Subgroups — #13054244

Order 1344.

436 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×D₇)⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10620670	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₃⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	#10620529	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620529	C₂	1	1	yes	maximal, characteristic, complemented
8	#89436	—	3	1	no	maximal, Hall
9	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
13	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₃×D₇	—	2	2	no	—
17	C₂×Dic₃×D₇	—	2	2	no	—
18	C₂×Dic₃×D₇	C₄	1	4	yes	complemented
19	C₃×(C₂×D₇)⋊C₄	—	2	2	no	—
20	(C₂×D₇)⋊Dic₃	—	2	2	no	—
21	C₂²×C₄×D₇	—	3	1	no	—
22	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
23	C₂×(C₂×D₇)⋊C₄	D₃	1	1	yes	characteristic
24	C₇×#95	—	3	1	no	—
25	C₇⋊#95	—	3	1	no	—
26	C₇⋊#95	—	3	1	no	—
27	C₇⋊#95	—	3	1	no	—
28	#4943	—	7	1	no	maximal, Hall
29	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
30	C₂²×C₃×D₇	—	2	1	no	—
31	C₂²×C₃×D₇	—	2	1	no	—
32	C₂²×C₃×D₇	—	2	1	no	—
33	C₂²×C₃×D₇	—	2	1	no	—
34	C₂²×C₃×D₇	C₂×C₄	1	2	yes	—
35	C₂²×C₃×D₇	D₄	1	2	yes	—
36	C₂²×C₃×D₇	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₇	—	2	1	no	—
38	C₂×C₃×C₄×C₇	—	2	1	no	—
39	C₂×C₃×C₄×C₇	—	2	1	no	—
40	C₂×Dic₂₁	—	2	1	no	—
41	C₂×Dic₂₁	—	2	1	no	—
42	C₂×Dic₂₁	—	2	1	no	—
43	C₂×Dic₂₁	—	2	1	no	—
44	C₂×Dic₂₁	—	2	1	no	—
45	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
46	C₂×C₇×Dic₃	—	2	1	no	—
47	C₂×C₇×Dic₃	—	2	1	no	—
48	C₂×C₇×Dic₃	—	2	1	no	—
49	C₂×C₇×Dic₃	—	2	1	no	—
50	C₂×C₇×Dic₃	—	2	1	no	—
51	C₂×C₇×Dic₃	C₂×C₄	1	2	yes	—
52	C₂×C₃×Dic₇	—	2	1	no	—
53	C₂×C₃×Dic₇	—	2	1	no	—
54	C₂×C₃×Dic₇	—	2	1	no	—
55	Dic₃×D₇	—	2	8	no	—
56	C₂³×D₇	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₄×C₇	—	3	1	no	—
58	C₂²×C₄×C₇	—	3	1	no	—
59	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
60	C₂²×Dic₇	—	3	1	no	—
61	C₂²×Dic₇	—	3	1	no	—
62	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄×D₇	—	3	4	no	—
64	C₂×C₄×D₇	—	6	2	no	—
65	C₂×C₄×D₇	—	6	2	no	—
66	(C₂×D₇)⋊C₄	—	2	2	no	—
67	(C₂×D₇)⋊C₄	—	6	2	no	—
68	C₂³×Dic₃	—	7	1	no	—
69	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
70	C₂×#217	—	7	1	no	—
71	C₃⋊#95	—	7	1	no	—
72	C₃⋊#95	—	7	1	no	—
73	C₃⋊#95	—	7	1	no	—
74	C₃⋊#95	D₇	1	1	yes	characteristic, complemented
75	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₇	#41	1	1	yes	characteristic
79	C₂²×C₃×C₇	#41	1	1	yes	characteristic
80	C₂²×C₃×C₇	#41	1	1	yes	characteristic
81	C₂²×C₃×C₇	#41	1	1	yes	characteristic
82	C₂×C₃×D₇	—	2	2	no	—
83	C₂×C₃×D₇	—	2	2	no	—
84	C₂×C₃×D₇	—	2	2	no	—
85	C₂×C₃×D₇	—	2	2	no	—
86	C₂×C₃×D₇	—	2	2	no	—
87	C₂×C₃×D₇	—	2	2	no	—
88	C₂×C₃×D₇	C₂²⋊C₄	1	4	yes	—
89	C₃×C₄×C₇	—	4	1	no	—
90	Dic₂₁	—	2	2	no	—
91	Dic₂₁	—	4	1	no	—
92	C₃×Dic₇	—	4	1	no	—
93	C₇×Dic₃	—	2	2	no	—
94	C₇×Dic₃	—	4	1	no	—
95	#381	—	21	1	no	Sylow
96	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
97	C₂²×D₇	—	2	1	no	—
98	C₂²×D₇	—	2	1	no	—
99	C₂²×D₇	—	2	1	no	—
100	C₂²×D₇	—	2	1	no	—
101	C₂²×D₇	C₄×D₃	1	2	yes	—
102	C₂²×D₇	D₁₂	1	2	yes	—
103	C₂²×D₇	#67	1	2	yes	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	3	2	no	—
