Subgroups — C₃×#90275 #13060926

Order 1344.

424 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	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6509	C₂	1	2	yes	maximal, complemented
3	C₃×#6461	C₂	1	4	yes	maximal, complemented
4	C₃×#6455	C₂	1	2	yes	maximal, complemented
5	C₃×#6438	C₂	1	2	yes	maximal
6	C₃×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
7	C₃×#6432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Dic₁₄	C₂	1	2	yes	maximal, complemented
9	#90275	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
13	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
14	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₃×C₄²×C₇	C₂²	1	2	yes	—
18	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
19	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
20	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
21	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
24	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄×Dic₇	C₂²	1	4	yes	—
26	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#6509	C₂×C₃	1	2	yes	complemented
28	#6461	C₂×C₃	1	4	yes	complemented
29	#6455	C₂×C₃	1	2	yes	complemented
30	#6438	C₂×C₃	1	2	yes	—
31	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic
32	#6432	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×Dic₁₄	C₂×C₃	1	2	yes	complemented
34	C₃×#556	—	7	1	no	maximal, Hall
35	C₂²×C₃×D₇	C₂³	1	2	yes	—
36	C₃×C₇×Q₈	—	2	2	no	—
37	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₇	C₂³	1	2	yes	—
43	C₂×C₃×Dic₇	C₂³	1	4	yes	—
44	C₃×D₂₈	—	2	2	no	—
45	C₃×C₄×D₇	—	2	4	no	—
46	C₃×Dic₁₄	—	2	4	no	—
47	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
50	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
51	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₄²×C₇	C₂²×C₃	1	2	yes	—
54	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
55	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
56	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
57	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
58	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
59	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
60	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
61	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
62	C₄×Dic₇	C₂²×C₃	1	4	yes	—
63	C₃×C₄⋊Q₈	—	7	1	no	—
64	C₃×#126	—	7	4	no	—
65	C₃×#124	—	7	2	no	—
66	C₃×C₂²⋊Q₈	—	7	2	no	—
67	C₃×C₄×Q₈	—	7	2	no	—
68	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
69	C₃×C₄×D₄	—	7	1	no	—
70	C₃×#117	—	7	2	no	—
71	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₃×D₇	—	2	2	no	—
73	C₂×C₃×D₇	—	2	2	no	—
74	C₂×C₃×D₇	—	2	2	no	—
75	C₃×C₄×C₇	—	2	2	no	—
76	C₃×C₄×C₇	—	2	2	no	—
77	C₃×C₄×C₇	—	2	1	no	—
78	C₃×C₄×C₇	#41	1	2	yes	—
79	C₃×C₄×C₇	#41	1	2	yes	—
80	C₃×Dic₇	—	2	4	no	—
81	C₃×Dic₇	—	2	2	no	—
82	#556	—	7	1	no	Sylow
83	C₂²×D₇	C₂³×C₃	1	2	yes	—
84	C₇×Q₈	—	2	2	no	—
85	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
86	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
87	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
90	C₂×Dic₇	C₂³×C₃	1	2	yes	—
91	C₂×Dic₇	C₂³×C₃	1	4	yes	—
92	D₂₈	—	2	2	no	—
93	C₄×D₇	—	2	4	no	—
94	Dic₁₄	—	2	4	no	—
95	C₂×C₃×Q₈	—	7	2	no	—
96	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
97	C₂×C₃×D₄	—	7	1	no	—
98	C₂²×C₃×C₄	—	7	2	no	—
99	C₃×C₄⋊C₄	—	7	4	no	—
100	C₃×C₄⋊C₄	—	7	2	no	—
101	C₃×C₄⋊C₄	—	7	1	no	—
102	C₃×C₄⋊C₄	—	7	2	no	—
103	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
104	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
105	C₃×C₂²⋊C₄	—	7	4	no	—
106	C₃×C₂²⋊C₄	—	7	2	no	—
107	C₃×C₂²⋊C₄	—	7	4	no	—
108	C₃×C₄²	—	7	4	no	—
109	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
110	C₃×C₄²	C₂×D₇	1	2	yes	—
