Subgroups — C₂×C₃×#6379 #13055370

Order 1344.

340 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6379	C₂	1	4	yes	maximal, direct product factor
5	C₂×#6379	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₄×C₇	—	2	1	no	—
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₂²⋊D₇	—	2	1	no	—
9	C₃×#1157	—	2	1	no	—
10	C₃×#1157	—	2	1	no	—
11	C₃×#1157	C₄	1	2	yes	complemented
12	C₃×#1157	C₂²	1	2	yes	—
13	C₂×C₃×D₂₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃×C₄×D₇	—	2	1	no	—
15	C₂×C₃×Dic₁₄	C₄	1	1	yes	characteristic, complemented
16	C₃×C₄²×C₇	—	2	1	no	—
17	C₃×C₄²×C₇	C₂²	1	2	yes	complemented
18	C₃×#1136	—	2	1	no	—
19	C₃×#1136	C₂²	1	2	yes	complemented
20	C₂×C₃×C₇⋊C₈	—	2	1	no	—
21	C₂×#1157	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#1136	C₂×C₃	1	1	yes	characteristic, complemented
24	#6379	C₂×C₃	1	4	yes	direct product factor
25	C₂×C₃×#104	—	7	1	no	maximal, Hall
26	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
27	C₂²×C₃×D₇	—	2	1	no	—
28	C₂×C₃×C₄×C₇	—	2	4	no	—
29	C₂×C₃×C₄×C₇	—	2	2	no	—
30	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₃×C₂²⋊D₇	—	2	2	no	—
37	C₃×C₂²⋊D₇	—	2	2	no	—
38	C₂×C₃×Dic₇	—	2	1	no	—
39	C₃×D₂₈	—	2	1	no	—
40	C₃×D₂₈	C₂×C₄	1	2	yes	complemented
41	C₃×C₄×D₇	—	2	2	no	—
42	C₃×C₄×D₇	—	2	2	no	—
43	C₃×Dic₁₄	—	2	1	no	—
44	C₃×Dic₁₄	C₂×C₄	1	2	yes	complemented
45	C₃×C₇⋊C₈	—	2	2	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₂²⋊D₇	—	2	1	no	—
49	#1157	—	2	1	no	—
50	#1157	—	2	1	no	—
51	#1157	C₃×C₄	1	2	yes	complemented
52	#1157	C₂²×C₃	1	2	yes	—
53	C₂×D₂₈	C₃×C₄	1	1	yes	characteristic, complemented
54	C₂×C₄×D₇	—	2	1	no	—
55	C₂×Dic₁₄	C₃×C₄	1	1	yes	characteristic, complemented
56	C₄²×C₇	—	2	1	no	—
57	C₄²×C₇	C₂²×C₃	1	2	yes	complemented
58	#1136	—	2	1	no	—
59	#1136	C₂²×C₃	1	2	yes	complemented
60	C₂×C₇⋊C₈	—	2	1	no	—
61	C₂×C₃×#41	—	7	1	no	—
62	C₂×C₃×M₁₆	—	7	1	no	—
63	C₂×C₃×C₄²	D₇	1	1	yes	characteristic, complemented
64	C₃×#104	—	7	4	no	—
65	C₂²×C₃×C₇	—	2	2	no	—
66	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×D₇	—	2	2	no	—
70	C₂×C₃×D₇	—	4	2	no	—
71	C₃×C₄×C₇	—	2	4	no	—
72	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₃×Dic₇	—	2	2	no	—
77	C₂×#104	—	7	1	no	Sylow
78	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
79	C₂²×D₇	—	2	1	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
88	C₂²⋊D₇	—	2	2	no	—
89	C₂²⋊D₇	—	2	2	no	—
90	C₂×Dic₇	—	2	1	no	—
91	D₂₈	—	2	1	no	—
92	D₂₈	C₂×C₃×C₄	1	2	yes	complemented
93	C₄×D₇	—	2	2	no	—
94	C₄×D₇	—	2	2	no	—
95	Dic₁₄	—	2	1	no	—
96	Dic₁₄	C₂×C₃×C₄	1	2	yes	complemented
97	C₇⋊C₈	—	2	2	no	—
98	C₃×#41	—	7	2	no	—
99	C₃×#41	—	7	2	no	—
100	C₃×#41	—	14	1	no	—
101	C₃×#41	—	14	1	no	—
102	C₂×C₃×Q₈	—	7	1	no	—
103	C₂×C₃×D₄	—	7	1	no	—
104	C₂×C₃×D₄	—	14	1	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	14	1	no	—
107	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
108	C₃×M₁₆	—	7	2	no	—
109	C₃×M₁₆	—	14	1	no	—
110	C₂×C₃×C₈	—	14	1	no	—
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₂×D₇	1	2	yes	complemented
113	C₂×C₃×C₇	—	2	1	no	—
114	C₂×C₃×C₇	—	2	1	no	—
115	C₂×C₃×C₇	#104	1	2	yes	complemented
116	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₃×D₇	—	4	2	no	—
118	C₂×#41	—	7	1	no	—
119	C₂×M₁₆	—	7	1	no	—
120	C₂×C₄²	C₃×D₇	1	1	yes	characteristic, complemented
121	#104	—	7	4	no	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
126	C₂×D₇	—	2	2	no	—
127	C₂×D₇	—	4	2	no	—
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
133	Dic₇	—	2	2	no	—
134	C₂³×C₃	—	14	1	no	—
135	C₂³×C₃	C₂²⋊D₇	1	1	yes	characteristic
136	C₃×Q₈	—	7	2	no	—
137	C₃×Q₈	—	14	1	no	—
138	C₃×D₄	—	7	2	no	—
139	C₃×D₄	—	14	2	no	—
140	C₃×D₄	—	14	2	no	—
141	C₃×D₄	—	14	1	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	14	1	no	—
145	C₂×C₃×C₄	—	14	2	no	—
146	C₂×C₃×C₄	—	14	2	no	—
147	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
148	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
149	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
