Subgroups — C₃×#90524 #13061174

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6509	C₂	1	2	yes	maximal, complemented
4	C₃×#6502	C₂	1	4	yes	maximal, complemented
5	C₃×#6501	C₂	1	2	yes	maximal, complemented
6	C₃×#6493	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6487	C₂	1	2	yes	maximal, complemented
9	#90524	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₇×#41	—	2	4	no	—
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	—
18	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
19	C₃×#1157	—	2	4	no	—
20	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
22	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
24	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
27	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
28	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
30	#6509	C₂×C₃	1	2	yes	complemented
31	#6502	C₂×C₃	1	4	yes	complemented
32	#6501	C₂×C₃	1	2	yes	complemented
33	#6493	C₂×C₃	1	2	yes	complemented
34	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
35	#6487	C₂×C₃	1	2	yes	complemented
36	C₃×#537	—	7	1	no	maximal, Hall
37	C₂³×C₃×C₇	C₂³	1	2	yes	—
38	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₃×D₇	C₂³	1	2	yes	—
40	C₃×C₇×Q₈	—	2	2	no	—
41	C₃×C₇×D₄	—	2	4	no	—
42	C₃×C₇×D₄	—	2	2	no	—
43	C₂×C₃×C₄×C₇	—	2	4	no	—
44	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
45	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₃×C₂²⋊D₇	—	2	4	no	—
49	C₃×C₂²⋊D₇	—	4	4	no	—
50	C₂×C₃×Dic₇	—	4	2	no	—
51	C₂×C₃×Dic₇	C₂³	1	2	yes	—
52	C₂×C₃×Dic₇	C₂³	1	4	yes	—
53	C₃×D₂₈	—	2	2	no	—
54	C₃×C₄×D₇	—	2	4	no	—
55	C₃×Dic₁₄	—	2	2	no	—
56	C₇×#41	—	2	4	no	—
57	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
60	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
61	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
63	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
64	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
65	#1157	—	2	4	no	—
66	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
68	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
69	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
70	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
71	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
72	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
73	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₃×#41	—	7	1	no	—
75	C₂×C₃×#41	D₇	1	1	yes	characteristic, complemented
76	C₃×C₂²⋊Q₈	—	7	2	no	—
77	C₃×C₂²⋊Q₈	—	7	2	no	—
78	C₃×#121	—	7	2	no	—
79	C₃×#121	—	7	4	no	—
80	C₃×#121	—	7	2	no	—
81	C₂×C₃×C₄⋊C₄	—	7	1	no	—
82	C₂²×C₃×C₇	—	2	1	no	—
83	C₂²×C₃×C₇	—	2	1	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	—	2	2	no	—
86	C₂²×C₃×C₇	—	2	2	no	—
87	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
88	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₃×D₇	—	2	2	no	—
90	C₂×C₃×D₇	—	2	2	no	—
91	C₂×C₃×D₇	—	2	2	no	—
92	C₃×C₄×C₇	—	2	2	no	—
93	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
95	C₃×Dic₇	—	2	2	no	—
96	C₃×Dic₇	—	2	4	no	—
97	#537	—	7	1	no	Sylow
98	C₂³×C₇	C₂³×C₃	1	2	yes	—
99	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
100	C₂²×D₇	C₂³×C₃	1	2	yes	—
101	C₇×Q₈	—	2	2	no	—
102	C₇×D₄	—	2	4	no	—
103	C₇×D₄	—	2	2	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
106	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
107	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂²⋊D₇	—	2	4	no	—
110	C₂²⋊D₇	—	4	4	no	—
111	C₂×Dic₇	—	4	2	no	—
112	C₂×Dic₇	C₂³×C₃	1	2	yes	—
113	C₂×Dic₇	C₂³×C₃	1	4	yes	—
114	D₂₈	—	2	2	no	—
115	C₄×D₇	—	2	4	no	—
116	Dic₁₄	—	2	2	no	—
117	C₃×#41	—	2	4	no	—
118	C₃×#41	—	14	4	no	—
119	C₂×C₃×Q₈	—	7	1	no	—
120	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
121	C₂×C₃×D₄	—	7	2	no	—
122	C₂×C₃×D₄	—	7	4	no	—
123	C₂×C₃×D₄	—	7	1	no	—
124	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
125	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
129	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	7	4	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	2	no	—
133	C₃×C₂²⋊C₄	—	7	4	no	—
134	C₃×C₂²⋊C₄	—	7	4	no	—
135	C₂×C₃×C₇	—	2	2	no	—
136	C₂×C₃×C₇	—	4	2	no	—
137	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₃×D₇	—	4	2	no	—
141	C₂×#41	—	7	1	no	—
142	C₂×#41	C₃×D₇	1	1	yes	characteristic, complemented
143	C₂²⋊Q₈	—	7	2	no	—
144	C₂²⋊Q₈	—	7	2	no	—
145	#121	—	7	2	no	—
146	#121	—	7	4	no	—
147	#121	—	7	2	no	—
148	C₂×C₄⋊C₄	—	7	1	no	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	—	2	2	no	—