108	C₂×C₄×C₇	—	6	1	no	—
109	C₂×C₄×C₇	—	6	1	no	—
110	C₂×C₄×C₇	—	6	1	no	—
111	C₂×C₄×C₇	—	6	1	no	—
112	C₂×C₄×C₇	—	6	1	no	—
113	C₂×Dic₇	—	2	1	no	—
114	C₂×Dic₇	—	2	1	no	—
115	C₂×Dic₇	—	2	1	no	—
116	C₂×Dic₇	—	3	2	no	—
117	C₂×Dic₇	—	6	1	no	—
118	C₂×Dic₇	—	6	1	no	—
119	C₂×Dic₇	—	6	1	no	—
120	C₂×Dic₇	—	6	1	no	—
121	C₂×Dic₇	—	6	1	no	—
122	C₄×D₇	—	6	8	no	—
123	C₂⁴×C₃	—	7	1	no	—
124	C₂²×C₃×C₄	—	7	1	no	—
125	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
126	C₂²×Dic₃	—	7	1	no	—
127	C₂²×Dic₃	—	7	4	no	—
128	C₂²×Dic₃	—	7	1	no	—
129	C₂²×Dic₃	—	14	2	no	—
130	C₂²×Dic₃	—	14	2	no	—
131	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
132	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
133	C₃×C₂²⋊C₄	—	14	2	no	—
134	#217	—	14	2	no	—
135	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₇	#117	1	1	yes	characteristic
137	C₂×C₃×C₇	#117	1	1	yes	characteristic
138	C₂×C₃×C₇	#123	1	1	yes	characteristic
139	C₂×C₃×C₇	#123	1	1	yes	characteristic
140	C₂×C₃×C₇	#123	1	1	yes	characteristic
141	C₂×C₃×C₇	#123	1	1	yes	characteristic
142	C₃×D₇	—	2	4	no	—
143	C₂³×C₄	—	21	1	no	—
144	C₂×C₂²⋊C₄	—	7	1	no	—
145	C₂×C₂²⋊C₄	—	21	1	no	—
146	#95	—	3	1	no	—
147	#95	—	21	1	no	—
148	#95	—	21	1	no	—
149	#95	—	21	1	no	—
150	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
152	C₂²×C₇	#237	1	1	yes	characteristic
153	C₂²×C₇	#237	1	1	yes	characteristic
154	C₂²×C₇	#237	1	1	yes	characteristic
155	C₂²×C₇	#237	1	1	yes	characteristic
156	C₂²×C₇	C₂×#67	1	1	yes	characteristic
157	C₂×D₇	—	2	2	no	—
158	C₂×D₇	—	2	2	no	—
159	C₂×D₇	—	2	2	no	—
160	C₂×D₇	—	2	2	no	—
161	C₂×D₇	—	2	2	no	—
162	C₂×D₇	—	2	2	no	—
163	C₂×D₇	(C₂×D₃)⋊C₄	1	4	yes	—
164	C₄×C₇	—	4	1	no	—
165	C₄×C₇	—	6	2	no	—
166	C₄×C₇	—	12	1	no	—
167	Dic₇	—	4	1	no	—
168	Dic₇	—	6	2	no	—
169	Dic₇	—	12	1	no	—
170	C₂³×C₃	—	7	2	no	—
171	C₂³×C₃	—	7	2	no	—
172	C₂³×C₃	—	7	2	no	—
173	C₂³×C₃	—	14	1	no	—
174	C₂³×C₃	—	14	1	no	—
175	C₂³×C₃	—	14	1	no	—
176	C₂³×C₃	—	14	1	no	—
177	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	14	1	no	—
182	C₂×C₃×C₄	—	14	1	no	—
183	C₂×C₃×C₄	—	14	1	no	—
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	2	1	no	—
186	C₂×Dic₃	—	2	1	no	—
187	C₂×Dic₃	—	2	1	no	—
188	C₂×Dic₃	—	2	1	no	—
189	C₂×Dic₃	—	7	2	no	—
190	C₂×Dic₃	—	14	8	no	—
191	C₂×Dic₃	—	14	1	no	—
192	C₂×Dic₃	—	14	1	no	—
193	C₂×Dic₃	—	14	1	no	—
194	C₂×Dic₃	—	14	1	no	—
195	C₂×Dic₃	—	14	1	no	—
196	C₂×Dic₃	C₄×D₇	1	2	yes	—
197	C₃×C₇	#381	1	1	yes	characteristic, complemented, Hall
198	C₂⁴	—	7	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	1	no	—
201	C₂²×C₄	—	7	1	no	—
202	C₂²×C₄	—	21	4	no	—
203	C₂²×C₄	—	21	1	no	—
204	C₂²×C₄	—	21	1	no	—
205	C₂²×C₄	—	42	2	no	—
206	C₂²×C₄	—	42	2	no	—
207	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
208	C₂²⋊C₄	—	14	2	no	—
209	C₂²⋊C₄	—	42	2	no	—
210	C₂×C₇	#877	1	1	yes	characteristic
211	C₂×C₇	#877	1	1	yes	characteristic
212	C₂×C₇	#886	1	1	yes	characteristic
213	C₂×C₇	#886	1	1	yes	characteristic
214	C₂×C₇	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
215	C₂×C₇	#935	1	1	yes	characteristic
216	C₂×C₇	#935	1	1	yes	characteristic
217	D₇	—	2	4	no	—
218	C₂²×C₃	—	7	4	no	—
