Subgroups — C₂×C₃×#6495 #13061141

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1142	C₂	1	2	yes	maximal, complemented
4	C₃×#6495	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#1141	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6495	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₃×C₇	—	2	1	no	—
9	C₂×C₃×C₇×D₄	—	2	1	no	—
10	C₂×C₃×C₇×D₄	—	2	2	no	—
11	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₇×D₄	C₂²	1	4	yes	complemented
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₇	—	2	1	no	—
15	C₃×#1157	—	2	2	no	—
16	C₃×#1157	C₂²	1	4	yes	complemented
17	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃×C₄×D₇	—	2	1	no	—
19	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
20	C₃×#1142	C₂²	1	8	yes	complemented
21	C₃×#1141	C₂²	1	8	yes	complemented
22	C₃×#1136	C₂²	1	4	yes	complemented
23	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	complemented
24	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#1142	C₂×C₃	1	2	yes	complemented
27	#6495	C₂×C₃	1	8	yes	direct product factor
28	C₂×#1141	C₂×C₃	1	2	yes	complemented
29	C₂×#1136	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×AGL(1,ℤ/8)	—	7	1	no	maximal, Hall
31	C₂³×C₃×C₇	—	2	2	no	—
32	C₂³×C₃×C₇	—	2	4	no	—
33	C₂³×C₃×C₇	—	4	4	no	—
34	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
35	C₂²×C₃×D₇	—	2	1	no	—
36	C₃×C₇×D₄	—	2	2	no	—
37	C₃×C₇×D₄	—	2	4	no	—
38	C₃×C₇×D₄	C₂³	1	4	yes	—
39	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
43	C₃×C₂²⋊D₇	—	2	4	no	—
44	C₂×C₃×Dic₇	—	2	1	no	—
45	C₃×D₂₈	—	2	1	no	—
46	C₃×D₂₈	C₂³	1	2	yes	complemented
47	C₃×C₄×D₇	—	2	4	no	—
48	C₃×Dic₁₄	—	2	1	no	—
49	C₃×Dic₁₄	C₂³	1	2	yes	complemented
50	C₃×C₇⋊C₈	C₂³	1	4	yes	complemented
51	C₂⁴×C₇	—	2	1	no	—
52	C₂×C₇×D₄	—	2	1	no	—
53	C₂×C₇×D₄	—	2	2	no	—
54	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
55	C₂×C₇×D₄	C₂²×C₃	1	4	yes	complemented
56	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₂²⋊D₇	—	2	1	no	—
58	#1157	—	2	2	no	—
59	#1157	C₂²×C₃	1	4	yes	complemented
60	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic, complemented
61	C₂×C₄×D₇	—	2	1	no	—
62	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
63	#1142	C₂²×C₃	1	8	yes	complemented
64	#1141	C₂²×C₃	1	8	yes	complemented
65	#1136	C₂²×C₃	1	4	yes	complemented
66	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	complemented
67	C₂×C₃×#41	—	7	1	no	—
68	C₂²×C₃×D₄	D₇	1	1	yes	characteristic, complemented
69	C₃×AGL(1,ℤ/8)	—	7	8	no	—
70	C₂×C₃×SD₁₆	—	7	2	no	—
71	C₂×C₃×D₈	—	7	2	no	—
72	C₂×C₃×M₁₆	—	7	1	no	—
73	C₂²×C₃×C₇	—	2	2	no	—
74	C₂²×C₃×C₇	—	2	4	no	—
75	C₂²×C₃×C₇	—	4	8	no	—
76	C₂²×C₃×C₇	—	4	4	no	—
77	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
78	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×D₇	—	2	2	no	—
80	C₂×C₃×D₇	—	4	2	no	—
81	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	C₃×Dic₇	—	2	2	no	—
85	C₂×AGL(1,ℤ/8)	—	7	1	no	Sylow
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	—	2	4	no	—
88	C₂³×C₇	—	4	4	no	—
89	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
90	C₂²×D₇	—	2	1	no	—
91	C₇×D₄	—	2	2	no	—
92	C₇×D₄	—	2	4	no	—
93	C₇×D₄	C₂³×C₃	1	4	yes	—
