Subgroups — #13053973

Order 1344.

380 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₇)⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₂×(C₂×D₇)⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	#10620670	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620529	C₂	1	1	yes	maximal, characteristic, complemented
8	#89712	—	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	C₄×C₇×Dic₃	—	2	2	no	—
17	C₃×(C₂×D₇)⋊C₄	C₄	1	4	yes	complemented
18	Dic₇⋊Dic₃	—	2	2	no	—
19	(C₂×D₇)⋊Dic₃	—	2	2	no	—
20	(C₂×D₇)⋊Dic₃	—	2	2	no	—
21	C₂×C₄²×C₇	—	3	1	no	—
22	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
23	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
24	C₂×(C₂×D₇)⋊C₄	D₃	1	1	yes	characteristic
25	C₂×Dic₇⋊C₄	—	3	1	no	—
26	C₇⋊#95	—	3	1	no	—
27	C₇⋊#95	—	3	1	no	—
28	#4951	—	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	—	2	1	no	—
35	C₂²×C₃×D₇	—	2	1	no	—
36	C₂²×C₃×D₇	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×C₇	—	2	1	no	—
38	C₂×C₃×C₄×C₇	—	2	1	no	—
39	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
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	D₄	1	2	yes	—
51	C₂×C₇×Dic₃	D₄	1	2	yes	—
52	C₂×C₃×Dic₇	—	2	1	no	—
53	C₂×C₃×Dic₇	—	2	1	no	—
54	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
55	C₂³×D₇	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₄×C₇	—	3	1	no	—
57	C₂²×C₄×C₇	—	3	1	no	—
58	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
59	C₂²×Dic₇	—	3	1	no	—
60	C₂²×Dic₇	—	3	1	no	—
61	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
62	C₄²×C₇	—	6	2	no	—
63	(C₂×D₇)⋊C₄	—	6	2	no	—
64	(C₂×D₇)⋊C₄	—	6	2	no	—
65	(C₂×D₇)⋊C₄	Dic₃	1	4	yes	complemented
66	Dic₇⋊C₄	—	6	2	no	—
67	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
68	C₂×#217	—	7	1	no	—
69	C₂×#217	—	7	1	no	—
70	C₂×C₄⋊Dic₃	—	7	1	no	—
71	C₂×C₄×Dic₃	D₇	1	1	yes	characteristic, complemented
72	C₃⋊#95	—	7	1	no	—
73	C₃⋊#95	—	7	1	no	—
74	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₇	#41	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	C₂×C₃×D₇	—	2	2	no	—
82	C₂×C₃×D₇	—	2	2	no	—
83	C₂×C₃×D₇	—	2	2	no	—
84	C₂×C₃×D₇	—	4	1	no	—
85	C₂×C₃×D₇	—	4	1	no	—
86	C₂×C₃×D₇	—	4	1	no	—
87	C₂×C₃×D₇	—	4	1	no	—
88	C₃×C₄×C₇	—	2	2	no	—
89	Dic₂₁	—	4	1	no	—
90	Dic₂₁	—	4	1	no	—
91	C₃×Dic₇	—	2	2	no	—
92	C₇×Dic₃	—	2	2	no	—
93	C₇×Dic₃	—	2	2	no	—
94	#388	—	21	1	no	Sylow
95	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
96	C₂²×D₇	—	2	1	no	—
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	—	2	1	no	—
102	C₂²×D₇	C₂×Dic₃	1	2	yes	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	3	2	no	—
106	C₂×C₄×C₇	—	3	2	no	—
107	C₂×C₄×C₇	—	6	1	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	C₂×Dic₃	1	2	yes	—
112	C₂×Dic₇	—	2	1	no	—
113	C₂×Dic₇	—	2	1	no	—
114	C₂×Dic₇	—	6	1	no	—
115	C₂×Dic₇	—	6	1	no	—
116	C₂×Dic₇	—	6	1	no	—
117	C₂×Dic₇	—	6	1	no	—
118	C₂×Dic₇	—	6	1	no	—
119	C₂×Dic₇	—	6	1	no	—
