Subgroups — C₃×#90221 #13060875

Order 1344.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6456	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6453	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6447	C₂	1	2	yes	maximal, complemented
9	C₃×#6446	C₂	1	2	yes	maximal, complemented
10	C₃×#6439	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×D₂₈	C₂	1	2	yes	maximal, complemented
12	#90221	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₃×D₂₈	C₂²	1	2	yes	—
17	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
21	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₇	C₂²	1	2	yes	—
25	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
28	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
29	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
32	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
34	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
37	#6493	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
39	#6458	C₂×C₃	1	1	yes	characteristic, complemented
40	#6456	C₂×C₃	1	1	yes	characteristic, complemented
41	D₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
42	#6453	C₂×C₃	1	1	yes	characteristic, complemented
43	#6447	C₂×C₃	1	2	yes	complemented
44	#6446	C₂×C₃	1	2	yes	complemented
45	#6439	C₂×C₃	1	2	yes	complemented
46	C₄×D₂₈	C₂×C₃	1	2	yes	complemented
47	C₃×#553	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₇	C₂³	1	2	yes	—
50	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	—	2	2	no	—
52	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
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	C₃×C₂²⋊D₇	—	2	2	no	—
57	C₃×C₂²⋊D₇	—	4	2	no	—
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₇	C₂³	1	2	yes	—
62	C₃×D₂₈	—	2	4	no	—
63	C₃×D₂₈	—	2	2	no	—
64	C₃×C₄×D₇	—	2	2	no	—
65	C₃×C₄×D₇	—	4	2	no	—
66	C₃×C₄×D₇	—	4	2	no	—
67	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
70	C₂×D₂₈	C₂²×C₃	1	2	yes	—
71	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
74	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
75	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄²×C₇	C₂²×C₃	1	2	yes	—
79	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
80	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
82	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
83	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
85	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
86	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
88	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
89	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
90	C₃×#126	—	7	2	no	—
91	C₃×#125	—	7	1	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×#121	—	7	1	no	—
94	C₃×#121	—	7	1	no	—
95	C₃×#121	—	7	2	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×C₄×D₄	—	7	1	no	—
98	C₃×C₄×D₄	—	7	2	no	—
99	C₃×#117	D₇	1	1	yes	characteristic, complemented
100	C₂×C₃×C₄⋊C₄	—	7	1	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	—	2	1	no	—
103	C₂²×C₃×C₇	—	2	1	no	—
104	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×D₇	—	2	1	no	—
106	C₂×C₃×D₇	—	2	1	no	—
107	C₂×C₃×D₇	—	2	2	no	—
108	C₂×C₃×D₇	—	2	2	no	—
109	C₂×C₃×D₇	—	2	2	no	—
110	C₂×C₃×D₇	#41	1	2	yes	—
111	C₃×C₄×C₇	—	2	2	no	—
112	C₃×C₄×C₇	—	2	1	no	—
113	C₃×C₄×C₇	—	2	2	no	—
114	C₃×C₄×C₇	#41	1	2	yes	—
115	C₃×Dic₇	—	2	1	no	—