94	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
98	C₂²⋊D₇	—	2	4	no	—
99	C₂×Dic₇	—	2	1	no	—
100	D₂₈	—	2	1	no	—
101	D₂₈	C₂³×C₃	1	2	yes	complemented
102	C₄×D₇	—	2	4	no	—
103	Dic₁₄	—	2	1	no	—
104	Dic₁₄	C₂³×C₃	1	2	yes	complemented
105	C₇⋊C₈	C₂³×C₃	1	4	yes	complemented
106	C₂⁴×C₃	—	2	1	no	—
107	C₃×#41	—	7	4	no	—
108	C₃×#41	—	14	2	no	—
109	C₂×C₃×Q₈	—	7	1	no	—
110	C₂×C₃×D₄	—	2	2	no	—
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂×C₃×D₄	—	7	1	no	—
113	C₂×C₃×D₄	—	14	1	no	—
114	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
115	C₂×C₃×D₄	C₂×D₇	1	4	yes	complemented
116	C₂²×C₃×C₄	—	14	1	no	—
117	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
118	C₃×SD₁₆	—	7	8	no	—
119	C₃×D₈	—	7	8	no	—
120	C₃×M₁₆	—	7	4	no	—
121	C₂×C₃×C₈	—	7	2	no	—
122	C₂×C₃×C₇	—	2	2	no	—
123	C₂×C₃×C₇	—	4	4	no	—
124	C₂×C₃×C₇	AGL(1,ℤ/8)	1	2	yes	complemented
125	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
126	C₃×D₇	—	4	2	no	—
127	C₂×#41	—	7	1	no	—
128	C₂²×D₄	C₃×D₇	1	1	yes	characteristic, complemented
129	AGL(1,ℤ/8)	—	7	8	no	—
130	C₂×SD₁₆	—	7	2	no	—
131	C₂×D₈	—	7	2	no	—
132	C₂×M₁₆	—	7	1	no	—
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	—	2	4	no	—
135	C₂²×C₇	—	4	8	no	—
136	C₂²×C₇	—	4	4	no	—
137	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
138	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×D₇	—	2	2	no	—
140	C₂×D₇	—	4	2	no	—
141	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
142	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
143	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
144	Dic₇	—	2	2	no	—
145	C₂³×C₃	—	2	4	no	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	—	4	4	no	—
148	C₂³×C₃	—	14	1	no	—
149	C₂³×C₃	C₂²⋊D₇	1	1	yes	characteristic
150	C₃×Q₈	—	7	2	no	—
151	C₃×Q₈	—	14	1	no	—
152	C₃×D₄	—	2	2	no	—
153	C₃×D₄	—	2	4	no	—
154	C₃×D₄	—	7	2	no	—
155	C₃×D₄	—	14	4	no	—
156	C₃×D₄	—	14	1	no	—
157	C₃×D₄	C₂²×D₇	1	4	yes	—
158	C₂×C₃×C₄	—	14	1	no	—
159	C₂×C₃×C₄	—	14	4	no	—
160	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
161	C₂×C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
163	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
164	C₃×C₈	—	7	4	no	—
165	C₃×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Hall
166	C₂⁴	—	2	1	no	—
167	#41	—	7	4	no	—
168	#41	—	14	2	no	—
169	C₂×Q₈	—	7	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	7	1	no	—
173	C₂×D₄	—	14	1	no	—
174	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
175	C₂×D₄	C₂×C₃×D₇	1	4	yes	complemented
176	C₂²×C₄	—	14	1	no	—
177	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
178	SD₁₆	—	7	8	no	—
179	D₈	—	7	8	no	—
180	M₁₆	—	7	4	no	—
181	C₂×C₈	—	7	2	no	—
182	C₂×C₇	—	2	2	no	—
183	C₂×C₇	—	4	4	no	—
184	C₂×C₇	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
185	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
186	D₇	—	4	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	4	8	no	—
190	C₂²×C₃	—	4	4	no	—
191	C₂²×C₃	—	14	2	no	—
192	C₂²×C₃	—	28	2	no	—
193	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
194	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
195	C₃×C₄	—	14	2	no	—
196	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
