Subgroups — C₃×#89438 #13055079

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×Dic₇	C₂	1	2	yes	maximal, complemented
4	C₃×C₇×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	#89438	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
11	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₇	—	2	4	no	—
13	C₂×C₃×C₄×D₇	C₄	1	4	yes	complemented
14	C₃×(C₂×D₇)⋊C₄	C₄	1	8	yes	complemented
15	C₃×C₄×Dic₇	—	2	4	no	—
16	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic
17	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	—
18	C₂×C₄×Dic₇	C₂×C₃	1	2	yes	complemented
19	C₇×#95	C₂×C₃	1	1	yes	characteristic, complemented
20	C₇⋊#95	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₄×C₂²⋊C₄	—	7	1	no	maximal, Hall
22	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
23	C₂²×C₃×D₇	—	2	2	no	—
24	C₂²×C₃×D₇	—	2	1	no	—
25	C₂²×C₃×D₇	—	2	1	no	—
26	C₂²×C₃×D₇	C₂×C₄	1	4	yes	—
27	C₂²×C₃×D₇	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₇	—	2	2	no	—
29	C₂×C₃×C₄×C₇	—	2	2	no	—
30	C₂×C₃×C₄×C₇	—	2	2	no	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
33	C₂×C₃×Dic₇	—	2	2	no	—
34	C₂×C₃×Dic₇	—	2	2	no	—
35	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
36	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	complemented
37	C₂×C₃×Dic₇	D₄	1	4	yes	—
38	C₃×C₄×D₇	—	2	8	no	—
39	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
42	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
43	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₄×D₇	—	2	4	no	—
45	C₂×C₄×D₇	C₃×C₄	1	4	yes	complemented
46	(C₂×D₇)⋊C₄	C₃×C₄	1	8	yes	complemented
47	C₄×Dic₇	—	2	4	no	—
48	C₂³×C₃×C₄	—	7	1	no	—
49	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
50	C₂×C₃×C₄²	—	7	2	no	—
51	C₃×#95	—	7	1	no	—
52	C₃×#95	D₇	1	1	yes	characteristic, complemented
53	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
56	C₂²×C₃×C₇	#41	1	1	yes	characteristic
57	C₂²×C₃×C₇	#41	1	1	yes	characteristic
58	C₂×C₃×D₇	—	2	2	no	—
59	C₂×C₃×D₇	—	2	2	no	—
60	C₂×C₃×D₇	—	2	4	no	—
61	C₂×C₃×D₇	—	2	4	no	—
62	C₂×C₃×D₇	C₄²	1	4	yes	complemented
63	C₃×C₄×C₇	—	2	2	no	—
64	C₃×C₄×C₇	—	2	4	no	—
65	C₃×Dic₇	—	2	4	no	—
66	C₃×Dic₇	C₂²⋊C₄	1	4	yes	complemented
67	C₄×C₂²⋊C₄	—	7	1	no	Sylow
68	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
69	C₂²×D₇	—	2	2	no	—
70	C₂²×D₇	—	2	1	no	—
71	C₂²×D₇	—	2	1	no	—
72	C₂²×D₇	C₂×C₃×C₄	1	4	yes	—
73	C₂²×D₇	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
79	C₂×Dic₇	—	2	2	no	—
80	C₂×Dic₇	—	2	2	no	—
81	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
82	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	complemented
83	C₂×Dic₇	C₃×D₄	1	4	yes	—
84	C₄×D₇	—	2	8	no	—
85	C₂⁴×C₃	—	7	1	no	—
86	C₂²×C₃×C₄	—	7	1	no	—
87	C₂²×C₃×C₄	—	7	2	no	—
88	C₂²×C₃×C₄	—	7	4	no	—
89	C₂²×C₃×C₄	—	14	4	no	—
90	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
92	C₃×C₂²⋊C₄	—	7	8	no	—
93	C₃×C₄²	—	14	4	no	—
94	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₃×C₇	#117	1	1	yes	characteristic
97	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
98	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
99	C₃×D₇	—	2	4	no	—
100	C₂³×C₄	—	7	1	no	—
101	C₂×C₂²⋊C₄	—	7	2	no	—
102	C₂×C₄²	—	7	2	no	—
103	#95	—	7	1	no	—
104	#95	C₃×D₇	1	1	yes	characteristic, complemented
105	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
106	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
107	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
108	C₂²×C₇	C₃×#41	1	1	yes	characteristic
109	C₂²×C₇	C₃×#41	1	1	yes	characteristic
110	C₂×D₇	—	2	2	no	—
111	C₂×D₇	—	2	2	no	—
112	C₂×D₇	—	2	4	no	—
113	C₂×D₇	—	2	4	no	—
114	C₂×D₇	C₃×C₄²	1	4	yes	complemented
115	C₄×C₇	—	2	2	no	—
116	C₄×C₇	—	2	4	no	—
117	Dic₇	—	2	4	no	—
118	Dic₇	C₃×C₂²⋊C₄	1	4	yes	complemented
119	C₂³×C₃	—	7	4	no	—
120	C₂³×C₃	—	7	2	no	—
121	C₂³×C₃	—	14	2	no	—
122	C₂³×C₃	—	14	1	no	—
123	C₂³×C₃	—	14	1	no	—
