Subgroups — C₃×#89783 #13055444

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×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6377	C₂	1	1	yes	maximal, characteristic, complemented
8	#89783	C₃	1	1	yes	maximal, characteristic, direct product factor, 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	C₃×#1136	—	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	C₂×C₃	1	1	yes	characteristic
23	C₂×#1136	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²⋊(C₇⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²⋊(C₇⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄⋊(C₇⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
28	#6377	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#435	—	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	C₃×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
49	C₇×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
50	C₇×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
51	C₇×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
52	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
53	#1136	—	2	2	no	—
54	C₂×C₇⋊C₈	—	2	2	no	—
55	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×M₁₆	—	7	1	no	—
60	C₂²×C₃×C₈	—	7	1	no	—
61	C₃×C₄×D₄	D₇	1	1	yes	characteristic
62	C₃×C₄⋊C₈	—	7	1	no	—
63	C₃×C₂²⋊C₈	—	7	1	no	—
64	C₃×C₂²⋊C₈	—	7	1	no	—
65	C₃×#97	—	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	—	7	1	no	Sylow
80	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂³×C₇	C₂×C₃×C₄	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×C₃×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₇⋊C₈	—	2	1	no	—
93	C₇⋊C₈	—	2	1	no	—
94	C₇⋊C₈	—	2	1	no	—
95	C₇⋊C₈	C₃×D₄	1	2	yes	complemented
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×M₁₆	—	14	2	no	—
100	C₂×C₃×C₈	—	7	1	no	—
101	C₂×C₃×C₈	—	7	1	no	—
102	C₂×C₃×C₈	—	7	1	no	—
103	C₂×C₃×C₈	—	7	1	no	—
104	C₂×C₃×C₈	—	14	2	no	—
105	C₃×C₄⋊C₄	Dic₇	1	1	yes	characteristic
106	C₃×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
107	C₃×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
108	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
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	—	7	1	no	—
115	C₂²×C₈	—	7	1	no	—
116	C₄×D₄	C₃×D₇	1	1	yes	characteristic
117	C₄⋊C₈	—	7	1	no	—
118	C₂²⋊C₈	—	7	1	no	—
119	C₂²⋊C₈	—	7	1	no	—
120	#97	—	7	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	C₃×M₁₆	1	2	yes	complemented
127	C₂²×C₇	C₂²×C₃×C₄	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	C₂×C₃×D₄	1	1	yes	characteristic
133	C₄×C₇	C₃×#41	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	C₃×Dic₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
154	M₁₆	—	14	2	no	—
155	C₂×C₈	—	7	1	no	—
156	C₂×C₈	—	7	1	no	—
157	C₂×C₈	—	7	1	no	—
158	C₂×C₈	—	7	1	no	—
159	C₂×C₈	—	14	2	no	—
160	C₄⋊C₄	C₃×Dic₇	1	1	yes	characteristic
161	C₂²⋊C₄	C₃×Dic₇	1	1	yes	characteristic
162	C₂²⋊C₄	C₃×Dic₇	1	1	yes	characteristic
163	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	—	4	1	no	—
166	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
167	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
168	C₂×C₇	C₃×#131	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×C₃×Dic₇	1	1	yes	characteristic
183	C₂³	C₂×C₃×Dic₇	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×C₃×Dic₇	1	1	yes	characteristic
190	C₂×C₄	C₂×C₃×Dic₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₃×Dic₇	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃×Dic₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
194	C₈	—	7	2	no	—
195	C₈	—	14	1	no	—
196	C₈	—	14	1	no	—
197	C₈	—	14	1	no	—
198	C₇	C₃×#435	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	C₃×#1136	1	2	yes	complemented
210	C₂²	C₂²×C₃×Dic₇	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	C₃×D₄×D₇	1	1	yes	characteristic
216	C₄	C₃×#1159	1	1	yes	characteristic
217	C₃	#89783	1	1	yes	characteristic, direct product factor, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	4	1	no	—
220	C₂	C₂×C₃×#1136	1	1	yes	characteristic
221	C₂	C₃×D₄×Dic₇	1	1	yes	characteristic
222	C₂	C₃×#6511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×C₇×D₄	C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#435
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#1136	C₂	semidirect
3. C₂²×C₃×C₇⋊C₈	C₂	semidirect
4. C₃×C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. C₃×C₂²⋊(C₇⋊C₈)	C₂	semidirect
6. C₃×C₄⋊(C₇⋊C₈)	C₂	semidirect
7. C₃×#6377	C₂	semidirect
8. #89783	C₃	direct (N × H)
23. C₂×#1136	C₂×C₃	semidirect
24. C₂²×C₇⋊C₈	C₂×C₃	semidirect
25. C₂²⋊(C₇⋊C₈)	C₂×C₃	semidirect
26. C₂²⋊(C₇⋊C₈)	C₂×C₃	semidirect
27. C₄⋊(C₇⋊C₈)	C₂×C₃	semidirect
28. #6377	C₂×C₃	semidirect
45. C₃×C₇⋊C₈	D₄	semidirect
71. C₂²×C₃×C₇	M₁₆	semidirect
95. C₇⋊C₈	C₃×D₄	semidirect
126. C₂²×C₇	C₃×M₁₆	semidirect
150. C₃×C₇	#435	semidirect
174. C₂²×C₃	#1136	semidirect
198. C₇	C₃×#435	semidirect
209. C₂²	C₃×#1136	semidirect
217. C₃	#89783	direct (N × H)

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×C₃×#1136	2
C₂²×C₃×C₇⋊C₈	2
C₃×C₂²⋊(C₇⋊C₈)	2
C₃×C₄⋊(C₇⋊C₈)	2
C₃×#6377	2
#89783	3
C₃×#435	7

Subgroups — C3×#89783 #13055444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89783 #13055444