Subgroups — C₃×#90215 #13060869

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 264 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	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6460	C₂	1	2	yes	maximal, complemented
5	C₃×#6458	C₂	1	2	yes	maximal, complemented
6	C₃×#6449	C₂	1	2	yes	maximal, complemented
7	C₃×#6445	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×D₂₈	C₂	1	4	yes	maximal, complemented
9	#90215	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₃×#1157	—	2	4	no	—
15	C₂×C₃×D₂₈	C₂²	1	4	yes	—
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₄×D₇	—	2	4	no	—
18	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₇	C₂²	1	2	yes	complemented
21	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
23	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₇	C₂²	1	2	yes	complemented
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
27	C₃×C₄⋊Dic₇	C₂²	1	4	yes	complemented
28	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
30	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
31	#6460	C₂×C₃	1	2	yes	complemented
32	#6458	C₂×C₃	1	2	yes	complemented
33	#6449	C₂×C₃	1	2	yes	complemented
34	#6445	C₂×C₃	1	2	yes	complemented
35	C₄×D₂₈	C₂×C₃	1	4	yes	complemented
36	C₃×#525	—	7	1	no	maximal, Hall
37	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
38	C₂²×C₃×D₇	—	2	2	no	—
39	C₂²×C₃×D₇	—	2	4	no	—
40	C₂²×C₃×D₇	C₂³	1	2	yes	—
41	C₂²×C₃×D₇	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
43	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
44	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₃×C₂²⋊D₇	—	2	4	no	—
47	C₂×C₃×Dic₇	—	2	2	no	—
48	C₂×C₃×Dic₇	C₂³	1	2	yes	—
49	C₂×C₃×Dic₇	C₂³	1	2	yes	—
50	C₃×D₂₈	—	2	8	no	—
51	C₃×D₂₈	—	2	2	no	—
52	C₃×C₄×D₇	—	2	4	no	—
53	C₃×C₄×D₇	—	2	4	no	—
54	C₃×C₄×D₇	—	2	4	no	—
55	C₃×Dic₁₄	—	2	2	no	—
56	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
59	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
60	#1157	—	2	4	no	—
61	C₂×D₂₈	C₂²×C₃	1	4	yes	—
62	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic, complemented
63	C₂×C₄×D₇	—	2	4	no	—
64	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
65	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
66	C₂×C₄×D₇	C₂²×C₃	1	2	yes	complemented
67	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
68	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
69	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
70	C₄²×C₇	C₂²×C₃	1	2	yes	complemented
71	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
73	C₄⋊Dic₇	C₂²×C₃	1	4	yes	complemented
74	C₂×C₃×#41	—	7	1	no	—
75	C₂³×C₃×C₄	—	7	1	no	—
76	C₃×#123	—	7	2	no	—
77	C₃×C₂²⋊Q₈	—	7	2	no	—
78	C₃×#121	—	7	2	no	—
79	C₃×#120	—	7	2	no	—
80	C₃×C₄×D₄	—	7	4	no	—
81	C₃×#117	D₇	1	1	yes	characteristic, complemented
82	C₂²×C₃×C₇	—	2	2	no	—
83	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
84	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₃×D₇	—	2	8	no	—
86	C₂×C₃×D₇	—	2	2	no	—
87	C₂×C₃×D₇	—	2	2	no	—
88	C₂×C₃×D₇	—	2	2	no	—
89	C₂×C₃×D₇	—	2	4	no	—
90	C₂×C₃×D₇	#41	1	4	yes	complemented
91	C₃×C₄×C₇	—	2	4	no	—
92	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
93	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₃×Dic₇	—	2	2	no	—
95	C₃×Dic₇	—	2	2	no	—
96	#525	—	7	1	no	Sylow
