Subgroups — C₃×#89900 #13055552

Order 1344.

244 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	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#1131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#89900	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₂²⋊D₇	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
12	C₃×C₇×M₁₆	—	2	2	no	—
13	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊Dic₇	C₄	1	1	yes	characteristic
16	C₃×(C₂×D₇)⋊C₄	C₄	1	1	yes	characteristic
17	C₃×Dic₇⋊C₄	C₄	1	1	yes	characteristic
18	C₃×C₄×Dic₇	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	C₃×#1131	—	2	2	no	—
22	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₄×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic
24	C₂×#1131	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²⋊(C₇⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
26	(C₂×D₇)⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	Dic₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#436	—	7	1	no	maximal, Hall
30	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×D₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	—	2	1	no	—
33	C₂×C₃×C₄×C₇	—	2	1	no	—
34	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₃×C₂²⋊D₇	—	2	2	no	—
37	C₂×C₃×Dic₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×Dic₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×Dic₇	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄×D₇	—	2	1	no	—
41	C₃×C₄×D₇	—	2	1	no	—
42	C₃×C₇×C₈	—	2	1	no	—
43	C₃×C₇×C₈	D₄	1	2	yes	complemented
44	C₃×C₇⋊C₈	—	2	1	no	—
45	C₃×C₇⋊C₈	—	2	1	no	—
46	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₂²⋊D₇	C₃×C₄	1	1	yes	characteristic
48	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
49	C₇×M₁₆	—	2	2	no	—
50	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²⋊Dic₇	C₃×C₄	1	1	yes	characteristic
53	(C₂×D₇)⋊C₄	C₃×C₄	1	1	yes	characteristic
54	Dic₇⋊C₄	C₃×C₄	1	1	yes	characteristic
55	C₄×Dic₇	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	#1131	—	2	2	no	—
59	C₂×C₃×M₁₆	—	7	1	no	—
60	C₂×C₃×M₁₆	D₇	1	1	yes	characteristic, complemented
61	C₃×C₄×D₄	—	7	1	no	—
62	C₃×C₄⋊C₈	—	7	1	no	—
63	C₃×C₂²⋊C₈	—	7	1	no	—
64	C₃×C₂²⋊C₈	—	7	1	no	—
65	C₃×C₄×C₈	—	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	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×D₇	—	2	1	no	—
71	C₂×C₃×D₇	—	2	1	no	—
72	C₂×C₃×D₇	—	2	1	no	—
73	C₃×C₄×C₇	—	2	1	no	—
74	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×C₇	#41	1	1	yes	characteristic
76	C₃×Dic₇	—	2	1	no	—
77	C₃×Dic₇	—	2	1	no	—
78	C₃×Dic₇	M₁₆	1	2	yes	complemented
79	#436	—	7	1	no	Sylow
80	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×D₇	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	—	2	1	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
86	C₂²⋊D₇	—	2	2	no	—
87	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
90	C₄×D₇	—	2	1	no	—
91	C₄×D₇	—	2	1	no	—
92	C₇×C₈	—	2	1	no	—
93	C₇×C₈	C₃×D₄	1	2	yes	complemented
94	C₇⋊C₈	—	2	1	no	—
95	C₇⋊C₈	—	2	1	no	—
96	C₂×C₃×D₄	—	7	1	no	—
97	C₂²×C₃×C₄	—	7	1	no	—
98	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	14	2	no	—
101	C₂×C₃×C₈	—	7	1	no	—
102	C₂×C₃×C₈	—	7	1	no	—
103	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	7	1	no	—
106	C₃×C₂²⋊C₄	—	7	1	no	—
107	C₃×C₂²⋊C₄	—	7	1	no	—
108	C₃×C₄²	—	7	1	no	—
109	C₂×C₃×C₇	—	4	1	no	—
110	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₃×C₇	#131	1	1	yes	characteristic
113	C₃×D₇	—	4	1	no	—
114	C₂×M₁₆	—	7	1	no	—
115	C₂×M₁₆	C₃×D₇	1	1	yes	characteristic, complemented
116	C₄×D₄	—	7	1	no	—
117	C₄⋊C₈	—	7	1	no	—
118	C₂²⋊C₈	—	7	1	no	—
119	C₂²⋊C₈	—	7	1	no	—
120	C₄×C₈	—	7	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	2	1	no	—
124	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×D₇	—	2	1	no	—
126	C₂×D₇	—	2	1	no	—
127	C₂×D₇	—	2	1	no	—
