Subgroups — #13252840

Order 1792.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂×#89342	C₂	1	4	yes	maximal
3	#11727148	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₇×#384	—	2	2	no	—
9	C₇×#384	C₄	1	4	yes	—
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₇⋊C₈	—	2	1	no	—
15	#89342	—	2	8	no	—
16	#89319	—	2	1	no	—
17	#9694	—	7	1	no	maximal, Sylow
18	C₂²×C₇×Q₈	—	2	2	no	—
19	C₇×C₄⋊Q₈	—	2	4	no	—
20	C₇×C₄⋊Q₈	—	2	4	no	—
21	C₇×C₄⋊Q₈	—	4	4	no	—
22	C₇×C₄⋊Q₈	—	4	2	no	—
23	C₇×C₄⋊Q₈	—	4	2	no	—
24	C₂×C₇×C₄⋊C₄	—	2	4	no	—
25	C₂×C₇×C₄⋊C₄	—	2	4	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₇⋊C₈	—	2	2	no	—
34	C₂²×C₇⋊C₈	—	2	2	no	—
35	C₄⋊(C₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₇⋊C₈)	—	4	4	no	—
38	C₄⋊(C₇⋊C₈)	D₄	1	4	yes	—
39	C₄×C₇⋊C₈	—	4	2	no	—
40	#2868	D₇	1	1	yes	characteristic
41	#1842	—	7	1	no	—
42	C₄×C₄⋊C₈	—	7	1	no	—
43	C₂×#332	—	7	4	no	—
44	C₂×C₇×Q₈	—	4	4	no	—
45	C₂×C₇×Q₈	—	4	2	no	—
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂×C₇×Q₈	—	4	4	no	—
48	C₂²×C₄×C₇	—	2	4	no	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₇×C₄⋊C₄	—	2	8	no	—
55	C₇×C₄⋊C₄	—	4	8	no	—
56	C₇×C₄⋊C₄	—	4	8	no	—
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₇×C₄⋊C₄	—	4	2	no	—
59	C₄²×C₇	—	2	4	no	—
60	C₄²×C₇	—	2	2	no	—
61	C₄²×C₇	D₈	1	2	yes	—
62	C₄²×C₇	SD₁₆	1	2	yes	—
63	C₄²×C₇	SD₁₆	1	4	yes	—
64	C₄²×C₇	Q₁₆	1	4	yes	—
65	C₄²×C₇	C₂×D₄	1	2	yes	—
66	C₄²×C₇	#41	1	2	yes	—
67	C₂×C₇⋊C₈	—	2	4	no	—
68	C₂×C₇⋊C₈	—	4	2	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₂×C₇⋊C₈	—	4	2	no	—
71	C₂×C₇⋊C₈	—	4	4	no	—
72	C₂×C₄⋊Q₈	—	2	1	no	—
73	C₂×C₄⋊Q₈	—	2	1	no	—
74	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
75	C₂×C₄⋊C₈	—	7	2	no	—
76	C₂×C₄⋊C₈	—	7	2	no	—
77	C₂×C₄⋊C₈	—	14	2	no	—
78	C₂×C₄×C₈	—	14	1	no	—
79	#384	—	2	2	no	—
80	#384	Dic₇	1	4	yes	—
81	C₄³	C₂×D₇	1	1	yes	characteristic
82	#336	—	14	1	no	—
83	#332	—	14	8	no	—
84	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₇×Q₈	—	8	4	no	—
86	C₇×Q₈	—	8	4	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	4	4	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	—	4	4	no	—
93	C₂×C₄×C₇	#101	1	4	yes	—
94	C₂×C₄×C₇	#102	1	4	yes	—
95	C₂×C₄×C₇	#103	1	4	yes	—
96	C₂×C₄×C₇	#103	1	4	yes	—
97	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₇	#121	1	2	yes	—
99	C₂×C₄×C₇	#124	1	1	yes	characteristic
100	C₂×C₄×C₇	#127	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₇⋊C₈	—	4	4	no	—
107	C₇⋊C₈	—	8	2	no	—
108	C₂²×Q₈	—	2	2	no	—
109	C₂²×C₈	—	14	2	no	—
110	C₂²×C₈	—	14	2	no	—
111	C₄⋊Q₈	—	2	4	no	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	4	4	no	—
114	C₄⋊Q₈	—	4	2	no	—
115	C₄⋊Q₈	—	4	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
120	C₂×C₄²	C₂×Dic₇	1	2	yes	—
121	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
124	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
