—
103	C₄×C₁₆	—	7	2	no	—
104	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
105	C₇×Q₈	—	8	4	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	—	4	4	no	—
108	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
113	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₇	—	4	4	no	—
115	C₂×Dic₇	—	8	4	no	—
116	C₇×C₈	C₄⋊Q₈	1	2	yes	—
117	C₇×C₈	C₂×Q₁₆	1	2	yes	—
118	C₇×C₈	C₂×Q₁₆	1	2	yes	—
119	C₇×C₈	C₂²×Q₈	1	2	yes	—
120	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	2	no	—
123	#125	—	2	2	no	—
124	#125	—	14	2	no	—
125	#125	—	14	2	no	—
126	#125	—	28	2	no	—
127	C₂²⋊Q₈	—	4	2	no	—
128	C₄×Q₈	—	4	2	no	—
129	#117	—	2	2	no	—
130	C₂×C₄⋊C₄	—	14	2	no	—
131	C₂×C₄⋊C₄	—	28	2	no	—
132	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
133	M₃₂	—	7	8	no	—
134	C₂×C₁₆	—	7	4	no	—
135	#107	—	2	2	no	—
136	#107	—	7	4	no	—
137	#107	—	14	2	no	—
138	#107	C₂²×D₇	1	4	yes	—
139	#106	—	2	2	no	—
140	#106	—	2	2	no	—
141	#106	—	14	2	no	—
142	#106	—	14	2	no	—
143	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
145	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	#500	1	2	yes	—
149	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	8	4	no	—
153	C₄×C₇	#500	1	2	yes	—
154	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
156	Dic₇	—	8	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂²×C₄	—	28	2	no	—
159	C₂²×C₄	#1141	1	2	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
161	C₂×C₈	#1144	1	4	yes	—
162	C₂×C₈	Q₈×D₇	1	4	yes	—
163	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
166	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	14	4	no	—
171	C₄⋊C₄	—	28	4	no	—
172	C₄⋊C₄	—	28	4	no	—
173	C₄⋊C₄	—	28	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₄²	—	4	2	no	—
176	C₄²	#1141	1	2	yes	—
177	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
178	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
179	C₁₆	—	7	8	no	—
180	C₂×C₇	—	2	2	no	—
181	C₂×C₇	C₂×#500	1	1	yes	characteristic
182	C₂×C₇	#3414	1	2	yes	—
183	C₂³	C₂×#1141	1	1	yes	characteristic
184	Q₈	—	8	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	28	4	no	—
188	C₂×C₄	—	56	4	no	—
189	C₂×C₄	C₂×#1141	1	2	yes	—
190	C₂×C₄	C₂×#1141	1	2	yes	—
191	C₂×C₄	C₂×#1141	1	1	yes	characteristic
192	C₂×C₄	C₂×#1144	1	2	yes	—
193	C₂×C₄	#6507	1	2	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
195	C₈	C₂×#1144	1	2	yes	—
196	C₈	C₂×#1144	1	2	yes	—
197	C₈	#6507	1	2	yes	—
198	C₈	C₂×Q₈×D₇	1	2	yes	—
199	C₇	#33732	1	1	yes	characteristic, complemented, Sylow
200	C₂²	—	2	2	no	—
201	C₂²	#89850	1	2	yes	—
202	C₂²	C₂²×#1141	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	56	4	no	—
207	C₄	#89850	1	2	yes	—
208	C₄	C₂²×#1144	1	1	yes	characteristic
209	C₄	C₂×#6507	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#89850	1	1	yes	characteristic
212	C₂	#11741339	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#3160	D₇
hypocenter	C₇	#33732
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #11729140	C₂	semidirect
4. #11729108	C₂	semidirect
53. #6386	Q₈	semidirect
199. C₇	#33732	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
C₂×#89455	2
#11729140	2
#11729108	2
#11728403	2
C₄×#6386	2
#33732	7