Subgroups — C₂×C₃×#6505 #13061153

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#1144	C₂	1	2	yes	maximal, complemented
4	C₃×#6505	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#1143	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6505	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₇×Q₈	—	2	1	no	—
9	C₂×C₃×C₇×Q₈	—	2	2	no	—
10	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₇×Q₈	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₄×C₇	—	2	1	no	—
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	—
17	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₇	—	2	1	no	—
19	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
20	C₃×#1144	C₂²	1	8	yes	complemented
21	C₃×#1143	C₂²	1	8	yes	complemented
22	C₃×#1136	C₂²	1	4	yes	complemented
23	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	—
24	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#1157	C₂×C₃	1	1	yes	characteristic
26	C₂×#1144	C₂×C₃	1	2	yes	complemented
27	#6505	C₂×C₃	1	8	yes	direct product factor
28	C₂×#1143	C₂×C₃	1	2	yes	complemented
29	C₂×#1136	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×#137	—	7	1	no	maximal, Hall
31	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
32	C₂²×C₃×D₇	—	2	1	no	—
33	C₃×C₇×Q₈	—	2	2	no	—
34	C₃×C₇×Q₈	—	2	4	no	—
35	C₃×C₇×Q₈	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₇	—	2	2	no	—
37	C₂×C₃×C₄×C₇	—	2	4	no	—
38	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
42	C₃×C₂²⋊D₇	—	2	4	no	—
43	C₂×C₃×Dic₇	—	2	1	no	—
44	C₃×D₂₈	—	2	1	no	—
45	C₃×D₂₈	C₂³	1	2	yes	—
46	C₃×C₄×D₇	—	2	4	no	—
47	C₃×Dic₁₄	—	2	1	no	—
48	C₃×Dic₁₄	C₂³	1	2	yes	—
49	C₃×C₇⋊C₈	C₂³	1	4	yes	—
50	C₂×C₇×Q₈	—	2	1	no	—
51	C₂×C₇×Q₈	—	2	2	no	—
52	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₇×Q₈	C₂²×C₃	1	4	yes	complemented
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₂²⋊D₇	—	2	1	no	—
57	#1157	—	2	2	no	—
58	#1157	C₂²×C₃	1	4	yes	—
59	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄×D₇	—	2	1	no	—
61	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
62	#1144	C₂²×C₃	1	8	yes	complemented
63	#1143	C₂²×C₃	1	8	yes	complemented
64	#1136	C₂²×C₃	1	4	yes	complemented
65	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	—
66	C₂×C₃×#41	—	7	1	no	—
67	C₂²×C₃×Q₈	D₇	1	1	yes	characteristic, complemented
68	C₃×#137	—	7	8	no	—
69	C₂×C₃×Q₁₆	—	7	2	no	—
70	C₂×C₃×SD₁₆	—	7	2	no	—
71	C₂×C₃×M₁₆	—	7	1	no	—
72	C₂²×C₃×C₇	—	2	2	no	—
73	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
74	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×D₇	—	2	2	no	—
76	C₂×C₃×D₇	—	4	2	no	—
77	C₃×C₄×C₇	—	2	4	no	—
78	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
80	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₃×Dic₇	—	2	2	no	—
82	C₂×#137	—	7	1	no	Sylow
83	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
84	C₂²×D₇	—	2	1	no	—
85	C₇×Q₈	—	2	2	no	—
86	C₇×Q₈	—	2	4	no	—
87	C₇×Q₈	C₂³×C₃	1	4	yes	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
92	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
94	C₂²⋊D₇	—	2	4	no	—
95	C₂×Dic₇	—	2	1	no	—
96	D₂₈	—	2	1	no	—
97	D₂₈	C₂³×C₃	1	2	yes	—
98	C₄×D₇	—	2	4	no	—
99	Dic₁₄	—	2	1	no	—
100	Dic₁₄	C₂³×C₃	1	2	yes	—
101	C₇⋊C₈	C₂³×C₃	1	4	yes	—
102	C₃×#41	—	7	4	no	—
103	C₃×#41	—	14	2	no	—
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	—	2	1	no	—
106	C₂×C₃×Q₈	—	7	1	no	—
107	C₂×C₃×Q₈	C₂×D₇	1	2	yes	—
108	C₂×C₃×Q₈	C₂×D₇	1	4	yes	complemented
109	C₂×C₃×D₄	—	7	1	no	—
110	C₂×C₃×D₄	—	14	1	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	14	1	no	—
113	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
114	C₃×Q₁₆	—	7	8	no	—
115	C₃×SD₁₆	—	7	8	no	—
116	C₃×M₁₆	—	7	4	no	—
117	C₂×C₃×C₈	—	7	2	no	—
118	C₂×C₃×C₇	—	2	2	no	—
119	C₂×C₃×C₇	#137	1	2	yes	complemented
120	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
121	C₃×D₇	—	4	2	no	—
122	C₂×#41	—	7	1	no	—
123	C₂²×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
124	#137	—	7	8	no	—
125	C₂×Q₁₆	—	7	2	no	—
126	C₂×SD₁₆	—	7	2	no	—
127	C₂×M₁₆	—	7	1	no	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
130	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×D₇	—	2	2	no	—
132	C₂×D₇	—	4	2	no	—
133	C₄×C₇	—	2	4	no	—
134	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