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
130	C₄×C₇	C₃×#41	1	1	yes	characteristic
131	Dic₇	—	2	1	no	—
132	Dic₇	—	2	1	no	—
133	Dic₇	C₃×M₁₆	1	2	yes	complemented
134	C₂³×C₃	—	7	1	no	—
135	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
136	C₃×D₄	—	14	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	7	1	no	—
140	C₂×C₃×C₄	—	7	1	no	—
141	C₂×C₃×C₄	—	7	1	no	—
142	C₂×C₃×C₄	—	14	1	no	—
143	C₂×C₃×C₄	—	14	1	no	—
144	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	14	1	no	—
148	C₃×C₈	—	14	1	no	—
149	C₃×C₈	C₂²⋊D₇	1	2	yes	complemented
150	C₃×C₇	#436	1	1	yes	characteristic, complemented, Hall
151	C₂×D₄	—	7	1	no	—
152	C₂²×C₄	—	7	1	no	—
153	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	14	2	no	—
156	C₂×C₈	—	7	1	no	—
157	C₂×C₈	—	7	1	no	—
158	C₂×C₈	C₂×C₃×D₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₃×D₇	1	1	yes	characteristic
160	C₄⋊C₄	—	7	1	no	—
161	C₂²⋊C₄	—	7	1	no	—
162	C₂²⋊C₄	—	7	1	no	—
163	C₄²	—	7	1	no	—
164	C₂×C₇	—	4	1	no	—
165	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
166	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
167	C₂×C₇	C₃×#131	1	1	yes	characteristic
168	D₇	—	4	1	no	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	14	1	no	—
173	C₂²×C₃	—	14	1	no	—
174	C₂²×C₃	—	14	1	no	—
175	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	7	2	no	—
178	C₃×C₄	—	14	1	no	—
179	C₃×C₄	—	14	1	no	—
180	C₃×C₄	#1157	1	1	yes	characteristic
181	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
182	C₂³	—	7	1	no	—
183	C₂³	C₃×C₄×D₇	1	1	yes	characteristic
184	D₄	—	14	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	7	1	no	—
188	C₂×C₄	—	7	1	no	—
189	C₂×C₄	—	7	1	no	—
190	C₂×C₄	—	14	1	no	—
191	C₂×C₄	—	14	1	no	—
192	C₂×C₄	C₃×C₄×D₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	14	1	no	—
196	C₈	—	14	1	no	—
197	C₈	C₃×C₂²⋊D₇	1	2	yes	complemented
198	C₇	C₃×#436	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₃	—	4	1	no	—
200	C₂×C₃	—	28	1	no	—
201	C₂×C₃	D₇×M₁₆	1	1	yes	characteristic
202	C₂×C₃	#6470	1	1	yes	characteristic
203	C₂×C₃	C₄×C₂²⋊D₇	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	14	1	no	—
208	C₂²	—	14	1	no	—
209	C₂²	—	14	1	no	—
210	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	7	2	no	—
213	C₄	—	14	1	no	—
214	C₄	—	14	1	no	—
215	C₄	C₃×#1157	1	1	yes	characteristic
216	C₄	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
217	C₃	#89900	1	1	yes	characteristic, direct product factor, Sylow
218	C₂	—	4	1	no	—
219	C₂	—	28	1	no	—
220	C₂	C₃×D₇×M₁₆	1	1	yes	characteristic
221	C₂	C₃×#6470	1	1	yes	characteristic
222	C₂	C₃×C₄×C₂²⋊D₇	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	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×C₂²⋊D₇	C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₇
hypocenter	C₇	C₃×#436
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×M₁₆	C₂	semidirect
3. C₂×C₃×#1131	C₂	semidirect
4. C₃×C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. C₃×(C₂×D₇)⋊C₈	C₂	semidirect
6. C₃×Dic₇⋊C₈	C₂	semidirect
7. C₃×C₈×Dic₇	C₂	semidirect
8. #89900	C₃	direct (N × H)
22. C₂×C₇×M₁₆	C₂×C₃	semidirect
24. C₂×#1131	C₂×C₃	semidirect
25. C₂²⋊(C₇⋊C₈)	C₂×C₃	semidirect
26. (C₂×D₇)⋊C₈	C₂×C₃	semidirect
27. Dic₇⋊C₈	C₂×C₃	semidirect
28. C₈×Dic₇	C₂×C₃	semidirect
43. C₃×C₇×C₈	D₄	semidirect
60. C₂×C₃×M₁₆	D₇	semidirect
78. C₃×Dic₇	M₁₆	semidirect
93. C₇×C₈	C₃×D₄	semidirect
115. C₂×M₁₆	C₃×D₇	semidirect
133. Dic₇	C₃×M₁₆	semidirect
149. C₃×C₈	C₂²⋊D₇	semidirect
150. C₃×C₇	#436	semidirect
197. C₈	C₃×C₂²⋊D₇	semidirect
198. C₇	C₃×#436	semidirect
217. C₃	#89900	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×M₁₆	2
C₃×C₄×C₂²⋊D₇	2
C₂×C₃×#1131	2
C₃×C₂²⋊(C₇⋊C₈)	2
C₃×(C₂×D₇)⋊C₈	2
C₃×Dic₇⋊C₈	2
C₃×C₈×Dic₇	2
#89900	3
C₃×#436	7

Subgroups — C3×#89900 #13055552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89900 #13055552