Subgroups — C₂×C₃×#6509 #13061155

Order 1344.

644 conjugacy classes of subgroups.

The subgroup lattice has 208 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	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
3	C₃×#6509	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×Dic₇⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂×#6509	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₇×Q₈	—	2	4	no	—
10	C₂×C₃×C₇×Q₈	C₂²	1	4	yes	complemented
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
15	C₂×C₃×C₄×D₇	—	2	4	no	—
16	C₂×C₃×C₄×D₇	C₂²	1	4	yes	—
17	C₃×(C₂×D₇)⋊C₄	C₂²	1	8	yes	—
18	C₃×C₄⋊Dic₇	C₂²	1	4	yes	complemented
19	C₃×Dic₇⋊C₄	C₂²	1	8	yes	complemented
20	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic
22	#6509	C₂×C₃	1	8	yes	direct product factor
23	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	—
24	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×Dic₇⋊C₄	C₂×C₃	1	2	yes	complemented
26	C₂×C₃×C₂²⋊Q₈	—	7	1	no	maximal, Hall
27	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
28	C₂²×C₃×D₇	—	2	2	no	—
29	C₂²×C₃×D₇	—	2	2	no	—
30	C₂²×C₃×D₇	Q₈	1	4	yes	complemented
31	C₂²×C₃×D₇	C₂³	1	2	yes	—
32	C₃×C₇×Q₈	—	2	8	no	—
33	C₂×C₃×C₄×C₇	—	2	4	no	—
34	C₂×C₃×C₄×C₇	D₄	1	4	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
37	C₂×C₃×Dic₇	—	2	2	no	—
38	C₂×C₃×Dic₇	—	2	4	no	—
39	C₂×C₃×Dic₇	C₂³	1	2	yes	—
40	C₂×C₃×Dic₇	C₂³	1	4	yes	—
41	C₃×C₄×D₇	—	2	8	no	—
42	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₇×Q₈	—	2	4	no	—
44	C₂×C₇×Q₈	C₂²×C₃	1	4	yes	complemented
45	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
47	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
48	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
49	C₂×C₄×D₇	—	2	4	no	—
50	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
51	(C₂×D₇)⋊C₄	C₂²×C₃	1	8	yes	—
52	C₄⋊Dic₇	C₂²×C₃	1	4	yes	complemented
53	Dic₇⋊C₄	C₂²×C₃	1	8	yes	complemented
54	C₂²×C₃×Q₈	D₇	1	1	yes	characteristic, complemented
55	C₂³×C₃×C₄	—	7	1	no	—
56	C₃×C₂²⋊Q₈	—	7	8	no	—
57	C₂×C₃×C₄⋊C₄	—	7	2	no	—
58	C₂×C₃×C₄⋊C₄	—	7	1	no	—
59	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
60	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
61	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₃×C₇	#41	1	2	yes	—
63	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₃×D₇	—	2	8	no	—
65	C₂×C₃×D₇	—	2	2	no	—
66	C₂×C₃×D₇	—	2	2	no	—
67	C₂×C₃×D₇	C₂×Q₈	1	4	yes	complemented
68	C₃×C₄×C₇	—	2	4	no	—
69	C₃×C₄×C₇	C₂×D₄	1	4	yes	—
70	C₃×Dic₇	—	2	2	no	—
71	C₃×Dic₇	—	2	4	no	—
72	C₂×C₂²⋊Q₈	—	7	1	no	Sylow
73	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
74	C₂²×D₇	—	2	2	no	—
75	C₂²×D₇	—	2	2	no	—
76	C₂²×D₇	C₃×Q₈	1	4	yes	complemented
77	C₂²×D₇	C₂³×C₃	1	2	yes	—
78	C₇×Q₈	—	2	8	no	—
79	C₂×C₄×C₇	—	2	4	no	—