111	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
114	C₃×D₇	—	4	2	no	—
115	C₄⋊Q₈	—	7	1	no	—
116	#126	—	7	4	no	—
117	#124	—	7	2	no	—
118	C₂²⋊Q₈	—	7	2	no	—
119	C₄×Q₈	—	7	2	no	—
120	C₄×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
121	C₄×D₄	—	7	1	no	—
122	#117	—	7	2	no	—
123	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
124	C₂×D₇	—	2	2	no	—
125	C₂×D₇	—	2	2	no	—
126	C₂×D₇	—	2	2	no	—
127	C₄×C₇	—	2	2	no	—
128	C₄×C₇	—	2	2	no	—
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	C₃×#41	1	2	yes	—
131	C₄×C₇	C₃×#41	1	2	yes	—
132	Dic₇	—	2	4	no	—
133	Dic₇	—	2	2	no	—
134	C₂³×C₃	—	7	2	no	—
135	C₃×Q₈	—	2	2	no	—
136	C₃×Q₈	—	14	4	no	—
137	C₃×D₄	—	14	2	no	—
138	C₂×C₃×C₄	—	7	2	no	—
139	C₂×C₃×C₄	—	7	4	no	—
140	C₂×C₃×C₄	—	14	4	no	—
141	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
142	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
144	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
146	C₃×C₇	#556	1	1	yes	characteristic, complemented, Hall
147	C₂×Q₈	—	7	2	no	—
148	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
149	C₂×D₄	—	7	1	no	—
150	C₂²×C₄	—	7	2	no	—
151	C₄⋊C₄	—	7	4	no	—
152	C₄⋊C₄	—	7	2	no	—
153	C₄⋊C₄	—	7	1	no	—
154	C₄⋊C₄	—	7	2	no	—
155	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
156	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
157	C₂²⋊C₄	—	7	4	no	—
158	C₂²⋊C₄	—	7	2	no	—
159	C₂²⋊C₄	—	7	4	no	—
160	C₄²	—	7	4	no	—
161	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
162	C₄²	C₂×C₃×D₇	1	2	yes	—
163	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
164	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
165	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
166	D₇	—	4	2	no	—
167	C₂²×C₃	—	14	2	no	—
168	C₂²×C₃	—	14	2	no	—
169	C₂²×C₃	—	14	2	no	—
170	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	14	2	no	—
175	C₃×C₄	—	14	4	no	—
176	C₃×C₄	#1157	1	2	yes	—
177	C₃×C₄	#1161	1	2	yes	—
178	C₂³	—	7	2	no	—
179	Q₈	—	2	2	no	—
180	Q₈	—	14	4	no	—
181	D₄	—	14	2	no	—
182	C₂×C₄	—	7	2	no	—
183	C₂×C₄	—	7	4	no	—
184	C₂×C₄	—	14	4	no	—
185	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
186	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
188	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
190	C₇	C₃×#556	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₃	—	28	2	no	—
192	C₂×C₃	C₂×#1157	1	1	yes	characteristic
193	C₂×C₃	C₂×#1161	1	1	yes	characteristic
194	C₂×C₃	#6554	1	1	yes	characteristic
195	C₂²	—	14	2	no	—
196	C₂²	—	14	2	no	—
197	C₂²	—	14	2	no	—
198	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	—	14	2	no	—
203	C₄	—	14	4	no	—
204	C₄	C₃×#1157	1	2	yes	—
205	C₄	C₃×#1161	1	2	yes	—
206	C₃	#90275	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	28	2	no	—
208	C₂	C₂×C₃×#1157	1	1	yes	characteristic
209	C₂	C₂×C₃×#1161	1	1	yes	characteristic
210	C₂	C₃×#6554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×D₂₈	C₂²
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₃×#556
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₇×Q₈	C₂	semidirect
2. C₃×#6509	C₂	semidirect
3. C₃×#6461	C₂	semidirect
4. C₃×#6455	C₂	semidirect
7. C₃×#6432	C₂	semidirect
8. C₃×C₄×Dic₁₄	C₂	semidirect
9. #90275	C₃	direct (N × H)
26. C₄×C₇×Q₈	C₂×C₃	semidirect
27. #6509	C₂×C₃	semidirect
28. #6461	C₂×C₃	semidirect
29. #6455	C₂×C₃	semidirect
32. #6432	C₂×C₃	semidirect
33. C₄×Dic₁₄	C₂×C₃	semidirect
68. C₃×C₄×Q₈	D₇	semidirect
120. C₄×Q₈	C₃×D₇	semidirect
146. C₃×C₇	#556	semidirect
190. C₇	C₃×#556	semidirect
206. C₃	#90275	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇×Q₈	2
C₃×#6509	2
C₃×#6461	2
C₃×#6455	2
C₃×#6438	2
C₃×C₄×D₂₈	2
C₃×#6432	2
C₃×C₄×Dic₁₄	2
#90275	3
C₃×#556	7

Subgroups — C3×#90275 #13060926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90275 #13060926