135	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
136	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
137	Dic₇	—	2	2	no	—
138	C₂³×C₃	—	14	1	no	—
139	C₂³×C₃	C₂²⋊D₇	1	1	yes	characteristic
140	C₃×Q₈	—	2	2	no	—
141	C₃×Q₈	—	2	4	no	—
142	C₃×Q₈	—	7	2	no	—
143	C₃×Q₈	—	14	1	no	—
144	C₃×Q₈	C₂²×D₇	1	4	yes	—
145	C₃×D₄	—	7	2	no	—
146	C₃×D₄	—	14	4	no	—
147	C₃×D₄	—	14	1	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	14	1	no	—
151	C₂×C₃×C₄	—	14	4	no	—
152	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
153	C₂×C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
155	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
156	C₃×C₈	—	7	4	no	—
157	C₃×C₇	C₂×#137	1	1	yes	characteristic, complemented, Hall
158	#41	—	7	4	no	—
159	#41	—	14	2	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	7	1	no	—
163	C₂×Q₈	C₂×C₃×D₇	1	2	yes	—
164	C₂×Q₈	C₂×C₃×D₇	1	4	yes	complemented
165	C₂×D₄	—	7	1	no	—
166	C₂×D₄	—	14	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	14	1	no	—
169	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
170	Q₁₆	—	7	8	no	—
171	SD₁₆	—	7	8	no	—
172	M₁₆	—	7	4	no	—
173	C₂×C₈	—	7	2	no	—
174	C₂×C₇	—	2	2	no	—
175	C₂×C₇	C₃×#137	1	2	yes	complemented
176	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
177	D₇	—	4	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	14	2	no	—
180	C₂²×C₃	—	28	2	no	—
181	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
182	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	—	14	2	no	—
185	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
186	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
187	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
188	C₂³	—	14	1	no	—
189	C₂³	C₃×C₂²⋊D₇	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	Q₈	—	2	4	no	—
192	Q₈	—	7	2	no	—
193	Q₈	—	14	1	no	—
194	Q₈	C₂²×C₃×D₇	1	4	yes	—
195	D₄	—	7	2	no	—
196	D₄	—	14	4	no	—
197	D₄	—	14	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	14	1	no	—
201	C₂×C₄	—	14	4	no	—
202	C₂×C₄	C₃×C₂²⋊D₇	1	1	yes	characteristic
203	C₂×C₄	C₃×C₂²⋊D₇	1	2	yes	—
204	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
206	C₈	—	7	4	no	—
207	C₇	C₂×C₃×#137	1	1	yes	characteristic, complemented, Sylow
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	28	2	no	—
210	C₂×C₃	#6505	1	2	yes	direct product factor
211	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	14	2	no	—
214	C₂²	—	28	2	no	—
215	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
216	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	14	2	no	—
219	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
220	C₄	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
221	C₄	C₂³×C₃×D₇	1	1	yes	characteristic
222	C₃	C₂×#6505	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	28	2	no	—
225	C₂	C₃×#6505	1	2	yes	direct product factor
226	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×Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×#1157	C₂
hypercenter	C₂²×C₃×Q₈	D₇
hypocenter	C₇	C₂×C₃×#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×Q₈	C₂	semidirect
3. C₂×C₃×#1144	C₂	semidirect
4. C₃×#6505	C₂	direct (N × H)
5. C₂×C₃×#1143	C₂	semidirect
6. C₂×C₃×#1136	C₂	semidirect
7. C₂×#6505	C₃	direct (N × H)
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
20. C₃×#1144	C₂²	semidirect
21. C₃×#1143	C₂²	semidirect
22. C₃×#1136	C₂²	semidirect
24. C₂²×C₇×Q₈	C₂×C₃	semidirect
26. C₂×#1144	C₂×C₃	semidirect
27. #6505	C₂×C₃	direct (N × H)
28. C₂×#1143	C₂×C₃	semidirect
29. C₂×#1136	C₂×C₃	semidirect
53. C₂×C₇×Q₈	C₂²×C₃	semidirect
62. #1144	C₂²×C₃	semidirect
63. #1143	C₂²×C₃	semidirect
64. #1136	C₂²×C₃	semidirect
67. C₂²×C₃×Q₈	D₇	semidirect
108. C₂×C₃×Q₈	C₂×D₇	semidirect
119. C₂×C₃×C₇	#137	semidirect
123. C₂²×Q₈	C₃×D₇	semidirect
157. C₃×C₇	C₂×#137	semidirect
164. C₂×Q₈	C₂×C₃×D₇	semidirect
175. C₂×C₇	C₃×#137	semidirect
207. C₇	C₂×C₃×#137	semidirect
210. C₂×C₃	#6505	direct (N × H)
222. C₃	C₂×#6505	direct (N × H)
225. C₂	C₃×#6505	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×Q₈	2
C₂×C₃×#1157	2
C₂×C₃×#1144	2
C₃×#6505	2
C₂×C₃×#1143	2
C₂×C₃×#1136	2
C₂×#6505	3
C₂×C₃×#137	7

Subgroups — C2×C3×#6505 #13061153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6505 #13061153