80	C₂×C₄×C₇	C₃×D₄	1	4	yes	—
81	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
82	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
83	C₂×Dic₇	—	2	2	no	—
84	C₂×Dic₇	—	2	4	no	—
85	C₂×Dic₇	C₂³×C₃	1	2	yes	—
86	C₂×Dic₇	C₂³×C₃	1	4	yes	—
87	C₄×D₇	—	2	8	no	—
88	C₂⁴×C₃	—	7	1	no	—
89	C₂×C₃×Q₈	—	2	4	no	—
90	C₂×C₃×Q₈	C₂×D₇	1	4	yes	complemented
91	C₂²×C₃×C₄	—	7	2	no	—
92	C₂²×C₃×C₄	—	7	1	no	—
93	C₂²×C₃×C₄	—	7	4	no	—
94	C₂²×C₃×C₄	—	14	4	no	—
95	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
96	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
97	C₃×C₄⋊C₄	—	7	8	no	—
98	C₃×C₄⋊C₄	—	7	4	no	—
99	C₃×C₂²⋊C₄	—	7	8	no	—
100	C₂×C₃×C₇	C₂²⋊Q₈	1	4	yes	complemented
101	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
104	C₃×D₇	—	2	4	no	—
105	C₂²×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
106	C₂³×C₄	—	7	1	no	—
107	C₂²⋊Q₈	—	7	8	no	—
108	C₂×C₄⋊C₄	—	7	2	no	—
109	C₂×C₄⋊C₄	—	7	1	no	—
110	C₂×C₂²⋊C₄	—	7	2	no	—
111	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
112	C₂²×C₇	C₂×C₃×Q₈	1	2	yes	—
113	C₂²×C₇	C₃×#41	1	2	yes	—
114	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
115	C₂×D₇	—	2	8	no	—
116	C₂×D₇	—	2	2	no	—
117	C₂×D₇	—	2	2	no	—
118	C₂×D₇	C₂×C₃×Q₈	1	4	yes	complemented
119	C₄×C₇	—	2	4	no	—
120	C₄×C₇	C₂×C₃×D₄	1	4	yes	—
121	Dic₇	—	2	2	no	—
122	Dic₇	—	2	4	no	—
123	C₂³×C₃	—	7	4	no	—
124	C₂³×C₃	—	7	2	no	—
125	C₂³×C₃	—	14	2	no	—
126	C₂³×C₃	—	14	2	no	—
127	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
128	C₃×Q₈	—	2	8	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	7	4	no	—
131	C₂×C₃×C₄	—	7	2	no	—
132	C₂×C₃×C₄	—	14	4	no	—
133	C₂×C₃×C₄	—	14	2	no	—
134	C₂×C₃×C₄	—	14	8	no	—
135	C₂×C₃×C₄	C₂²⋊D₇	1	4	yes	—
136	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
137	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
138	C₃×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
139	C₂⁴	—	7	1	no	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	C₂×C₃×D₇	1	4	yes	complemented
142	C₂²×C₄	—	7	2	no	—
143	C₂²×C₄	—	7	1	no	—
144	C₂²×C₄	—	7	4	no	—
145	C₂²×C₄	—	14	4	no	—
146	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
147	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
148	C₄⋊C₄	—	7	8	no	—
149	C₄⋊C₄	—	7	4	no	—
150	C₂²⋊C₄	—	7	8	no	—
151	C₂×C₇	C₃×C₂²⋊Q₈	1	4	yes	complemented
152	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
153	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
154	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
155	D₇	—	2	4	no	—
156	C₂²×C₃	—	7	4	no	—
157	C₂²×C₃	—	14	8	no	—
158	C₂²×C₃	—	14	2	no	—
159	C₂²×C₃	—	14	2	no	—
160	C₂²×C₃	Q₈×D₇	1	2	yes	—
161	C₂²×C₃	#1161	1	2	yes	—
162	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
163	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	14	4	no	—
166	C₃×C₄	—	14	2	no	—
