Subgroups — C₃×#90332 #13060970

Order 1344.

424 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×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic
5	C₃×#6489	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6461	C₂	1	2	yes	maximal, complemented
7	C₃×#6455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6450	C₂	1	2	yes	maximal, complemented
10	C₃×#6448	C₂	1	2	yes	maximal
11	C₃×#6442	C₂	1	2	yes	maximal, complemented
12	#90332	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
17	C₂×C₃×Dic₁₄	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×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
32	C₃×C₄×Dic₇	C₂²	1	2	yes	—
33	C₃×C₄×Dic₇	C₂²	1	2	yes	—
34	C₃×C₄×Dic₇	C₂²	1	2	yes	—
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	#6509	C₂×C₃	1	1	yes	characteristic, complemented
38	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic
40	#6489	C₂×C₃	1	1	yes	characteristic, complemented
41	#6461	C₂×C₃	1	2	yes	complemented
42	#6455	C₂×C₃	1	1	yes	characteristic, complemented
43	#6451	C₂×C₃	1	1	yes	characteristic, complemented
44	#6450	C₂×C₃	1	2	yes	complemented
45	#6448	C₂×C₃	1	2	yes	—
46	#6442	C₂×C₃	1	2	yes	complemented
47	C₃×#556	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	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	—	2	2	no	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₃×C₂²⋊D₇	—	2	2	no	—
57	C₂×C₃×Dic₇	C₂³	1	2	yes	—
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₇	—	2	2	no	—
63	C₃×Dic₁₄	—	2	4	no	—
64	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
69	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
73	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
74	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
78	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
82	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₄×Dic₇	C₂²×C₃	1	2	yes	—
84	C₄×Dic₇	C₂²×C₃	1	2	yes	—
85	C₄×Dic₇	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₃×C₄⋊Q₈	—	7	1	no	—
88	C₃×#126	—	7	2	no	—
89	C₃×#126	—	7	2	no	—
90	C₃×#124	—	7	2	no	—
91	C₃×C₂²⋊Q₈	—	7	1	no	—
92	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
93	C₃×C₄×Q₈	—	7	1	no	—
94	C₃×C₄×Q₈	—	7	2	no	—
95	C₃×C₄×D₄	—	7	1	no	—
96	C₃×#117	—	7	1	no	—
97	C₃×#117	—	7	1	no	—
98	C₂²×C₃×C₇	—	2	1	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₃×D₇	—	2	1	no	—
103	C₂×C₃×D₇	—	2	1	no	—
104	C₂×C₃×D₇	—	2	1	no	—
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	#41	1	2	yes	—
109	C₃×Dic₇	—	2	2	no	—
110	C₃×Dic₇	—	2	1	no	—
111	C₃×Dic₇	—	2	2	no	—
112	C₃×Dic₇	—	2	1	no	—
113	C₃×Dic₇	#41	1	2	yes	—
114	#556	—	7	1	no	Sylow
115	C₂³×C₇	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	—	2	2	no	—
119	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
121	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂²⋊D₇	—	2	2	no	—
124	C₂×Dic₇	C₂³×C₃	1	2	yes	—
125	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
129	C₄×D₇	—	2	2	no	—
130	Dic₁₄	—	2	4	no	—
131	C₂×C₃×Q₈	—	7	2	no	—
132	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
133	C₂×C₃×D₄	—	7	1	no	—
134	C₂²×C₃×C₄	—	7	1	no	—
135	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	7	1	no	—
137	C₃×C₄⋊C₄	—	7	2	no	—
138	C₃×C₄⋊C₄	—	7	2	no	—
139	C₃×C₄⋊C₄	—	7	1	no	—
140	C₃×C₄⋊C₄	—	7	1	no	—
141	C₃×C₄⋊C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
143	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	1	no	—
146	C₃×C₂²⋊C₄	—	7	2	no	—
147	C₃×C₂²⋊C₄	—	7	1	no	—
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
150	C₃×C₄²	—	7	2	no	—
151	C₃×C₄²	—	7	2	no	—
152	C₃×C₄²	—	7	1	no	—
153	C₃×C₄²	—	7	2	no	—
154	C₂×C₃×C₇	—	4	1	no	—
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
158	C₃×D₇	—	4	1	no	—
159	C₄⋊Q₈	—	7	1	no	—
160	#126	—	7	2	no	—
161	#126	—	7	2	no	—
162	#124	—	7	2	no	—
163	C₂²⋊Q₈	—	7	1	no	—
164	C₂²⋊Q₈	C₃×D₇	1	1	yes	characteristic, complemented
165	C₄×Q₈	—	7	1	no	—
166	C₄×Q₈	—	7	2	no	—
167	C₄×D₄	—	7	1	no	—
168	#117	—	7	1	no	—
169	#117	—	7	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×D₇	—	2	1	no	—