97	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
98	C₂²×D₇	—	2	2	no	—
99	C₂²×D₇	—	2	4	no	—
100	C₂²×D₇	C₂³×C₃	1	2	yes	—
101	C₂²×D₇	C₂³×C₃	1	2	yes	—
102	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
103	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
104	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
106	C₂²⋊D₇	—	2	4	no	—
107	C₂×Dic₇	—	2	2	no	—
108	C₂×Dic₇	C₂³×C₃	1	2	yes	—
109	C₂×Dic₇	C₂³×C₃	1	2	yes	—
110	D₂₈	—	2	8	no	—
111	D₂₈	—	2	2	no	—
112	C₄×D₇	—	2	4	no	—
113	C₄×D₇	—	2	4	no	—
114	C₄×D₇	—	2	4	no	—
115	Dic₁₄	—	2	2	no	—
116	C₂⁴×C₃	—	7	1	no	—
117	C₃×#41	—	14	4	no	—
118	C₂×C₃×Q₈	—	7	1	no	—
119	C₂×C₃×D₄	—	7	2	no	—
120	C₂×C₃×D₄	—	7	4	no	—
121	C₂×C₃×D₄	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	1	no	—
123	C₂²×C₃×C₄	—	7	2	no	—
124	C₂²×C₃×C₄	—	7	2	no	—
125	C₂²×C₃×C₄	—	7	2	no	—
126	C₂²×C₃×C₄	—	14	4	no	—
127	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
128	C₃×C₄⋊C₄	—	7	4	no	—
129	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	complemented
130	C₃×C₂²⋊C₄	—	7	4	no	—
131	C₃×C₂²⋊C₄	—	7	4	no	—
132	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
133	C₃×C₄²	C₂×D₇	1	2	yes	complemented
134	C₂×C₃×C₇	—	2	2	no	—
135	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₃×C₇	C₂×#41	1	2	yes	—
137	C₃×D₇	—	2	4	no	—
138	C₃×D₇	—	4	2	no	—
139	C₂×#41	—	7	1	no	—
140	C₂³×C₄	—	7	1	no	—
141	#123	—	7	2	no	—
142	C₂²⋊Q₈	—	7	2	no	—
143	#121	—	7	2	no	—
144	#120	—	7	2	no	—
145	C₄×D₄	—	7	4	no	—
146	#117	C₃×D₇	1	1	yes	characteristic, complemented
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
149	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
150	C₂×D₇	—	2	8	no	—
151	C₂×D₇	—	2	2	no	—
152	C₂×D₇	—	2	2	no	—
153	C₂×D₇	—	2	2	no	—
154	C₂×D₇	—	2	4	no	—
155	C₂×D₇	C₃×#41	1	4	yes	complemented
156	C₄×C₇	—	2	4	no	—
157	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
158	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
159	Dic₇	—	2	2	no	—
160	Dic₇	—	2	2	no	—
161	C₂³×C₃	—	7	2	no	—
162	C₂³×C₃	—	7	2	no	—
163	C₂³×C₃	—	14	4	no	—
164	C₂³×C₃	—	14	2	no	—
165	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
166	C₃×Q₈	—	14	2	no	—
167	C₃×D₄	—	14	8	no	—
168	C₃×D₄	—	14	4	no	—
169	C₃×D₄	—	14	2	no	—
170	C₂×C₃×C₄	—	7	2	no	—
171	C₂×C₃×C₄	—	7	2	no	—
172	C₂×C₃×C₄	—	14	2	no	—
173	C₂×C₃×C₄	—	14	4	no	—
174	C₂×C₃×C₄	—	14	4	no	—
175	C₂×C₃×C₄	—	14	4	no	—
176	C₂×C₃×C₄	D₂₈	1	4	yes	complemented
177	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
178	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
180	C₃×C₇	#525	1	1	yes	characteristic, complemented, Hall
181	C₂⁴	—	7	1	no	—
182	#41	—	14	4	no	—
183	C₂×Q₈	—	7	1	no	—
184	C₂×D₄	—	7	2	no	—
185	C₂×D₄	—	7	4	no	—
186	C₂×D₄	—	7	1	no	—
187	C₂²×C₄	—	7	1	no	—
188	C₂²×C₄	—	7	2	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₂²×C₄	—	14	4	no	—
192	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
193	C₄⋊C₄	—	7	4	no	—
194	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	complemented
195	C₂²⋊C₄	—	7	4	no	—
196	C₂²⋊C₄	—	7	4	no	—
197	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
198	C₄²	C₂×C₃×D₇	1	2	yes	complemented
199	C₂×C₇	—	2	2	no	—
200	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₂×C₇	C₂×C₃×#41	1	2	yes	—
202	D₇	—	2	4	no	—
203	D₇	—	4	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	7	4	no	—