125	C₄⋊C₈	—	7	4	no	—
126	C₄⋊C₈	—	14	4	no	—
127	C₄⋊C₈	—	14	2	no	—
128	C₄⋊C₈	—	28	4	no	—
129	C₄×C₈	—	28	2	no	—
130	C₂²×C₇	#390	1	1	yes	characteristic
131	C₂²×C₇	C₂×#101	1	2	yes	—
132	C₂²×C₇	C₂×#102	1	1	yes	characteristic
133	C₂²×C₇	C₂×#103	1	2	yes	—
134	C₂²×C₇	#418	1	1	yes	characteristic
135	C₄×C₇	—	2	4	no	—
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	—	8	4	no	—
138	C₄×C₇	#332	1	8	yes	complemented
139	C₄×C₇	#459	1	2	yes	—
140	C₄×C₇	#460	1	2	yes	—
141	C₄×C₇	#487	1	2	yes	—
142	C₄×C₇	#492	1	1	yes	characteristic
143	C₄×C₇	#494	1	1	yes	characteristic
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
150	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
151	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
154	C₂×C₈	—	14	4	no	—
155	C₂×C₈	—	28	2	no	—
156	C₂×C₈	—	28	2	no	—
157	C₂×C₈	—	28	2	no	—
158	C₂×C₈	—	28	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	4	8	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	#1141	1	2	yes	—
167	C₄²	#1142	1	2	yes	—
168	C₄²	#1143	1	4	yes	—
169	C₄²	#1144	1	4	yes	—
170	C₄²	D₄×D₇	1	2	yes	—
171	C₄²	#1159	1	2	yes	—
172	C₂×C₇	C₂×#332	1	2	yes	—
173	C₂×C₇	#1549	1	2	yes	—
174	C₂×C₇	#1956	1	1	yes	characteristic
175	C₂×C₇	#1966	1	1	yes	characteristic
176	C₂×C₇	#1982	1	1	yes	characteristic
177	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
178	Q₈	—	8	4	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	#6406	1	4	yes	—
187	C₂×C₄	#6409	1	4	yes	—
188	C₂×C₄	#6409	1	4	yes	—
189	C₂×C₄	#6410	1	4	yes	—
190	C₂×C₄	C₂×#1141	1	1	yes	characteristic
191	C₂×C₄	#6495	1	2	yes	—
192	C₂×C₄	C₂×#1142	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₇	1	2	yes	—
194	C₂×C₄	#6499	1	1	yes	characteristic
195	C₂×C₄	#6501	1	2	yes	—
196	C₂×C₄	#6503	1	1	yes	characteristic
197	C₂×C₄	C₂×#1143	1	2	yes	—
198	C₂×C₄	C₂×#1144	1	2	yes	—
199	C₈	—	28	4	no	—
200	C₈	—	56	2	no	—
201	C₇	#9694	1	1	yes	characteristic, complemented, Sylow
202	C₂²	C₂×#6406	1	1	yes	characteristic
203	C₂²	#89987	1	1	yes	characteristic
204	C₂²	#89993	1	1	yes	characteristic
205	C₂²	C₂×#6409	1	2	yes	—
206	C₂²	C₂×#6410	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	#89342	1	8	yes	complemented
211	C₄	#89844	1	2	yes	—
212	C₄	#89847	1	2	yes	—
213	C₄	#89853	1	2	yes	—
214	C₄	#89857	1	1	yes	characteristic
215	C₄	#89864	1	1	yes	characteristic
216	C₂	#11729515	1	1	yes	characteristic
217	C₂	C₂×#89342	1	2	yes	—
218	C₂	#11729529	1	2	yes	—
219	C₂	#11729539	1	1	yes	characteristic
220	C₂	#11729543	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2868	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2868	D₇
hypocenter	C₇	#9694
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
138. C₄×C₇	#332	semidirect
201. C₇	#9694	semidirect
210. C₄	#89342	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2868	2
C₂×#89342	2
#11727148	2
C₄×C₄⋊(C₇⋊C₈)	2
#9694	7