Subgroups — C₃×#90272 #13060923

Order 1344.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 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	1	yes	maximal, characteristic, complemented
3	C₃×#6509	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6457	C₂	1	2	yes	maximal
5	C₃×#6455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6452	C₂	1	2	yes	maximal, complemented
8	C₃×#6439	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
10	C₃×#6435	C₂	1	2	yes	maximal, complemented
11	C₃×#6433	C₂	1	1	yes	maximal, characteristic, complemented
12	#90272	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₇	C₂²	1	2	yes	—
22	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
24	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
33	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
34	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₃×C₄×Dic₇	C₂²	1	2	yes	—
36	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	#6509	C₂×C₃	1	1	yes	characteristic, complemented
38	#6509	C₂×C₃	1	1	yes	characteristic, complemented
39	#6457	C₂×C₃	1	2	yes	—
40	#6455	C₂×C₃	1	1	yes	characteristic, complemented
41	D₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
42	#6452	C₂×C₃	1	2	yes	complemented
43	#6439	C₂×C₃	1	2	yes	complemented
44	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic
45	#6435	C₂×C₃	1	2	yes	complemented
46	#6433	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×#552	—	7	1	no	maximal, Hall
48	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
50	C₃×C₇×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
57	C₂×C₃×Dic₇	C₂³	1	2	yes	—
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	2	yes	—
60	C₃×D₂₈	—	2	2	no	—
61	C₃×C₄×D₇	—	2	4	no	—
62	C₃×C₄×D₇	—	2	2	no	—
63	C₃×C₄×D₇	—	2	2	no	—
64	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
72	C₄²×C₇	C₂²×C₃	1	2	yes	—
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
82	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
85	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	2	yes	—
87	C₃×#126	—	7	2	no	—
88	C₃×#125	—	7	2	no	—
89	C₃×#125	—	7	1	no	—
90	C₃×#123	—	7	2	no	—
91	C₃×C₂²⋊Q₈	—	7	1	no	—
92	C₃×C₂²⋊Q₈	—	7	1	no	—
93	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
94	C₃×C₄×D₄	—	7	1	no	—
95	C₃×#117	—	7	1	no	—
96	C₃×#117	—	7	2	no	—
97	C₂×C₃×C₄⋊C₄	—	7	1	no	—
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×D₇	—	2	1	no	—
100	C₂×C₃×D₇	—	2	1	no	—
101	C₂×C₃×D₇	—	2	1	no	—
102	C₂×C₃×D₇	—	2	1	no	—
103	C₂×C₃×D₇	—	2	1	no	—
104	C₂×C₃×D₇	#41	1	2	yes	—
105	C₃×C₄×C₇	—	2	2	no	—
106	C₃×C₄×C₇	—	2	1	no	—
107	C₃×C₄×C₇	—	2	2	no	—
108	C₃×C₄×C₇	—	2	1	no	—
109	C₃×C₄×C₇	#41	1	2	yes	—
110	C₃×Dic₇	—	2	1	no	—
111	C₃×Dic₇	—	2	1	no	—
112	C₃×Dic₇	—	2	2	no	—
113	C₃×Dic₇	—	2	2	no	—
114	#552	—	7	1	no	Sylow
115	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
116	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
117	C₇×Q₈	—	2	2	no	—
118	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
119	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂³×C₃	1	2	yes	—
125	C₂×Dic₇	C₂³×C₃	1	2	yes	—
126	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
127	D₂₈	—	2	2	no	—
128	C₄×D₇	—	2	4	no	—
129	C₄×D₇	—	2	2	no	—
130	C₄×D₇	—	2	2	no	—
131	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
132	C₂×C₃×D₄	—	7	1	no	—
133	C₂²×C₃×C₄	—	7	2	no	—
134	C₂²×C₃×C₄	—	7	1	no	—
135	C₂²×C₃×C₄	—	7	1	no	—
136	C₃×C₄⋊C₄	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	2	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	1	no	—
142	C₃×C₄⋊C₄	—	7	1	no	—
143	C₃×C₄⋊C₄	—	7	1	no	—
144	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
145	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
146	C₃×C₂²⋊C₄	—	7	1	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	1	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₄²	—	7	2	no	—
152	C₃×C₄²	C₂×D₇	1	2	yes	—
153	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
154	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
157	C₃×D₇	—	2	2	no	—
158	C₃×D₇	—	4	1	no	—
159	#126	—	7	2	no	—
160	#125	—	7	2	no	—
161	#125	—	7	1	no	—
162	#123	—	7	2	no	—
163	C₂²⋊Q₈	—	7	1	no	—
164	C₂²⋊Q₈	—	7	1	no	—
165	C₄×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