124	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	7	2	no	—
129	C₂×C₃×C₄	—	7	4	no	—
130	C₂×C₃×C₄	—	7	4	no	—
131	C₂×C₃×C₄	—	14	8	no	—
132	C₂×C₃×C₄	—	14	2	no	—
133	C₂×C₃×C₄	—	14	2	no	—
134	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
135	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
136	C₃×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
137	C₂⁴	—	7	1	no	—
138	C₂²×C₄	—	7	1	no	—
139	C₂²×C₄	—	7	2	no	—
140	C₂²×C₄	—	7	4	no	—
141	C₂²×C₄	—	14	4	no	—
142	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
144	C₂²⋊C₄	—	7	8	no	—
145	C₄²	—	14	4	no	—
146	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
147	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₇	C₃×#117	1	1	yes	characteristic
149	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
150	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
151	D₇	—	2	4	no	—
152	C₂²×C₃	—	7	4	no	—
153	C₂²×C₃	—	14	2	no	—
154	C₂²×C₃	—	14	4	no	—
155	C₂²×C₃	—	14	2	no	—
156	C₂²×C₃	—	14	4	no	—
157	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
158	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
159	C₂²×C₃	D₄×D₇	1	2	yes	—
160	C₂²×C₃	#1159	1	1	yes	characteristic
161	C₂²×C₃	#1161	1	1	yes	characteristic
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	7	4	no	—
165	C₃×C₄	—	14	4	no	—
166	C₂³	—	7	4	no	—
167	C₂³	—	7	2	no	—
168	C₂³	—	14	2	no	—
169	C₂³	—	14	1	no	—
170	C₂³	—	14	1	no	—
171	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	7	2	no	—
176	C₂×C₄	—	7	4	no	—
177	C₂×C₄	—	7	4	no	—
178	C₂×C₄	—	14	8	no	—
179	C₂×C₄	—	14	2	no	—
180	C₂×C₄	—	14	2	no	—
181	C₂×C₄	C₃×C₄×D₇	1	2	yes	—
182	C₂×C₄	C₃×C₄×D₇	1	4	yes	complemented
183	C₇	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₃	—	14	4	no	—
185	C₂×C₃	C₄²×D₇	1	1	yes	characteristic
186	C₂×C₃	D₇×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₃	(C₂²⋊D₇)⋊C₄	1	2	yes	—
188	C₂×C₃	#6455	1	1	yes	characteristic
189	C₂×C₃	#6456	1	2	yes	—
190	C₂²	—	7	4	no	—
191	C₂²	—	14	2	no	—
192	C₂²	—	14	4	no	—
193	C₂²	—	14	2	no	—
194	C₂²	—	14	4	no	—
195	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
196	C₂²	C₂×C₃×C₄×D₇	1	2	yes	—
197	C₂²	C₃×D₄×D₇	1	2	yes	—
198	C₂²	C₃×#1159	1	1	yes	characteristic
199	C₂²	C₃×#1161	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	7	4	no	—
203	C₄	—	14	4	no	—
204	C₃	#89438	1	1	yes	characteristic, direct product factor, Sylow
205	C₂	—	14	4	no	—
206	C₂	C₃×C₄²×D₇	1	1	yes	characteristic
207	C₂	C₃×D₇×C₂²⋊C₄	1	1	yes	characteristic
208	C₂	C₃×(C₂²⋊D₇)⋊C₄	1	2	yes	—
209	C₂	C₃×#6455	1	1	yes	characteristic
210	C₂	C₃×#6456	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₃×C₇×#95	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₃×#95	D₇
hypocenter	C₇	C₃×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄×Dic₇	C₂	semidirect
4. C₃×C₇×#95	C₂	semidirect
5. C₃×C₇⋊#95	C₂	semidirect
6. #89438	C₃	direct (N × H)
13. C₂×C₃×C₄×D₇	C₄	semidirect
14. C₃×(C₂×D₇)⋊C₄	C₄	semidirect
18. C₂×C₄×Dic₇	C₂×C₃	semidirect
19. C₇×#95	C₂×C₃	semidirect
20. C₇⋊#95	C₂×C₃	semidirect
32. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
36. C₂×C₃×Dic₇	C₂×C₄	semidirect
45. C₂×C₄×D₇	C₃×C₄	semidirect
46. (C₂×D₇)⋊C₄	C₃×C₄	semidirect
52. C₃×#95	D₇	semidirect
62. C₂×C₃×D₇	C₄²	semidirect
66. C₃×Dic₇	C₂²⋊C₄	semidirect
78. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
82. C₂×Dic₇	C₂×C₃×C₄	semidirect
104. #95	C₃×D₇	semidirect
114. C₂×D₇	C₃×C₄²	semidirect
118. Dic₇	C₃×C₂²⋊C₄	semidirect
135. C₂×C₃×C₄	C₄×D₇	semidirect
136. C₃×C₇	C₄×C₂²⋊C₄	semidirect
182. C₂×C₄	C₃×C₄×D₇	semidirect
183. C₇	C₃×C₄×C₂²⋊C₄	semidirect
204. C₃	#89438	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×D₇	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×C₃×C₄×Dic₇	2
C₃×C₇×#95	2
C₃×C₇⋊#95	2
#89438	3
C₃×C₄×C₂²⋊C₄	7

Subgroups — C3×#89438 #13055079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89438 #13055079