Subgroups — C₃×#90244 #13060880

Order 1344.

624 conjugacy classes of subgroups.

The subgroup lattice has 248 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	2	yes	maximal, complemented
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6503	C₂	1	2	yes	maximal, complemented
4	C₃×#6493	C₂	1	2	yes	maximal, complemented
5	C₃×#6456	C₂	1	2	yes	maximal, complemented
6	C₃×#6447	C₂	1	4	yes	maximal, complemented
7	C₃×#6437	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#90244	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	—
13	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	—
14	C₃×#1157	—	2	8	no	—
15	C₂×C₃×D₂₈	C₂²	1	2	yes	—
16	C₂×C₃×D₂₈	C₂²	1	2	yes	—
17	C₂×C₃×D₂₈	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₇	C₂²	1	4	yes	—
19	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
24	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
25	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄×Dic₇	C₂²	1	2	yes	—
27	C₂×#1157	C₂×C₃	1	2	yes	complemented
28	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
29	#6503	C₂×C₃	1	2	yes	complemented
30	#6493	C₂×C₃	1	2	yes	complemented
31	#6456	C₂×C₃	1	2	yes	complemented
32	#6447	C₂×C₃	1	4	yes	complemented
33	#6437	C₂×C₃	1	1	yes	characteristic, complemented
34	C₄×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×#550	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	2	yes	—
37	C₂²×C₃×D₇	C₂³	1	4	yes	—
38	C₃×C₇×D₄	—	2	2	no	—
39	C₂×C₃×C₄×C₇	—	2	4	no	—
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₃×C₂²⋊D₇	—	2	8	no	—
45	C₃×C₂²⋊D₇	—	4	4	no	—
46	C₂×C₃×Dic₇	C₂³	1	2	yes	—
47	C₂×C₃×Dic₇	C₂³	1	2	yes	—
48	C₃×D₂₈	—	2	4	no	—
49	C₃×D₂₈	—	2	4	no	—
50	C₃×D₂₈	—	4	2	no	—
51	C₃×C₄×D₇	—	2	8	no	—
52	C₃×Dic₁₄	D₄	1	4	yes	complemented
53	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
55	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
56	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
57	#1157	—	2	8	no	—
58	C₂×D₂₈	C₂²×C₃	1	2	yes	—
59	C₂×D₂₈	C₂²×C₃	1	2	yes	—
60	C₂×D₂₈	C₂²×C₃	1	2	yes	—
61	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
62	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
63	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
65	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
66	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
67	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
68	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
69	C₄×Dic₇	C₂²×C₃	1	2	yes	—
70	C₂×C₃×#41	—	7	2	no	—
71	C₃×#127	—	7	2	no	—
72	C₃×#127	—	7	1	no	—
73	C₃×#121	—	7	2	no	—
74	C₃×#121	—	7	4	no	—
75	C₃×C₄×Q₈	—	7	1	no	—
76	C₃×C₄×D₄	—	7	2	no	—
77	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
78	C₂²×C₃×C₇	—	2	2	no	—
79	C₂²×C₃×C₇	—	2	2	no	—
80	C₂²×C₃×C₇	—	2	2	no	—
81	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₃×D₇	—	2	4	no	—
83	C₂×C₃×D₇	—	2	4	no	—
84	C₂×C₃×D₇	—	2	4	no	—
85	C₃×C₄×C₇	—	2	2	no	—
86	C₃×C₄×C₇	—	2	1	no	—
87	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
88	C₃×C₄×C₇	#41	1	2	yes	complemented
89	C₃×Dic₇	—	2	2	no	—
90	C₃×Dic₇	C₂×D₄	1	4	yes	—
91	#550	—	7	1	no	Sylow
92	C₂³×C₇	C₂³×C₃	1	2	yes	—
93	C₂²×D₇	C₂³×C₃	1	4	yes	—
94	C₇×D₄	—	2	2	no	—
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
98	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
100	C₂²⋊D₇	—	2	8	no	—
101	C₂²⋊D₇	—	4	4	no	—
102	C₂×Dic₇	C₂³×C₃	1	2	yes	—
103	C₂×Dic₇	C₂³×C₃	1	2	yes	—
104	D₂₈	—	2	4	no	—
105	D₂₈	—	2	4	no	—
106	D₂₈	—	4	2	no	—
107	C₄×D₇	—	2	8	no	—
108	Dic₁₄	C₃×D₄	1	4	yes	complemented
109	C₃×#41	—	14	8	no	—
110	C₂×C₃×Q₈	—	7	1	no	—
111	C₂×C₃×D₄	—	7	4	no	—
112	C₂×C₃×D₄	—	7	4	no	—
113	C₂×C₃×D₄	—	7	2	no	—
114	C₂×C₃×D₄	—	7	2	no	—
115	C₂×C₃×D₄	—	7	2	no	—
116	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	7	4	no	—
118	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
119	C₃×C₄⋊C₄	—	7	2	no	—
120	C₃×C₄⋊C₄	—	7	1	no	—
121	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	—	7	4	no	—
123	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
124	C₃×C₄²	—	7	2	no	—
125	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
126	C₂×C₃×C₇	—	4	2	no	—
127	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₃×D₇	—	4	4	no	—
131	C₂×#41	—	7	2	no	—
132	#127	—	7	2	no	—
133	#127	—	7	1	no	—
134	#121	—	7	2	no	—
135	#121	—	7	4	no	—
136	C₄×Q₈	—	7	1	no	—