166	C₄×D₄	—	7	1	no	—
167	#117	—	7	1	no	—
168	#117	—	7	2	no	—
169	C₂×C₄⋊C₄	—	7	1	no	—
170	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
171	C₂×D₇	—	2	1	no	—
172	C₂×D₇	—	2	1	no	—
173	C₂×D₇	—	2	1	no	—
174	C₂×D₇	—	2	1	no	—
175	C₂×D₇	—	2	1	no	—
176	C₂×D₇	C₃×#41	1	2	yes	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	C₃×#41	1	2	yes	—
182	Dic₇	—	2	1	no	—
183	Dic₇	—	2	1	no	—
184	Dic₇	—	2	2	no	—
185	Dic₇	—	2	2	no	—
186	C₂³×C₃	—	7	1	no	—
187	C₂³×C₃	—	7	1	no	—
188	C₃×Q₈	—	2	2	no	—
189	C₃×D₄	—	14	2	no	—
190	C₂×C₃×C₄	—	7	1	no	—
191	C₂×C₃×C₄	—	7	2	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	14	4	no	—
195	C₂×C₃×C₄	—	14	2	no	—
196	C₂×C₃×C₄	—	14	2	no	—
197	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
198	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
200	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
202	C₃×C₇	#552	1	1	yes	characteristic, complemented, Hall
203	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
204	C₂×D₄	—	7	1	no	—
205	C₂²×C₄	—	7	2	no	—
206	C₂²×C₄	—	7	1	no	—
207	C₂²×C₄	—	7	1	no	—
208	C₄⋊C₄	—	7	2	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₄⋊C₄	—	7	2	no	—
212	C₄⋊C₄	—	7	2	no	—
213	C₄⋊C₄	—	7	1	no	—
214	C₄⋊C₄	—	7	1	no	—
215	C₄⋊C₄	—	7	1	no	—
216	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
218	C₂²⋊C₄	—	7	1	no	—
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	—	7	1	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₄²	—	7	2	no	—
224	C₄²	C₂×C₃×D₇	1	2	yes	—
225	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
226	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
227	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
228	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
229	D₇	—	2	2	no	—
230	D₇	—	4	1	no	—
231	C₂²×C₃	—	7	2	no	—
232	C₂²×C₃	—	14	1	no	—
233	C₂²×C₃	—	14	1	no	—
234	C₂²×C₃	—	14	1	no	—
235	C₂²×C₃	—	14	1	no	—
236	C₂²×C₃	—	14	1	no	—
237	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	14	1	no	—
243	C₃×C₄	—	14	1	no	—
244	C₃×C₄	—	14	2	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	#1157	1	2	yes	—
247	C₂³	—	7	1	no	—
248	C₂³	—	7	1	no	—
249	Q₈	—	2	2	no	—
250	D₄	—	14	2	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	7	2	no	—
253	C₂×C₄	—	7	1	no	—
254	C₂×C₄	—	7	2	no	—
255	C₂×C₄	—	14	4	no	—
256	C₂×C₄	—	14	2	no	—
257	C₂×C₄	—	14	2	no	—
258	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
259	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
261	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
263	C₇	C₃×#552	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	—	14	2	no	—
265	C₂×C₃	—	28	1	no	—
266	C₂×C₃	C₂×#1157	1	1	yes	characteristic
267	C₂×C₃	#6551	1	1	yes	characteristic
268	C₂×C₃	D₇×#41	1	1	yes	characteristic
269	C₂²	—	7	2	no	—
270	C₂²	—	14	1	no	—
271	C₂²	—	14	1	no	—
272	C₂²	—	14	1	no	—
273	C₂²	—	14	1	no	—
274	C₂²	—	14	1	no	—
275	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	1	no	—
280	C₄	—	14	1	no	—
281	C₄	—	14	1	no	—
282	C₄	—	14	2	no	—
283	C₄	—	14	2	no	—
284	C₄	C₃×#1157	1	2	yes	—
285	C₃	#90272	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	14	2	no	—
287	C₂	—	28	1	no	—
288	C₂	C₂×C₃×#1157	1	1	yes	characteristic
289	C₂	C₃×#6551	1	1	yes	characteristic
290	C₂	C₃×D₇×#41	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×#552
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×#6509	C₂	semidirect
5. C₃×#6455	C₂	semidirect
6. C₃×D₇×C₄⋊C₄	C₂	semidirect
7. C₃×#6452	C₂	semidirect
8. C₃×#6439	C₂	semidirect
10. C₃×#6435	C₂	semidirect
11. C₃×#6433	C₂	semidirect
12. #90272	C₃	direct (N × H)
36. C₄×C₇×Q₈	C₂×C₃	semidirect
37. #6509	C₂×C₃	semidirect
38. #6509	C₂×C₃	semidirect
40. #6455	C₂×C₃	semidirect
41. D₇×C₄⋊C₄	C₂×C₃	semidirect
42. #6452	C₂×C₃	semidirect
43. #6439	C₂×C₃	semidirect
45. #6435	C₂×C₃	semidirect
46. #6433	C₂×C₃	semidirect
93. C₃×C₄×Q₈	D₇	semidirect
165. C₄×Q₈	C₃×D₇	semidirect
202. C₃×C₇	#552	semidirect
263. C₇	C₃×#552	semidirect
285. C₃	#90272	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×#6457	2
C₃×#6455	2
C₃×D₇×C₄⋊C₄	2
C₃×#6452	2
C₃×#6439	2
C₃×C₄×D₂₈	2
C₃×#6435	2
C₃×#6433	2
#90272	3
C₃×#552	7

Subgroups — C3×#90272 #13060923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90272 #13060923