167	C₃×C₄	C₂×C₂²⋊D₇	1	4	yes	—
168	C₂³	—	7	4	no	—
169	C₂³	—	7	2	no	—
170	C₂³	—	14	2	no	—
171	C₂³	—	14	2	no	—
172	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	7	4	no	—
176	C₂×C₄	—	7	2	no	—
177	C₂×C₄	—	14	4	no	—
178	C₂×C₄	—	14	2	no	—
179	C₂×C₄	—	14	8	no	—
180	C₂×C₄	C₃×C₂²⋊D₇	1	4	yes	—
181	C₂×C₄	C₂²×C₃×D₇	1	4	yes	—
182	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
183	C₇	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₃	—	14	4	no	—
185	C₂×C₃	#6509	1	4	yes	direct product factor
186	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
187	C₂×C₃	C₂×#1161	1	1	yes	characteristic
188	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
189	C₂²	—	7	4	no	—
190	C₂²	—	14	8	no	—
191	C₂²	—	14	2	no	—
192	C₂²	—	14	2	no	—
193	C₂²	C₃×Q₈×D₇	1	2	yes	—
194	C₂²	C₃×#1161	1	2	yes	—
195	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
196	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	14	4	no	—
199	C₄	—	14	2	no	—
200	C₄	C₂×C₃×C₂²⋊D₇	1	4	yes	—
201	C₃	C₂×#6509	1	1	yes	characteristic, direct product factor, Sylow
202	C₂	—	14	4	no	—
203	C₂	C₃×#6509	1	4	yes	direct product factor
204	C₂	C₂×C₃×Q₈×D₇	1	1	yes	characteristic
205	C₂	C₂×C₃×#1161	1	1	yes	characteristic
206	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂²×C₃×C₇×Q₈	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
hypercenter	C₂²×C₃×Q₈	D₇
hypocenter	C₇	C₂×C₃×C₂²⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ 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. C₃×#6509	C₂	direct (N × H)
5. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
6. C₂×C₃×Dic₇⋊C₄	C₂	semidirect
7. C₂×#6509	C₃	direct (N × H)
10. C₂×C₃×C₇×Q₈	C₂²	semidirect
18. C₃×C₄⋊Dic₇	C₂²	semidirect
19. C₃×Dic₇⋊C₄	C₂²	semidirect
20. C₂²×C₇×Q₈	C₂×C₃	semidirect
22. #6509	C₂×C₃	direct (N × H)
24. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
25. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
30. C₂²×C₃×D₇	Q₈	semidirect
44. C₂×C₇×Q₈	C₂²×C₃	semidirect
52. C₄⋊Dic₇	C₂²×C₃	semidirect
53. Dic₇⋊C₄	C₂²×C₃	semidirect
54. C₂²×C₃×Q₈	D₇	semidirect
67. C₂×C₃×D₇	C₂×Q₈	semidirect
76. C₂²×D₇	C₃×Q₈	semidirect
90. C₂×C₃×Q₈	C₂×D₇	semidirect
100. C₂×C₃×C₇	C₂²⋊Q₈	semidirect
105. C₂²×Q₈	C₃×D₇	semidirect
118. C₂×D₇	C₂×C₃×Q₈	semidirect
138. C₃×C₇	C₂×C₂²⋊Q₈	semidirect
141. C₂×Q₈	C₂×C₃×D₇	semidirect
151. C₂×C₇	C₃×C₂²⋊Q₈	semidirect
183. C₇	C₂×C₃×C₂²⋊Q₈	semidirect
185. C₂×C₃	#6509	direct (N × H)
201. C₃	C₂×#6509	direct (N × H)
203. C₂	C₃×#6509	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
C₂²×C₃×C₄×D₇	2
C₃×#6509	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×C₃×C₄⋊Dic₇	2
C₂×C₃×Dic₇⋊C₄	2
C₂×#6509	3
C₂×C₃×C₂²⋊Q₈	7

Subgroups — C2×C3×#6509 #13061155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6509 #13061155