Subgroups — #13509284

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#11735887	C₂	1	1	yes	maximal, characteristic
3	#11729139	C₂	1	2	yes	maximal, complemented
4	#11729138	C₂	1	2	yes	maximal, complemented
5	#11729108	C₂	1	4	yes	maximal, complemented
6	#11728402	C₂	1	4	yes	maximal
7	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	—	2	1	no	—
9	#90161	—	2	1	no	—
10	C₇×#500	C₂²	1	2	yes	—
11	C₇×#499	—	2	1	no	—
12	C₇×#499	C₂²	1	2	yes	—
13	C₇×#498	—	2	1	no	—
14	C₇×#427	—	2	1	no	—
15	C₇×#427	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#89878	—	2	1	no	—
18	#89878	C₂²	1	2	yes	—
19	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
20	#89455	—	2	1	no	—
21	#89455	C₂²	1	2	yes	—
22	#89454	C₂²	1	2	yes	—
23	#89452	—	2	1	no	—
24	#89285	C₂²	1	8	yes	—
25	#89284	C₂²	1	8	yes	—
26	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
29	#33726	—	7	1	no	maximal, Sylow
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊Q₈	—	2	2	no	—
33	C₇×#125	—	2	2	no	—
34	C₇×C₂²⋊Q₈	—	4	2	no	—
35	C₄×C₇×Q₈	—	4	2	no	—
36	C₇×#117	—	2	2	no	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	#6489	—	2	2	no	—
39	#6487	—	4	2	no	—
40	#6433	—	2	2	no	—
41	#6432	—	2	2	no	—
42	C₄×Dic₁₄	—	4	2	no	—
43	C₇×#107	—	2	2	no	—
44	C₇×#107	C₂³	1	4	yes	—
45	C₇×#106	—	2	2	no	—
46	C₇×#106	—	2	2	no	—
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	D₄	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	#6392	—	2	2	no	—
52	#6392	C₂³	1	4	yes	—
53	#6391	—	2	2	no	—
54	#6391	—	2	2	no	—
55	C₂×C₇⋊C₁₆	Q₈	1	8	yes	complemented
56	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
58	#3160	—	7	1	no	—
59	#3160	D₇	1	1	yes	characteristic
60	#2254	—	7	2	no	—
61	#2253	—	7	4	no	—
62	#2252	—	7	2	no	—
63	#2158	—	7	4	no	—
64	C₂×C₄×C₁₆	—	7	1	no	—
65	C₂×C₇×Q₈	—	4	2	no	—
66	C₂²×C₄×C₇	D₈	1	2	yes	—
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₁₄	—	4	2	no	—
69	C₂×C₇×C₈	Q₁₆	1	4	yes	—
70	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
74	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
75	C₇×C₄⋊C₄	—	2	4	no	—
76	C₇×C₄⋊C₄	—	4	4	no	—
77	C₇×C₄⋊C₄	—	4	2	no	—
78	C₇×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₇	—	4	2	no	—
80	C₄²×C₇	D₈	1	2	yes	—
81	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²⋊Dic₇	—	4	2	no	—
84	C₄⋊Dic₇	—	2	4	no	—
85	C₄⋊Dic₇	—	4	2	no	—
86	Dic₇⋊C₄	—	4	4	no	—
87	C₄×Dic₇	—	4	2	no	—
88	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
89	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
90	#533	—	2	1	no	—
91	#533	—	14	1	no	—
92	C₂²×C₁₆	—	7	2	no	—
93	#500	—	7	2	no	—
94	#500	C₂×D₇	1	2	yes	—
95	#499	—	2	1	no	—
96	#499	—	7	2	no	—
97	#499	—	14	1	no	—
98	#499	C₂×D₇	1	2	yes	—
99	#498	—	2	1	no	—
100	#498	—	14	1	no	—
101	#427	—	2	1	no	—
102	#427	—	7	2	no	—
103	#427	—	14	1	no	—
104	#427	C₂×D₇	1	2	yes	—
105	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
106	#367	—	7	8	no	—
107	#366	—	7	8	no	—
108	C₄×C₁₆	—	7	1	no	—
109	C₄×C₁₆	—	7	2	no	—
110	C₄×C₁₆	—	7	1	no	—
111	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
112	C₇×Q₈	—	8	4	no	—