206	C₂²×C₃	—	14	8	no	—
207	C₂²×C₃	—	14	2	no	—
208	C₂²×C₃	—	14	2	no	—
209	C₂²×C₃	—	14	2	no	—
210	C₂²×C₃	—	14	4	no	—
211	C₂²×C₃	C₂×D₂₈	1	2	yes	—
212	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	14	2	no	—
215	C₃×C₄	—	14	2	no	—
216	C₃×C₄	C₂×D₂₈	1	2	yes	—
217	C₃×C₄	C₂×D₂₈	1	2	yes	—
218	C₂³	—	7	2	no	—
219	C₂³	—	7	2	no	—
220	C₂³	—	14	4	no	—
221	C₂³	—	14	2	no	—
222	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
223	Q₈	—	14	2	no	—
224	D₄	—	14	8	no	—
225	D₄	—	14	4	no	—
226	D₄	—	14	2	no	—
227	C₂×C₄	—	7	2	no	—
228	C₂×C₄	—	7	2	no	—
229	C₂×C₄	—	14	2	no	—
230	C₂×C₄	—	14	4	no	—
231	C₂×C₄	—	14	4	no	—
232	C₂×C₄	—	14	4	no	—
233	C₂×C₄	C₃×D₂₈	1	4	yes	complemented
234	C₂×C₄	C₂²×C₃×D₇	1	4	yes	—
235	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
237	C₇	C₃×#525	1	1	yes	characteristic, complemented, Sylow
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	14	4	no	—
240	C₂×C₃	—	28	2	no	—
241	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
242	C₂×C₃	D₇×#41	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	7	4	no	—
245	C₂²	—	14	8	no	—
246	C₂²	—	14	2	no	—
247	C₂²	—	14	2	no	—
248	C₂²	—	14	2	no	—
249	C₂²	—	14	4	no	—
250	C₂²	C₂×C₃×D₂₈	1	2	yes	—
251	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	14	2	no	—
254	C₄	—	14	2	no	—
255	C₄	C₂×C₃×D₂₈	1	2	yes	—
256	C₄	C₂×C₃×D₂₈	1	2	yes	—
257	C₃	#90215	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	2	2	no	—
259	C₂	—	14	4	no	—
260	C₂	—	28	2	no	—
261	C₂	C₂²×C₃×D₂₈	1	1	yes	characteristic
262	C₂	C₃×D₇×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#525
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#1157	C₂	semidirect
2. C₂²×C₃×C₄×D₇	C₂	semidirect
3. C₃×C₇×#117	C₂	semidirect
4. C₃×#6460	C₂	semidirect
5. C₃×#6458	C₂	semidirect
6. C₃×#6449	C₂	semidirect
7. C₃×#6445	C₂	semidirect
8. C₃×C₄×D₂₈	C₂	semidirect
9. #90215	C₃	direct (N × H)
16. C₂×C₃×D₂₈	C₂²	semidirect
20. C₂×C₃×C₄×D₇	C₂²	semidirect
21. C₂×C₃×Dic₁₄	C₂²	semidirect
22. C₃×C₇×C₄⋊C₄	C₂²	semidirect
24. C₃×C₄²×C₇	C₂²	semidirect
26. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
27. C₃×C₄⋊Dic₇	C₂²	semidirect
28. C₂×#1157	C₂×C₃	semidirect
29. C₂²×C₄×D₇	C₂×C₃	semidirect
30. C₇×#117	C₂×C₃	semidirect
31. #6460	C₂×C₃	semidirect
32. #6458	C₂×C₃	semidirect
33. #6449	C₂×C₃	semidirect
34. #6445	C₂×C₃	semidirect
35. C₄×D₂₈	C₂×C₃	semidirect
42. C₂×C₃×C₄×C₇	D₄	semidirect
62. C₂×D₂₈	C₂²×C₃	semidirect
66. C₂×C₄×D₇	C₂²×C₃	semidirect
67. C₂×Dic₁₄	C₂²×C₃	semidirect
68. C₇×C₄⋊C₄	C₂²×C₃	semidirect
70. C₄²×C₇	C₂²×C₃	semidirect
72. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
73. C₄⋊Dic₇	C₂²×C₃	semidirect
81. C₃×#117	D₇	semidirect
90. C₂×C₃×D₇	#41	semidirect
102. C₂×C₄×C₇	C₃×D₄	semidirect
129. C₃×C₄⋊C₄	C₂×D₇	semidirect
133. C₃×C₄²	C₂×D₇	semidirect
146. #117	C₃×D₇	semidirect
155. C₂×D₇	C₃×#41	semidirect
176. C₂×C₃×C₄	D₂₈	semidirect
180. C₃×C₇	#525	semidirect
194. C₄⋊C₄	C₂×C₃×D₇	semidirect
198. C₄²	C₂×C₃×D₇	semidirect
233. C₂×C₄	C₃×D₂₈	semidirect
237. C₇	C₃×#525	semidirect
257. C₃	#90215	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
C₂²×C₃×C₄×D₇	2
C₃×C₇×#117	2
C₃×#6460	2
C₃×#6458	2
C₃×#6449	2
C₃×#6445	2
C₃×C₄×D₂₈	2
#90215	3
C₃×#525	7

Subgroups — C3×#90215 #13060869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90215 #13060869