Subgroups — #13056648

Order 1344.

248 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	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#66627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620669	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620669	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊(C₃⋊(C₇⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620642	C₂	1	1	yes	maximal, characteristic, complemented
8	#89783	—	3	1	no	maximal, Hall
9	C₂×C₃×C₇×D₄	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₃×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₃×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₃×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
16	#66627	—	2	2	no	—
17	C₂×C₃⋊(C₇⋊C₈)	—	2	2	no	—
18	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₇×D₄	D₃	1	1	yes	characteristic
23	C₂×#1136	—	3	1	no	—
24	C₂²×C₇⋊C₈	—	3	1	no	—
25	C₂²⋊(C₇⋊C₈)	—	3	1	no	—
26	C₂²⋊(C₇⋊C₈)	—	3	1	no	—
27	C₄⋊(C₇⋊C₈)	—	3	1	no	—
28	#6377	—	3	1	no	—
29	#5011	—	7	1	no	maximal, Hall
30	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
32	C₃×C₇×D₄	—	2	2	no	—
33	C₂×C₃×C₄×C₇	—	2	1	no	—
34	C₂×C₃×C₄×C₇	—	2	1	no	—
35	C₂×C₃×C₄×C₇	—	2	1	no	—
36	C₂×C₃×C₄×C₇	—	2	1	no	—
37	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₃⋊(C₇⋊C₈)	—	2	1	no	—
43	C₃⋊(C₇⋊C₈)	—	2	1	no	—
44	C₃⋊(C₇⋊C₈)	—	2	1	no	—
45	C₃⋊(C₇⋊C₈)	D₄	1	2	yes	complemented
46	C₂×C₇×D₄	Dic₃	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
49	C₇×C₄⋊C₄	Dic₃	1	1	yes	characteristic
50	C₇×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
51	C₇×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
52	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
53	#1136	—	6	2	no	—
54	C₂×C₇⋊C₈	—	3	1	no	—
55	C₂×C₇⋊C₈	—	3	1	no	—
56	C₂×C₇⋊C₈	—	3	1	no	—
57	C₂×C₇⋊C₈	—	3	1	no	—
58	C₂×C₇⋊C₈	—	6	2	no	—
59	C₃×C₄×D₄	D₇	1	1	yes	characteristic
60	C₂×#208	—	7	1	no	—
61	C₂²×C₃⋊C₈	—	7	1	no	—
62	#830	—	7	1	no	—
63	#830	—	7	1	no	—
64	C₄⋊(C₃⋊C₈)	—	7	1	no	—
65	#803	—	7	1	no	—
66	C₂²×C₃×C₇	—	2	1	no	—
67	C₂²×C₃×C₇	—	2	1	no	—
68	C₂²×C₃×C₇	—	2	1	no	—
69	C₂²×C₃×C₇	—	2	1	no	—
70	C₂²×C₃×C₇	—	2	1	no	—
71	C₂²×C₃×C₇	M₁₆	1	2	yes	complemented
72	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₄×C₇	—	2	1	no	—
74	C₃×C₄×C₇	—	2	1	no	—
75	C₃×C₄×C₇	—	2	1	no	—
76	C₃×C₄×C₇	—	2	1	no	—
77	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×C₇	#41	1	1	yes	characteristic
79	#435	—	21	1	no	Sylow
80	C₂³×C₇	C₂×Dic₃	1	1	yes	characteristic
81	C₂³×C₇	C₂×Dic₃	1	1	yes	characteristic
82	C₇×D₄	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
92	C₇⋊C₈	—	3	2	no	—
93	C₇⋊C₈	—	6	1	no	—
94	C₇⋊C₈	—	6	1	no	—
95	C₇⋊C₈	—	6	1	no	—
96	C₂×C₃×D₄	Dic₇	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
99	C₃×C₄⋊C₄	Dic₇	1	1	yes	characteristic
100	C₃×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
101	C₃×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
102	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
103	#208	—	14	2	no	—
104	C₂×C₃⋊C₈	—	7	1	no	—
105	C₂×C₃⋊C₈	—	7	1	no	—
106	C₂×C₃⋊C₈	—	7	1	no	—
107	C₂×C₃⋊C₈	—	7	1	no	—
108	C₂×C₃⋊C₈	—	14	2	no	—
109	C₂×C₃×C₇	—	2	2	no	—
110	C₂×C₃×C₇	—	4	1	no	—
111	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₃×C₇	#131	1	1	yes	characteristic
114	C₂×M₁₆	—	21	1	no	—
115	C₂²×C₈	—	21	1	no	—
116	C₄×D₄	D₂₁	1	1	yes	characteristic
117	C₄⋊C₈	—	21	1	no	—
118	C₂²⋊C₈	—	21	1	no	—