219	C₂²×C₃	—	14	2	no	—
220	C₂²×C₃	—	14	2	no	—
221	C₂²×C₃	—	14	2	no	—
222	C₂²×C₃	—	14	2	no	—
223	C₂²×C₃	—	14	2	no	—
224	C₂²×C₃	—	14	2	no	—
225	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
226	C₂²×C₃	#1157	1	1	yes	characteristic
227	C₂²×C₃	#1157	1	1	yes	characteristic
228	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
229	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
230	C₂²×C₃	#1159	1	1	yes	characteristic
231	C₂²×C₃	#1161	1	1	yes	characteristic
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	28	1	no	—
234	Dic₃	—	2	2	no	—
235	Dic₃	—	4	1	no	—
236	Dic₃	—	14	2	no	—
237	Dic₃	—	28	1	no	—
238	C₂³	—	7	2	no	—
239	C₂³	—	7	2	no	—
240	C₂³	—	7	2	no	—
241	C₂³	—	14	1	no	—
242	C₂³	—	14	1	no	—
243	C₂³	—	14	1	no	—
244	C₂³	—	14	1	no	—
245	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	3	2	no	—
250	C₂×C₄	—	6	1	no	—
251	C₂×C₄	—	6	1	no	—
252	C₂×C₄	—	6	1	no	—
253	C₂×C₄	—	6	1	no	—
254	C₂×C₄	—	6	1	no	—
255	C₂×C₄	—	14	1	no	—
256	C₂×C₄	—	14	1	no	—
257	C₂×C₄	—	14	1	no	—
258	C₂×C₄	—	21	2	no	—
259	C₂×C₄	—	42	8	no	—
260	C₂×C₄	—	42	1	no	—
261	C₂×C₄	—	42	1	no	—
262	C₂×C₄	—	42	1	no	—
263	C₂×C₄	—	42	1	no	—
264	C₂×C₄	—	42	1	no	—
265	C₇	#4943	1	1	yes	characteristic, complemented, Sylow
266	C₂×C₃	—	14	4	no	—
267	C₂×C₃	#6435	1	1	yes	characteristic
268	C₂×C₃	D₇×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₃	#6446	1	1	yes	characteristic
270	C₂×C₃	#6446	1	1	yes	characteristic
271	C₂×C₃	#6455	1	1	yes	characteristic
272	C₂×C₃	#6457	1	1	yes	characteristic
273	C₂×C₃	#6457	1	1	yes	characteristic
274	C₂²	—	7	4	no	—
275	C₂²	—	14	2	no	—
276	C₂²	—	14	2	no	—
277	C₂²	—	14	2	no	—
278	C₂²	—	14	2	no	—
279	C₂²	—	14	2	no	—
280	C₂²	—	14	2	no	—
281	C₂²	#66669	1	1	yes	characteristic
282	C₂²	#66670	1	1	yes	characteristic
283	C₂²	C₄×D₃×D₇	1	1	yes	characteristic
284	C₂²	D₇×D₁₂	1	1	yes	characteristic
285	C₂²	#66683	1	1	yes	characteristic
286	C₂²	#66684	1	1	yes	characteristic
287	C₂²	D₇×#67	1	1	yes	characteristic
288	C₄	—	4	1	no	—
289	C₄	—	6	2	no	—
290	C₄	—	12	1	no	—
291	C₄	—	28	1	no	—
292	C₄	—	42	2	no	—
293	C₄	—	84	1	no	—
294	C₃	#89436	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	14	4	no	—
296	C₂	#10620955	1	1	yes	characteristic
297	C₂	#10620956	1	1	yes	characteristic
298	C₂	#10620969	1	1	yes	characteristic
299	C₂	#10620975	1	1	yes	characteristic
300	C₂	#10620976	1	1	yes	characteristic
301	C₂	D₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
302	C₂	#10621110	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₃×D₇
commutator	C₂²×C₃×C₇	C₂²×C₄
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle, cosocle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#381
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #10620670	C₂	semidirect
5. C₇×C₃⋊#95	C₂	semidirect
6. #10620529	C₂	semidirect
7. #10620529	C₂	semidirect
18. C₂×Dic₃×D₇	C₄	semidirect
74. C₃⋊#95	D₇	semidirect
197. C₃×C₇	#381	semidirect
265. C₇	#4943	semidirect
294. C₃	#89436	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₃×D₇	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×(C₂×D₇)⋊Dic₃	2
#10620670	2
C₇×C₃⋊#95	2
#10620529	2
#89436	3
#4943	7