116	C₃×Dic₇	—	2	1	no	—
117	C₃×Dic₇	—	2	1	no	—
118	C₃×Dic₇	—	2	2	no	—
119	#553	—	7	1	no	Sylow
120	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
121	C₂²×D₇	C₂³×C₃	1	2	yes	—
122	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
127	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
128	C₂²⋊D₇	—	2	2	no	—
129	C₂²⋊D₇	—	4	2	no	—
130	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
131	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
132	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
133	C₂×Dic₇	C₂³×C₃	1	2	yes	—
134	D₂₈	—	2	4	no	—
135	D₂₈	—	2	2	no	—
136	C₄×D₇	—	2	2	no	—
137	C₄×D₇	—	4	2	no	—
138	C₄×D₇	—	4	2	no	—
139	C₂×C₃×D₄	—	7	1	no	—
140	C₂×C₃×D₄	—	7	2	no	—
141	C₂×C₃×D₄	—	7	2	no	—
142	C₂×C₃×D₄	—	7	1	no	—
143	C₂²×C₃×C₄	—	7	1	no	—
144	C₂²×C₃×C₄	—	7	2	no	—
145	C₂²×C₃×C₄	—	7	2	no	—
146	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	7	2	no	—
148	C₃×C₄⋊C₄	—	7	1	no	—
149	C₃×C₄⋊C₄	—	7	2	no	—
150	C₃×C₄⋊C₄	—	7	1	no	—
151	C₃×C₄⋊C₄	—	7	2	no	—
152	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
153	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	—	7	1	no	—
155	C₃×C₂²⋊C₄	—	7	1	no	—
156	C₃×C₂²⋊C₄	—	7	2	no	—
157	C₃×C₂²⋊C₄	—	7	2	no	—
158	C₃×C₂²⋊C₄	—	7	2	no	—
159	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
160	C₃×C₄²	—	7	1	no	—
161	C₃×C₄²	C₂×D₇	1	2	yes	—
162	C₂×C₃×C₇	—	4	1	no	—
163	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₃×D₇	—	2	2	no	—
167	C₃×D₇	—	4	2	no	—
168	#126	—	7	2	no	—
169	#125	—	7	1	no	—
170	#123	—	7	2	no	—
171	#121	—	7	1	no	—
172	#121	—	7	1	no	—
173	#121	—	7	2	no	—
174	C₄×D₄	—	7	1	no	—
175	C₄×D₄	—	7	1	no	—
176	C₄×D₄	—	7	2	no	—
177	#117	C₃×D₇	1	1	yes	characteristic, complemented
178	C₂×C₄⋊C₄	—	7	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	—	2	1	no	—
181	C₂²×C₇	—	2	1	no	—
182	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
183	C₂×D₇	—	2	1	no	—
184	C₂×D₇	—	2	1	no	—
185	C₂×D₇	—	2	2	no	—
186	C₂×D₇	—	2	2	no	—
187	C₂×D₇	—	2	2	no	—
188	C₂×D₇	C₃×#41	1	2	yes	—
189	C₄×C₇	—	2	2	no	—
190	C₄×C₇	—	2	1	no	—
191	C₄×C₇	—	2	2	no	—
192	C₄×C₇	C₃×#41	1	2	yes	—
193	Dic₇	—	2	1	no	—
194	Dic₇	—	2	1	no	—
195	Dic₇	—	2	1	no	—
196	Dic₇	—	2	2	no	—
197	C₂³×C₃	—	7	1	no	—
198	C₂³×C₃	—	7	2	no	—
199	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
200	C₃×D₄	—	14	4	no	—
201	C₃×D₄	—	14	2	no	—
202	C₃×D₄	—	14	2	no	—
203	C₃×D₄	—	28	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	7	1	no	—
206	C₂×C₃×C₄	—	7	1	no	—
207	C₂×C₃×C₄	—	7	1	no	—
208	C₂×C₃×C₄	—	7	2	no	—
209	C₂×C₃×C₄	—	14	2	no	—
210	C₂×C₃×C₄	—	28	2	no	—
211	C₂×C₃×C₄	—	28	2	no	—
212	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
214	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
215	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
216	C₃×C₇	#553	1	1	yes	characteristic, complemented, Hall
217	C₂×D₄	—	7	1	no	—
218	C₂×D₄	—	7	2	no	—
219	C₂×D₄	—	7	2	no	—
220	C₂×D₄	—	7	1	no	—
221	C₂²×C₄	—	7	1	no	—
222	C₂²×C₄	—	7	2	no	—
223	C₂²×C₄	—	7	2	no	—
224	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
225	C₄⋊C₄	—	7	2	no	—
226	C₄⋊C₄	—	7	1	no	—
227	C₄⋊C₄	—	7	2	no	—
228	C₄⋊C₄	—	7	1	no	—
229	C₄⋊C₄	—	7	2	no	—
230	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
231	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
232	C₂²⋊C₄	—	7	1	no	—
233	C₂²⋊C₄	—	7	1	no	—