119	C₂²⋊C₈	—	21	1	no	—
120	#97	—	21	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	2	1	no	—
124	C₂²×C₇	—	2	1	no	—
125	C₂²×C₇	—	2	1	no	—
126	C₂²×C₇	#208	1	2	yes	complemented
127	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	—	2	1	no	—
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	D₃×D₄	1	1	yes	characteristic
133	C₄×C₇	#237	1	1	yes	characteristic
134	C₂³×C₃	C₂×Dic₇	1	1	yes	characteristic
135	C₂³×C₃	C₂×Dic₇	1	1	yes	characteristic
136	C₃×D₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
146	C₃⋊C₈	—	7	2	no	—
147	C₃⋊C₈	—	14	1	no	—
148	C₃⋊C₈	—	14	1	no	—
149	C₃⋊C₈	—	14	1	no	—
150	C₃×C₇	#435	1	1	yes	characteristic, complemented, Hall
151	C₂×D₄	Dic₂₁	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
154	M₁₆	—	42	2	no	—
155	C₂×C₈	—	21	1	no	—
156	C₂×C₈	—	21	1	no	—
157	C₂×C₈	—	21	1	no	—
158	C₂×C₈	—	21	1	no	—
159	C₂×C₈	—	42	2	no	—
160	C₄⋊C₄	Dic₂₁	1	1	yes	characteristic
161	C₂²⋊C₄	Dic₂₁	1	1	yes	characteristic
162	C₂²⋊C₄	Dic₂₁	1	1	yes	characteristic
163	C₄²	C₂×D₂₁	1	1	yes	characteristic
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	—	4	1	no	—
166	C₂×C₇	C₂×#208	1	1	yes	characteristic
167	C₂×C₇	D₄×Dic₃	1	1	yes	characteristic
168	C₂×C₇	#948	1	1	yes	characteristic
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	#1136	1	2	yes	complemented
175	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	D₄×D₇	1	1	yes	characteristic
181	C₃×C₄	#1159	1	1	yes	characteristic
182	C₂³	C₂×Dic₂₁	1	1	yes	characteristic
183	C₂³	C₂×Dic₂₁	1	1	yes	characteristic
184	D₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	C₂×Dic₂₁	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₂₁	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₂₁	1	1	yes	characteristic
192	C₂×C₄	C₂×Dic₂₁	1	1	yes	characteristic
193	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
194	C₈	—	21	2	no	—
195	C₈	—	42	1	no	—
196	C₈	—	42	1	no	—
197	C₈	—	42	1	no	—
198	C₇	#5011	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₃	—	2	2	no	—
200	C₂×C₃	—	4	1	no	—
201	C₂×C₃	C₂×#1136	1	1	yes	characteristic
202	C₂×C₃	D₄×Dic₇	1	1	yes	characteristic
203	C₂×C₃	#6511	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	#66627	1	2	yes	complemented
210	C₂²	C₂²×Dic₂₁	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	D₄×D₂₁	1	1	yes	characteristic
216	C₄	#66730	1	1	yes	characteristic
217	C₃	#89783	1	1	yes	characteristic, complemented, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	4	1	no	—
220	C₂	C₂×#66627	1	1	yes	characteristic
221	C₂	D₄×Dic₂₁	1	1	yes	characteristic
222	C₂	#10621431	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₁
commutator, socle	C₂²×C₃×C₇	C₂²×C₄
Fitting	C₃×C₄×C₇×D₄	C₂
prime radical	C₂×C₃×C₇×D₄	C₄
cosocle	C₂×C₃×C₄×C₇	C₂³
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	#435
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#66627	C₂	semidirect
3. C₂²×C₃⋊(C₇⋊C₈)	C₂	semidirect
4. #10620669	C₂	semidirect
5. #10620669	C₂	semidirect
6. C₄⋊(C₃⋊(C₇⋊C₈))	C₂	semidirect
7. #10620642	C₂	semidirect
45. C₃⋊(C₇⋊C₈)	D₄	semidirect
71. C₂²×C₃×C₇	M₁₆	semidirect
126. C₂²×C₇	#208	semidirect
150. C₃×C₇	#435	semidirect
174. C₂²×C₃	#1136	semidirect
198. C₇	#5011	semidirect
209. C₂²	#66627	semidirect
217. C₃	#89783	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇×D₄	2
C₂×#66627	2
C₂²×C₃⋊(C₇⋊C₈)	2
#10620669	2
C₄⋊(C₃⋊(C₇⋊C₈))	2
#10620642	2
#89783	3
#5011	7