120	C₂×Dic₇	C₂×Dic₃	1	2	yes	—
121	C₂⁴×C₃	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	1	no	—
123	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
124	C₂²×Dic₃	—	7	1	no	—
125	C₂²×Dic₃	—	7	1	no	—
126	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
127	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
128	C₃×C₂²⋊C₄	—	7	4	no	—
129	#217	—	14	2	no	—
130	#217	—	14	2	no	—
131	C₄⋊Dic₃	—	14	2	no	—
132	C₄×Dic₃	—	2	2	no	—
133	C₂×C₃×C₇	#117	1	1	yes	characteristic
134	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₃×C₇	#121	1	1	yes	characteristic
137	C₂×C₃×C₇	#123	1	1	yes	characteristic
138	C₂×C₃×C₇	#124	1	1	yes	characteristic
139	C₂×C₃×C₇	#126	1	1	yes	characteristic
140	C₃×D₇	—	4	2	no	—
141	C₂×C₄⋊C₄	—	21	1	no	—
142	C₂×C₂²⋊C₄	—	7	1	no	—
143	C₂×C₂²⋊C₄	—	21	1	no	—
144	C₂×C₂²⋊C₄	—	21	1	no	—
145	C₂×C₄²	—	3	1	no	—
146	#95	—	21	1	no	—
147	#95	—	21	1	no	—
148	C₂²×C₇	#235	1	1	yes	characteristic
149	C₂²×C₇	#235	1	1	yes	characteristic
150	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
151	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
152	C₂²×C₇	#237	1	1	yes	characteristic
153	C₂²×C₇	#237	1	1	yes	characteristic
154	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
155	C₂×D₇	—	2	2	no	—
156	C₂×D₇	—	2	2	no	—
157	C₂×D₇	—	2	2	no	—
158	C₂×D₇	—	4	1	no	—
159	C₂×D₇	—	4	1	no	—
160	C₂×D₇	—	4	1	no	—
161	C₂×D₇	—	4	1	no	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	6	2	no	—
164	C₄×C₇	—	6	2	no	—
165	Dic₇	—	2	2	no	—
166	Dic₇	—	12	1	no	—
167	Dic₇	—	12	1	no	—
168	C₂³×C₃	—	7	2	no	—
169	C₂³×C₃	—	14	1	no	—
170	C₂³×C₃	—	14	1	no	—
171	C₂³×C₃	—	14	1	no	—
172	C₂³×C₃	—	14	1	no	—
173	C₂³×C₃	—	14	1	no	—
174	C₂³×C₃	—	14	1	no	—
175	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	7	2	no	—
179	C₂×C₃×C₄	—	14	1	no	—
180	C₂×C₃×C₄	—	14	1	no	—
181	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	—	2	1	no	—
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	2	1	no	—
186	C₂×Dic₃	—	14	1	no	—
187	C₂×Dic₃	—	14	1	no	—
188	C₂×Dic₃	—	14	1	no	—
189	C₂×Dic₃	—	14	1	no	—
190	C₂×Dic₃	—	14	1	no	—
191	C₂×Dic₃	—	14	1	no	—
192	C₂×Dic₃	D₂₈	1	2	yes	—
193	C₂×Dic₃	C₂²⋊D₇	1	2	yes	—
194	C₃×C₇	#388	1	1	yes	characteristic, complemented, Hall
195	C₂⁴	—	7	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	7	1	no	—
199	C₂²×C₄	—	21	1	no	—
200	C₂²×C₄	—	21	1	no	—
201	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
202	C₄⋊C₄	—	42	2	no	—
203	C₂²⋊C₄	—	7	4	no	—
204	C₂²⋊C₄	—	42	2	no	—
205	C₂²⋊C₄	—	42	2	no	—
206	C₄²	—	6	2	no	—
207	C₂×C₇	#879	1	1	yes	characteristic
208	C₂×C₇	#883	1	1	yes	characteristic
209	C₂×C₇	#884	1	1	yes	characteristic
210	C₂×C₇	#885	1	1	yes	characteristic
211	C₂×C₇	#926	1	1	yes	characteristic
212	C₂×C₇	D₄×Dic₃	1	1	yes	characteristic
213	C₂×C₇	D₄×Dic₃	1	1	yes	characteristic
214	D₇	—	4	2	no	—
215	C₂²×C₃	—	14	2	no	—
216	C₂²×C₃	—	14	2	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₂²×C₃	—	28	1	no	—