234	C₂²⋊C₄	—	7	2	no	—
235	C₂²⋊C₄	—	7	2	no	—
236	C₂²⋊C₄	—	7	2	no	—
237	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
238	C₄²	—	7	1	no	—
239	C₄²	C₂×C₃×D₇	1	2	yes	—
240	C₂×C₇	—	4	1	no	—
241	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
242	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
243	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
244	D₇	—	2	2	no	—
245	D₇	—	4	2	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	7	2	no	—
250	C₂²×C₃	—	14	1	no	—
251	C₂²×C₃	—	14	1	no	—
252	C₂²×C₃	—	14	2	no	—
253	C₂²×C₃	—	14	2	no	—
254	C₂²×C₃	—	14	2	no	—
255	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	14	1	no	—
260	C₃×C₄	—	14	1	no	—
261	C₃×C₄	—	14	1	no	—
262	C₃×C₄	—	14	2	no	—
263	C₃×C₄	#1157	1	2	yes	—
264	C₂³	—	7	1	no	—
265	C₂³	—	7	2	no	—
266	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
267	D₄	—	14	4	no	—
268	D₄	—	14	2	no	—
269	D₄	—	14	2	no	—
270	D₄	—	28	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	7	1	no	—
273	C₂×C₄	—	7	1	no	—
274	C₂×C₄	—	7	1	no	—
275	C₂×C₄	—	7	2	no	—
276	C₂×C₄	—	14	2	no	—
277	C₂×C₄	—	28	2	no	—
278	C₂×C₄	—	28	2	no	—
279	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
281	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
282	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
283	C₇	C₃×#553	1	1	yes	characteristic, complemented, Sylow
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	14	2	no	—
286	C₂×C₃	—	28	2	no	—
287	C₂×C₃	C₂×#1157	1	1	yes	characteristic
288	C₂×C₃	D₇×#41	1	1	yes	characteristic
289	C₂×C₃	#6553	1	1	yes	characteristic
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	7	2	no	—
294	C₂²	—	14	1	no	—
295	C₂²	—	14	1	no	—
296	C₂²	—	14	2	no	—
297	C₂²	—	14	2	no	—
298	C₂²	—	14	2	no	—
299	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	14	1	no	—
304	C₄	—	14	1	no	—
305	C₄	—	14	1	no	—
306	C₄	—	14	2	no	—
307	C₄	C₃×#1157	1	2	yes	—
308	C₃	#90221	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	4	1	no	—
310	C₂	—	14	2	no	—
311	C₂	—	28	2	no	—
312	C₂	C₂×C₃×#1157	1	1	yes	characteristic
313	C₂	C₃×D₇×#41	1	1	yes	characteristic
314	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×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#553
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#117	C₂	semidirect
2. C₃×#6493	C₂	semidirect
3. C₃×C₄×C₂²⋊D₇	C₂	semidirect
4. C₃×#6458	C₂	semidirect
5. C₃×#6456	C₂	semidirect
6. C₃×D₇×C₄⋊C₄	C₂	semidirect
7. C₃×#6453	C₂	semidirect
8. C₃×#6447	C₂	semidirect
9. C₃×#6446	C₂	semidirect
10. C₃×#6439	C₂	semidirect
11. C₃×C₄×D₂₈	C₂	semidirect
12. #90221	C₃	direct (N × H)
36. C₇×#117	C₂×C₃	semidirect
37. #6493	C₂×C₃	semidirect
38. C₄×C₂²⋊D₇	C₂×C₃	semidirect
39. #6458	C₂×C₃	semidirect
40. #6456	C₂×C₃	semidirect
41. D₇×C₄⋊C₄	C₂×C₃	semidirect
42. #6453	C₂×C₃	semidirect
43. #6447	C₂×C₃	semidirect
44. #6446	C₂×C₃	semidirect
45. #6439	C₂×C₃	semidirect
46. C₄×D₂₈	C₂×C₃	semidirect
99. C₃×#117	D₇	semidirect
177. #117	C₃×D₇	semidirect
216. C₃×C₇	#553	semidirect
283. C₇	C₃×#553	semidirect
308. C₃	#90221	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×#117	2
C₃×#6493	2
C₃×C₄×C₂²⋊D₇	2
C₃×#6458	2
C₃×#6456	2
C₃×D₇×C₄⋊C₄	2
C₃×#6453	2
C₃×#6447	2
C₃×#6446	2
C₃×#6439	2
C₃×C₄×D₂₈	2
#90221	3
C₃×#553	7

Subgroups — C3×#90221 #13060875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90221 #13060875