197	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
198	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	14	1	no	—
203	C₂³	C₃×C₂²⋊D₇	1	1	yes	characteristic
204	Q₈	—	7	2	no	—
205	Q₈	—	14	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	4	no	—
208	D₄	—	7	2	no	—
209	D₄	—	14	4	no	—
210	D₄	—	14	1	no	—
211	D₄	C₂²×C₃×D₇	1	4	yes	—
212	C₂×C₄	—	14	1	no	—
213	C₂×C₄	—	14	4	no	—
214	C₂×C₄	C₃×C₂²⋊D₇	1	1	yes	characteristic
215	C₂×C₄	C₃×C₂²⋊D₇	1	2	yes	—
216	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
218	C₈	—	7	4	no	—
219	C₇	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	4	4	no	—
222	C₂×C₃	—	28	2	no	—
223	C₂×C₃	#6495	1	2	yes	direct product factor
224	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	8	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	14	2	no	—
230	C₂²	—	28	2	no	—
231	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
232	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
233	C₄	—	14	2	no	—
234	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
235	C₄	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
236	C₄	C₂³×C₃×D₇	1	1	yes	characteristic
237	C₃	C₂×#6495	1	1	yes	characteristic, direct product factor, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	4	4	no	—
240	C₂	—	28	2	no	—
241	C₂	C₃×#6495	1	2	yes	direct product factor
242	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₇
Fitting	C₂²×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₂²×C₃×D₄	D₇
hypocenter	C₇	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂×C₃×#1157	C₂	semidirect
3. C₂×C₃×#1142	C₂	semidirect
4. C₃×#6495	C₂	direct (N × H)
5. C₂×C₃×#1141	C₂	semidirect
6. C₂×C₃×#1136	C₂	semidirect
7. C₂×#6495	C₃	direct (N × H)
12. C₂×C₃×C₇×D₄	C₂²	semidirect
16. C₃×#1157	C₂²	semidirect
17. C₂×C₃×D₂₈	C₂²	semidirect
19. C₂×C₃×Dic₁₄	C₂²	semidirect
20. C₃×#1142	C₂²	semidirect
21. C₃×#1141	C₂²	semidirect
22. C₃×#1136	C₂²	semidirect
23. C₂×C₃×C₇⋊C₈	C₂²	semidirect
24. C₂²×C₇×D₄	C₂×C₃	semidirect
25. C₂×#1157	C₂×C₃	semidirect
26. C₂×#1142	C₂×C₃	semidirect
27. #6495	C₂×C₃	direct (N × H)
28. C₂×#1141	C₂×C₃	semidirect
29. C₂×#1136	C₂×C₃	semidirect
46. C₃×D₂₈	C₂³	semidirect
49. C₃×Dic₁₄	C₂³	semidirect
50. C₃×C₇⋊C₈	C₂³	semidirect
55. C₂×C₇×D₄	C₂²×C₃	semidirect
59. #1157	C₂²×C₃	semidirect
60. C₂×D₂₈	C₂²×C₃	semidirect
62. C₂×Dic₁₄	C₂²×C₃	semidirect
63. #1142	C₂²×C₃	semidirect
64. #1141	C₂²×C₃	semidirect
65. #1136	C₂²×C₃	semidirect
66. C₂×C₇⋊C₈	C₂²×C₃	semidirect
68. C₂²×C₃×D₄	D₇	semidirect
101. D₂₈	C₂³×C₃	semidirect
104. Dic₁₄	C₂³×C₃	semidirect
105. C₇⋊C₈	C₂³×C₃	semidirect
115. C₂×C₃×D₄	C₂×D₇	semidirect
124. C₂×C₃×C₇	AGL(1,ℤ/8)	semidirect
128. C₂²×D₄	C₃×D₇	semidirect
165. C₃×C₇	C₂×AGL(1,ℤ/8)	semidirect
175. C₂×D₄	C₂×C₃×D₇	semidirect
184. C₂×C₇	C₃×AGL(1,ℤ/8)	semidirect
219. C₇	C₂×C₃×AGL(1,ℤ/8)	semidirect
223. C₂×C₃	#6495	direct (N × H)
237. C₃	C₂×#6495	direct (N × H)
241. C₂	C₃×#6495	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×C₇×D₄	2
C₂×C₃×#1157	2
C₂×C₃×#1142	2
C₃×#6495	2
C₂×C₃×#1141	2
C₂×C₃×#1136	2
C₂×#6495	3
C₂×C₃×AGL(1,ℤ/8)	7

Subgroups — C2×C3×#6495 #13061141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6495 #13061141