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
155	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
156	C₂×D₇	—	2	2	no	—
157	C₂×D₇	—	2	2	no	—
158	C₂×D₇	—	2	2	no	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
161	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
162	Dic₇	—	2	2	no	—
163	Dic₇	—	2	4	no	—
164	C₂³×C₃	—	7	2	no	—
165	C₂³×C₃	C₂²×D₇	1	2	yes	—
166	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
167	C₃×Q₈	—	2	2	no	—
168	C₃×Q₈	—	14	2	no	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	2	2	no	—
171	C₃×D₄	—	14	4	no	—
172	C₃×D₄	—	14	2	no	—
173	C₃×D₄	—	28	4	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	7	4	no	—
176	C₂×C₃×C₄	—	7	2	no	—
177	C₂×C₃×C₄	—	14	4	no	—
178	C₂×C₃×C₄	—	28	2	no	—
179	C₂×C₃×C₄	C₂²⋊D₇	1	4	yes	complemented
180	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
183	C₃×C₇	#537	1	1	yes	characteristic, complemented, Hall
184	#41	—	2	4	no	—
185	#41	—	14	4	no	—
186	C₂×Q₈	—	7	1	no	—
187	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
188	C₂×D₄	—	7	2	no	—
189	C₂×D₄	—	7	4	no	—
190	C₂×D₄	—	7	1	no	—
191	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
192	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
193	C₂²×C₄	—	7	2	no	—
194	C₂²×C₄	—	7	2	no	—
195	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
196	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
197	C₄⋊C₄	—	7	4	no	—
198	C₄⋊C₄	—	7	2	no	—
199	C₄⋊C₄	—	7	2	no	—
200	C₂²⋊C₄	—	7	4	no	—
201	C₂²⋊C₄	—	7	4	no	—
202	C₂×C₇	—	2	2	no	—
203	C₂×C₇	—	4	2	no	—
204	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
205	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
207	D₇	—	4	2	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	14	2	no	—
214	C₂²×C₃	—	14	2	no	—
215	C₂²×C₃	—	14	2	no	—
216	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
217	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	14	4	no	—
220	C₃×C₄	—	14	2	no	—
221	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
222	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
223	C₂³	—	7	2	no	—
224	C₂³	C₂²×C₃×D₇	1	2	yes	—
225	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	—	14	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	14	4	no	—
231	D₄	—	14	2	no	—
232	D₄	—	28	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	7	4	no	—
235	C₂×C₄	—	7	2	no	—
236	C₂×C₄	—	14	4	no	—
237	C₂×C₄	—	28	2	no	—
238	C₂×C₄	C₃×C₂²⋊D₇	1	4	yes	complemented
239	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
240	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
242	C₇	C₃×#537	1	1	yes	characteristic, complemented, Sylow
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	—	28	2	no	—
246	C₂×C₃	#6553	1	1	yes	characteristic
247	C₂×C₃	#6554	1	1	yes	characteristic
248	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	14	2	no	—
255	C₂²	—	14	2	no	—
256	C₂²	—	14	2	no	—
257	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
258	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	14	4	no	—
261	C₄	—	14	2	no	—
262	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
263	C₄	C₂×C₃×C₂²⋊D₇	1	2	yes	—
264	C₃	#90524	1	1	yes	characteristic, direct product factor, Sylow
265	C₂	—	2	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	28	2	no	—
268	C₂	C₃×#6553	1	1	yes	characteristic
269	C₂	C₃×#6554	1	1	yes	characteristic
270	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₃×#41	D₇
hypocenter	C₇	C₃×#537
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₂×C₃×#1157	C₂	semidirect
3. C₃×#6509	C₂	semidirect
4. C₃×#6502	C₂	semidirect
5. C₃×#6501	C₂	semidirect
6. C₃×#6493	C₂	semidirect
7. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
8. C₃×#6487	C₂	semidirect
9. #90524	C₃	direct (N × H)
28. C₂×C₇×#41	C₂×C₃	semidirect
29. C₂×#1157	C₂×C₃	semidirect
30. #6509	C₂×C₃	semidirect
31. #6502	C₂×C₃	semidirect
32. #6501	C₂×C₃	semidirect
33. #6493	C₂×C₃	semidirect
34. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
35. #6487	C₂×C₃	semidirect
44. C₂×C₃×C₄×C₇	D₄	semidirect
75. C₂×C₃×#41	D₇	semidirect
105. C₂×C₄×C₇	C₃×D₄	semidirect
142. C₂×#41	C₃×D₇	semidirect
179. C₂×C₃×C₄	C₂²⋊D₇	semidirect
183. C₃×C₇	#537	semidirect
238. C₂×C₄	C₃×C₂²⋊D₇	semidirect
242. C₇	C₃×#537	semidirect
264. C₃	#90524	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×C₇×#41	2
C₂×C₃×#1157	2
C₃×#6509	2
C₃×#6502	2
C₃×#6501	2
C₃×#6493	2
C₂×C₃×C₄⋊Dic₇	2
C₃×#6487	2
#90524	3
C₃×#537	7

Subgroups — C3×#90524 #13061174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90524 #13061174