175	C₂×D₇	—	2	1	no	—
176	C₂×D₇	—	2	1	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	C₃×#41	1	2	yes	—
181	Dic₇	—	2	2	no	—
182	Dic₇	—	2	1	no	—
183	Dic₇	—	2	2	no	—
184	Dic₇	—	2	1	no	—
185	Dic₇	C₃×#41	1	2	yes	—
186	C₂³×C₃	—	7	1	no	—
187	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
188	C₃×Q₈	—	2	2	no	—
189	C₃×Q₈	—	14	4	no	—
190	C₃×D₄	—	14	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	2	no	—
194	C₂×C₃×C₄	—	7	1	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	14	2	no	—
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	2	yes	—
201	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
202	C₃×C₇	#556	1	1	yes	characteristic, complemented, Hall
203	C₂×Q₈	—	7	2	no	—
204	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
205	C₂×D₄	—	7	1	no	—
206	C₂²×C₄	—	7	1	no	—
207	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
208	C₄⋊C₄	—	7	1	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	2	no	—
211	C₄⋊C₄	—	7	1	no	—
212	C₄⋊C₄	—	7	1	no	—
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
216	C₂²⋊C₄	—	7	2	no	—
217	C₂²⋊C₄	—	7	1	no	—
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	1	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
222	C₄²	—	7	2	no	—
223	C₄²	—	7	2	no	—
224	C₄²	—	7	1	no	—
225	C₄²	—	7	2	no	—
226	C₂×C₇	—	4	1	no	—
227	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
228	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
229	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
230	D₇	—	4	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	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	1	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	7	2	no	—
242	C₃×C₄	—	14	1	no	—
243	C₃×C₄	—	14	2	no	—
244	C₃×C₄	—	14	1	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	#1159	1	2	yes	—
247	C₂³	—	7	1	no	—
248	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	14	4	no	—
251	D₄	—	14	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	7	1	no	—
254	C₂×C₄	—	7	2	no	—
255	C₂×C₄	—	7	1	no	—
256	C₂×C₄	—	7	1	no	—
257	C₂×C₄	—	7	2	no	—
258	C₂×C₄	—	14	2	no	—
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	2	yes	—
262	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
263	C₇	C₃×#556	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	—	28	1	no	—
266	C₂×C₃	C₂×#1159	1	1	yes	characteristic
267	C₂×C₃	#6551	1	1	yes	characteristic
268	C₂×C₃	D₇×#41	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	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	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	7	2	no	—
280	C₄	—	14	1	no	—
281	C₄	—	14	2	no	—
282	C₄	—	14	1	no	—
283	C₄	—	14	2	no	—
284	C₄	C₃×#1159	1	2	yes	—
285	C₃	#90332	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	4	1	no	—
287	C₂	—	28	1	no	—
288	C₂	C₂×C₃×#1159	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×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×C₂²⋊D₇	C₂²
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#556
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
2. C₃×#6509	C₂	semidirect
3. C₃×Q₈×Dic₇	C₂	semidirect
5. C₃×#6489	C₂	semidirect
6. C₃×#6461	C₂	semidirect
7. C₃×#6455	C₂	semidirect
8. C₃×#6451	C₂	semidirect
9. C₃×#6450	C₂	semidirect
11. C₃×#6442	C₂	semidirect
12. #90332	C₃	direct (N × H)
36. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
37. #6509	C₂×C₃	semidirect
38. Q₈×Dic₇	C₂×C₃	semidirect
40. #6489	C₂×C₃	semidirect
41. #6461	C₂×C₃	semidirect
42. #6455	C₂×C₃	semidirect
43. #6451	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
46. #6442	C₂×C₃	semidirect
92. C₃×C₂²⋊Q₈	D₇	semidirect
164. C₂²⋊Q₈	C₃×D₇	semidirect
202. C₃×C₇	#556	semidirect
263. C₇	C₃×#556	semidirect
285. C₃	#90332	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×C₇×C₂²⋊Q₈	2
C₃×#6509	2
C₃×Q₈×Dic₇	2
C₃×C₄×C₂²⋊D₇	2
C₃×#6489	2
C₃×#6461	2
C₃×#6455	2
C₃×#6451	2
C₃×#6450	2
C₃×#6448	2
C₃×#6442	2
#90332	3
C₃×#556	7

Subgroups — C3×#90332 #13060970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90332 #13060970