219	C₂²×C₃	—	28	1	no	—
220	C₂²×C₃	—	28	1	no	—
221	C₂²×C₃	—	28	1	no	—
222	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
223	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
224	C₂²×C₃	#1157	1	1	yes	characteristic
225	C₂²×C₃	#1157	1	1	yes	characteristic
226	C₂²×C₃	#1157	1	1	yes	characteristic
227	C₂²×C₃	#1157	1	1	yes	characteristic
228	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	14	2	no	—
231	Dic₃	—	2	2	no	—
232	Dic₃	—	2	2	no	—
233	Dic₃	—	28	1	no	—
234	Dic₃	—	28	1	no	—
235	C₂³	—	7	2	no	—
236	C₂³	—	14	1	no	—
237	C₂³	—	14	1	no	—
238	C₂³	—	14	1	no	—
239	C₂³	—	14	1	no	—
240	C₂³	—	14	1	no	—
241	C₂³	—	14	1	no	—
242	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	6	1	no	—
248	C₂×C₄	—	6	1	no	—
249	C₂×C₄	—	6	1	no	—
250	C₂×C₄	—	6	1	no	—
251	C₂×C₄	—	7	2	no	—
252	C₂×C₄	—	14	1	no	—
253	C₂×C₄	—	14	1	no	—
254	C₂×C₄	—	42	1	no	—
255	C₂×C₄	—	42	1	no	—
256	C₂×C₄	—	42	1	no	—
257	C₂×C₄	—	42	1	no	—
258	C₂×C₄	—	42	1	no	—
259	C₂×C₄	—	42	1	no	—
260	C₂×C₄	Dic₃×D₇	1	2	yes	—
261	C₇	#4951	1	1	yes	characteristic, complemented, Sylow
262	C₂×C₃	—	28	2	no	—
263	C₂×C₃	#6435	1	1	yes	characteristic
264	C₂×C₃	C₄×D₂₈	1	1	yes	characteristic
265	C₂×C₃	#6438	1	1	yes	characteristic
266	C₂×C₃	#6439	1	1	yes	characteristic
267	C₂×C₃	C₄×C₂²⋊D₇	1	1	yes	characteristic
268	C₂×C₃	#6492	1	1	yes	characteristic
269	C₂×C₃	#6493	1	1	yes	characteristic
270	C₂²	—	14	2	no	—
271	C₂²	—	14	2	no	—
272	C₂²	—	14	2	no	—
273	C₂²	—	28	1	no	—
274	C₂²	—	28	1	no	—
275	C₂²	—	28	1	no	—
276	C₂²	—	28	1	no	—
277	C₂²	#66669	1	1	yes	characteristic
278	C₂²	#66675	1	1	yes	characteristic
279	C₂²	#66675	1	1	yes	characteristic
280	C₂²	D₃×D₂₈	1	1	yes	characteristic
281	C₂²	C₂×Dic₃×D₇	1	1	yes	characteristic
282	C₂²	#66683	1	1	yes	characteristic
283	C₂²	D₃×C₂²⋊D₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	6	2	no	—
286	C₄	—	6	2	no	—
287	C₄	—	14	2	no	—
288	C₄	—	84	1	no	—
289	C₄	—	84	1	no	—
290	C₃	#89712	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	28	2	no	—
292	C₂	#10620915	1	1	yes	characteristic
293	C₂	#10620919	1	1	yes	characteristic
294	C₂	#10620925	1	1	yes	characteristic
295	C₂	Dic₃×D₂₈	1	1	yes	characteristic
296	C₂	#10621014	1	1	yes	characteristic
297	C₂	#10621016	1	1	yes	characteristic
298	C₂	Dic₃×C₂²⋊D₇	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	#388
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
1. C₂×C₄×C₇×Dic₃	C₂	semidirect
3. C₂×Dic₇⋊Dic₃	C₂	semidirect
6. #10620670	C₂	semidirect
7. #10620529	C₂	semidirect
17. C₃×(C₂×D₇)⋊C₄	C₄	semidirect
65. (C₂×D₇)⋊C₄	Dic₃	semidirect
71. C₂×C₄×Dic₃	D₇	semidirect
194. C₃×C₇	#388	semidirect
261. C₇	#4951	semidirect
290. C₃	#89712	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×Dic₃	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×Dic₇⋊Dic₃	2
C₂×(C₂×D₇)⋊Dic₃	2
#10620670	2
#10620529	2
#89712	3
#4951	7