113	C₂×C₄×C₇	—	2	4	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₇	—	4	4	no	—
122	Dic₁₄	—	8	4	no	—
123	C₇×C₈	C₄⋊Q₈	1	2	yes	—
124	C₇×C₈	C₂×Q₁₆	1	2	yes	—
125	C₇×C₈	C₂×Q₁₆	1	2	yes	—
126	C₇×C₈	C₂²×Q₈	1	2	yes	—
127	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	14	2	no	—
131	#125	—	2	2	no	—
132	#125	—	14	2	no	—
133	C₂²⋊Q₈	—	4	2	no	—
134	C₂²⋊Q₈	—	28	2	no	—
135	C₄×Q₈	—	4	2	no	—
136	C₄×Q₈	—	28	2	no	—
137	#117	—	2	2	no	—
138	#117	—	14	2	no	—
139	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
140	C₂×C₁₆	—	7	8	no	—
141	C₂×C₁₆	—	7	2	no	—
142	C₂×C₁₆	—	7	2	no	—
143	#107	—	2	2	no	—
144	#107	—	7	4	no	—
145	#107	—	14	2	no	—
146	#107	C₂²×D₇	1	4	yes	—
147	#106	—	2	2	no	—
148	#106	—	2	2	no	—
149	#106	—	14	2	no	—
150	#106	—	14	2	no	—
151	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
153	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	#500	1	2	yes	—
157	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	8	4	no	—
160	C₄×C₇	#500	1	2	yes	—
161	C₄×C₇	#508	1	4	yes	—
162	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
164	Dic₇	—	8	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	28	2	no	—
167	C₂²×C₄	#1141	1	2	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
169	C₂×C₈	#1144	1	4	yes	—
170	C₂×C₈	Q₈×D₇	1	4	yes	—
171	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
174	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	14	4	no	—
179	C₄⋊C₄	—	28	4	no	—
180	C₄⋊C₄	—	28	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	28	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	28	2	no	—
185	C₄²	#1141	1	2	yes	—
186	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
188	C₁₆	—	7	4	no	—
189	C₁₆	—	7	4	no	—
190	C₂×C₇	—	2	2	no	—
191	C₂×C₇	C₂×#500	1	1	yes	characteristic
192	C₂×C₇	C₂×#508	1	2	yes	—
193	C₂³	C₂×#1141	1	1	yes	characteristic
194	Q₈	—	8	4	no	—
195	Q₈	—	56	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	28	4	no	—
199	C₂×C₄	C₂×#1141	1	2	yes	—
200	C₂×C₄	C₂×#1141	1	2	yes	—
201	C₂×C₄	C₂×#1141	1	1	yes	characteristic
202	C₂×C₄	C₂×#1144	1	2	yes	—
203	C₂×C₄	#6507	1	2	yes	—
204	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
205	C₈	C₂×#1144	1	2	yes	—
206	C₈	C₂×#1144	1	2	yes	—
207	C₈	#6507	1	2	yes	—
208	C₈	C₂×Q₈×D₇	1	2	yes	—
209	C₇	#33726	1	1	yes	characteristic, complemented, Sylow
210	C₂²	—	2	2	no	—
211	C₂²	#89850	1	2	yes	—
212	C₂²	C₂²×#1141	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	56	4	no	—
216	C₄	#89850	1	2	yes	—
217	C₄	#89966	1	4	yes	—
218	C₄	C₂²×#1144	1	1	yes	characteristic
219	C₄	C₂×#6507	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#89850	1	1	yes	characteristic
222	C₂	C₂×#89966	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#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	#33726
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #11729139	C₂	semidirect
4. #11729138	C₂	semidirect
5. #11729108	C₂	semidirect
55. C₂×C₇⋊C₁₆	Q₈	semidirect
209. C₇	#33726	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
#11735887	2
#11729139	2
#11729138	2
#11729108	2
#11728402	2
C₂×C₄×C₇⋊C₁₆	2
#33726	7