150	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
153	C₃×C₈	—	14	2	no	—
154	C₃×C₇	C₂×#104	1	1	yes	characteristic, complemented, Hall
155	#41	—	7	2	no	—
156	#41	—	7	2	no	—
157	#41	—	14	1	no	—
158	#41	—	14	1	no	—
159	C₂×Q₈	—	7	1	no	—
160	C₂×D₄	—	7	1	no	—
161	C₂×D₄	—	14	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	14	1	no	—
164	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
165	M₁₆	—	7	2	no	—
166	M₁₆	—	14	1	no	—
167	C₂×C₈	—	14	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	C₂×C₃×D₇	1	2	yes	complemented
170	C₂×C₇	—	2	1	no	—
171	C₂×C₇	—	2	1	no	—
172	C₂×C₇	C₃×#104	1	2	yes	complemented
173	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
174	D₇	—	4	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	14	2	no	—
177	C₂²×C₃	—	28	2	no	—
178	C₂²×C₃	(C₂×D₇)⋊C₄	1	1	yes	characteristic
179	C₂²×C₃	(C₂×D₇)⋊C₄	1	1	yes	characteristic
180	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	14	2	no	—
183	C₃×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
184	C₃×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
185	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
186	C₃×C₄	C₂×D₂₈	1	1	yes	characteristic
187	C₂³	—	14	1	no	—
188	C₂³	C₃×C₂²⋊D₇	1	1	yes	characteristic
189	Q₈	—	7	2	no	—
190	Q₈	—	14	1	no	—
191	D₄	—	7	2	no	—
192	D₄	—	14	2	no	—
193	D₄	—	14	2	no	—
194	D₄	—	14	1	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	14	1	no	—
198	C₂×C₄	—	14	2	no	—
199	C₂×C₄	—	14	2	no	—
200	C₂×C₄	C₃×C₄×D₇	1	1	yes	characteristic
201	C₂×C₄	C₃×C₄×D₇	1	1	yes	characteristic
202	C₂×C₄	C₃×D₂₈	1	1	yes	characteristic
203	C₂×C₄	C₃×D₂₈	1	1	yes	characteristic
204	C₂×C₄	C₃×C₂²⋊D₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
206	C₈	—	14	2	no	—
207	C₇	C₂×C₃×#104	1	1	yes	characteristic, complemented, Sylow
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	28	2	no	—
211	C₂×C₃	#6379	1	2	yes	direct product factor
212	C₂×C₃	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	14	2	no	—
215	C₂²	—	28	2	no	—
216	C₂²	C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
217	C₂²	C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
218	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	14	2	no	—
221	C₄	C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
222	C₄	C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
223	C₄	C₂×C₃×C₄×D₇	1	1	yes	characteristic
224	C₄	C₂×C₃×D₂₈	1	1	yes	characteristic
225	C₃	C₂×#6379	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	28	2	no	—
229	C₂	C₃×#6379	1	2	yes	direct product factor
230	C₂	C₂×C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₂₈
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂²×C₃×C₇	C₂²⋊C₄
prime radical	C₂×C₃×#1157	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₇
hypocenter	C₇	C₂×C₃×#104
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₇	C₂	semidirect
3. C₂×C₃×#1136	C₂	semidirect
4. C₃×#6379	C₂	direct (N × H)
5. C₂×#6379	C₃	direct (N × H)
11. C₃×#1157	C₄	semidirect
13. C₂×C₃×D₂₈	C₄	semidirect
15. C₂×C₃×Dic₁₄	C₄	semidirect
17. C₃×C₄²×C₇	C₂²	semidirect
19. C₃×#1136	C₂²	semidirect
22. C₂×C₄²×C₇	C₂×C₃	semidirect
23. C₂×#1136	C₂×C₃	semidirect
24. #6379	C₂×C₃	direct (N × H)
40. C₃×D₂₈	C₂×C₄	semidirect
44. C₃×Dic₁₄	C₂×C₄	semidirect
51. #1157	C₃×C₄	semidirect
53. C₂×D₂₈	C₃×C₄	semidirect
55. C₂×Dic₁₄	C₃×C₄	semidirect
57. C₄²×C₇	C₂²×C₃	semidirect
59. #1136	C₂²×C₃	semidirect
63. C₂×C₃×C₄²	D₇	semidirect
92. D₂₈	C₂×C₃×C₄	semidirect
96. Dic₁₄	C₂×C₃×C₄	semidirect
112. C₃×C₄²	C₂×D₇	semidirect
115. C₂×C₃×C₇	#104	semidirect
120. C₂×C₄²	C₃×D₇	semidirect
154. C₃×C₇	C₂×#104	semidirect
169. C₄²	C₂×C₃×D₇	semidirect
172. C₂×C₇	C₃×#104	semidirect
207. C₇	C₂×C₃×#104	semidirect
211. C₂×C₃	#6379	direct (N × H)
225. C₃	C₂×#6379	direct (N × H)
229. C₂	C₃×#6379	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×#1157	2
C₂×C₃×C₄²×C₇	2
C₂×C₃×#1136	2
C₃×#6379	2
C₂×#6379	3
C₂×C₃×#104	7

Subgroups — C2×C3×#6379 #13055370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6379 #13055370