137	C₄×D₄	—	7	2	no	—
138	C₄×D₄	C₃×D₇	1	1	yes	characteristic, complemented
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	—	2	2	no	—
142	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
143	C₂×D₇	—	2	4	no	—
144	C₂×D₇	—	2	4	no	—
145	C₂×D₇	—	2	4	no	—
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	—	2	1	no	—
148	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
149	C₄×C₇	C₃×#41	1	2	yes	complemented
150	Dic₇	—	2	2	no	—
151	Dic₇	C₂×C₃×D₄	1	4	yes	—
152	C₂³×C₃	—	7	4	no	—
153	C₂³×C₃	C₂²×D₇	1	2	yes	—
154	C₃×Q₈	—	7	4	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	—	14	8	no	—
157	C₃×D₄	—	14	4	no	—
158	C₃×D₄	—	14	4	no	—
159	C₃×D₄	—	28	2	no	—
160	C₃×D₄	—	28	4	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	7	2	no	—
163	C₂×C₃×C₄	—	7	2	no	—
164	C₂×C₃×C₄	—	14	8	no	—
165	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
167	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
169	C₃×C₇	#550	1	1	yes	characteristic, complemented, Hall
170	#41	—	14	8	no	—
171	C₂×Q₈	—	7	1	no	—
172	C₂×D₄	—	7	4	no	—
173	C₂×D₄	—	7	4	no	—
174	C₂×D₄	—	7	2	no	—
175	C₂×D₄	—	7	2	no	—
176	C₂×D₄	—	7	2	no	—
177	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
178	C₂²×C₄	—	7	4	no	—
179	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
180	C₄⋊C₄	—	7	2	no	—
181	C₄⋊C₄	—	7	1	no	—
182	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
183	C₂²⋊C₄	—	7	4	no	—
184	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
185	C₄²	—	7	2	no	—
186	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
187	C₂×C₇	—	4	2	no	—
188	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
190	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
191	D₇	—	4	4	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	14	4	no	—
196	C₂²×C₃	—	14	4	no	—
197	C₂²×C₃	—	14	4	no	—
198	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	7	4	no	—
202	C₃×C₄	—	14	2	no	—
203	C₃×C₄	#1157	1	2	yes	complemented
204	C₃×C₄	D₄×D₇	1	2	yes	—
205	C₂³	—	7	4	no	—
206	C₂³	C₂²×C₃×D₇	1	2	yes	—
207	Q₈	—	7	4	no	—
208	D₄	—	2	2	no	—
209	D₄	—	14	8	no	—
210	D₄	—	14	4	no	—
211	D₄	—	14	4	no	—
212	D₄	—	28	2	no	—
213	D₄	—	28	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	7	2	no	—
216	C₂×C₄	—	7	2	no	—
217	C₂×C₄	—	14	8	no	—
218	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
220	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
222	C₇	C₃×#550	1	1	yes	characteristic, complemented, Sylow
223	C₂×C₃	—	4	2	no	—
224	C₂×C₃	—	28	4	no	—
225	C₂×C₃	C₂×#1157	1	1	yes	characteristic
226	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
227	C₂×C₃	#6553	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	14	4	no	—
232	C₂²	—	14	4	no	—
233	C₂²	—	14	4	no	—
234	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	7	4	no	—
238	C₄	—	14	2	no	—
239	C₄	C₃×#1157	1	2	yes	complemented
240	C₄	C₃×D₄×D₇	1	2	yes	—
241	C₃	#90244	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	4	2	no	—
243	C₂	—	28	4	no	—
244	C₂	C₂×C₃×#1157	1	1	yes	characteristic
245	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
246	C₂	C₃×#6553	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×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×#550
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#1157	C₂	semidirect
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. C₃×#6503	C₂	semidirect
4. C₃×#6493	C₂	semidirect
5. C₃×#6456	C₂	semidirect
6. C₃×#6447	C₂	semidirect
7. C₃×#6437	C₂	semidirect
8. C₃×C₄×Dic₁₄	C₂	semidirect
9. #90244	C₃	direct (N × H)
27. C₂×#1157	C₂×C₃	semidirect
28. C₄×C₇×D₄	C₂×C₃	semidirect
29. #6503	C₂×C₃	semidirect
30. #6493	C₂×C₃	semidirect
31. #6456	C₂×C₃	semidirect
32. #6447	C₂×C₃	semidirect
33. #6437	C₂×C₃	semidirect
34. C₄×Dic₁₄	C₂×C₃	semidirect
52. C₃×Dic₁₄	D₄	semidirect
77. C₃×C₄×D₄	D₇	semidirect
88. C₃×C₄×C₇	#41	semidirect
108. Dic₁₄	C₃×D₄	semidirect
138. C₄×D₄	C₃×D₇	semidirect
149. C₄×C₇	C₃×#41	semidirect
169. C₃×C₇	#550	semidirect
203. C₃×C₄	#1157	semidirect
222. C₇	C₃×#550	semidirect
239. C₄	C₃×#1157	semidirect
241. C₃	#90244	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
C₃×C₄×C₇×D₄	2
C₃×#6503	2
C₃×#6493	2
C₃×#6456	2
C₃×#6447	2
C₃×#6437	2
C₃×C₄×Dic₁₄	2
#90244	3
C₃×#550	7

Subgroups — C3×#90244 #13060880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90244 #13060880