Subgroups — #13056577

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×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66631	C₂	1	1	yes	maximal, characteristic
3	C₂×#66631	C₂	1	1	yes	maximal, characteristic
4	C₂×#66631	C₂	1	1	yes	maximal, characteristic
5	C₂×#66629	C₂	1	1	yes	maximal, characteristic, complemented
6	#10620670	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620670	C₂	1	1	yes	maximal, characteristic, complemented
8	#89712	—	3	1	no	maximal, Hall
9	C₂³×D₂₁	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
16	C₃×C₄²×C₇	—	2	2	no	—
17	#66631	—	2	2	no	—
18	#66631	—	2	2	no	—
19	#66631	C₄	1	4	yes	complemented
20	#66629	—	2	2	no	—
21	C₂×C₄²×C₇	D₃	1	1	yes	characteristic, complemented
22	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
23	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
24	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
25	C₂×Dic₇⋊C₄	—	3	1	no	—
26	C₇⋊#95	—	3	1	no	—
27	C₇⋊#95	—	3	1	no	—
28	#4940	—	7	1	no	maximal, Hall
29	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
30	C₂²×D₂₁	—	2	1	no	—
31	C₂²×D₂₁	—	2	1	no	—
32	C₂²×D₂₁	—	2	1	no	—
33	C₂²×D₂₁	—	2	1	no	—
34	C₂²×D₂₁	—	2	1	no	—
35	C₂²×D₂₁	—	2	1	no	—
36	C₂²×D₂₁	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	—	2	1	no	—
40	C₂×C₃×C₄×C₇	—	2	1	no	—
41	C₂×C₃×C₄×C₇	—	2	1	no	—
42	C₂×C₃×C₄×C₇	—	2	1	no	—
43	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
45	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
46	C₂×Dic₂₁	—	2	1	no	—
47	C₂×Dic₂₁	—	2	1	no	—
48	C₂×Dic₂₁	—	2	1	no	—
49	C₂×Dic₂₁	—	2	1	no	—
50	C₂×Dic₂₁	—	2	1	no	—
51	C₂×Dic₂₁	—	2	1	no	—
52	C₂×Dic₂₁	—	2	1	no	—
53	C₂×Dic₂₁	—	2	1	no	—
54	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
55	C₂³×D₇	—	3	1	no	—
56	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
62	C₄²×C₇	—	2	2	no	—
63	(C₂×D₇)⋊C₄	—	3	4	no	—
64	(C₂×D₇)⋊C₄	—	6	2	no	—
65	(C₂×D₇)⋊C₄	—	6	2	no	—
66	Dic₇⋊C₄	—	6	2	no	—
67	C₂×C₃×C₄²	D₇	1	1	yes	characteristic, complemented
68	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
69	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
70	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
71	C₂×Dic₃⋊C₄	—	7	1	no	—
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×D₂₁	—	2	2	no	—
82	C₂×D₂₁	—	2	2	no	—
83	C₂×D₂₁	—	2	2	no	—
84	C₂×D₂₁	—	4	1	no	—
85	C₂×D₂₁	—	4	1	no	—
86	C₂×D₂₁	—	4	1	no	—
87	C₂×D₂₁	—	4	1	no	—
88	C₃×C₄×C₇	—	2	2	no	—
89	C₃×C₄×C₇	—	2	2	no	—
90	C₃×C₄×C₇	—	2	2	no	—
91	Dic₂₁	—	2	2	no	—
92	Dic₂₁	—	4	1	no	—
93	Dic₂₁	—	4	1	no	—
94	#388	—	21	1	no	Sylow
95	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
96	C₂²×D₇	—	3	2	no	—
97	C₂²×D₇	—	6	1	no	—
98	C₂²×D₇	—	6	1	no	—
99	C₂²×D₇	—	6	1	no	—
100	C₂²×D₇	—	6	1	no	—
101	C₂²×D₇	—	6	1	no	—
102	C₂²×D₇	—	6	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
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	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
110	C₂×C₄×C₇	D₁₂	1	2	yes	—
111	C₂×C₄×C₇	#67	1	2	yes	—
112	C₂×Dic₇	—	3	2	no	—
113	C₂×Dic₇	—	6	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	—	6	1	no	—
121	C₂³×D₃	—	7	1	no	—
122	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
125	C₂²×Dic₃	—	7	1	no	—
126	C₂²×Dic₃	—	7	1	no	—
127	C₂²×Dic₃	—	7	1	no	—
128	C₃×C₄²	—	2	2	no	—
129	(C₂×D₃)⋊C₄	—	7	4	no	—
130	(C₂×D₃)⋊C₄	—	14	2	no	—
131	(C₂×D₃)⋊C₄	—	14	2	no	—
132	Dic₃⋊C₄	—	14	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	D₂₁	—	4	2	no	—
141	C₂×C₄⋊C₄	—	21	1	no	—
142	C₂×C₂²⋊C₄	—	21	1	no	—
143	C₂×C₂²⋊C₄	—	21	1	no	—
144	C₂×C₂²⋊C₄	—	21	1	no	—
145	C₂×C₄²	D₂₁	1	1	yes	characteristic, complemented
146	#95	—	21	1	no	—
147	#95	—	21	1	no	—
148	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
150	C₂²×C₇	#235	1	1	yes	characteristic
151	C₂²×C₇	#235	1	1	yes	characteristic
152	C₂²×C₇	#235	1	1	yes	characteristic
153	C₂²×C₇	#235	1	1	yes	characteristic
154	C₂²×C₇	C₂×#67	1	1	yes	characteristic
155	C₂×D₇	—	6	2	no	—
156	C₂×D₇	—	6	2	no	—
157	C₂×D₇	—	6	2	no	—
158	C₂×D₇	—	12	1	no	—
159	C₂×D₇	—	12	1	no	—
160	C₂×D₇	—	12	1	no	—
161	C₂×D₇	—	12	1	no	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	2	no	—
165	Dic₇	—	6	2	no	—
166	Dic₇	—	12	1	no	—
167	Dic₇	—	12	1	no	—
168	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
169	C₂²×D₃	—	7	2	no	—
170	C₂²×D₃	—	14	1	no	—
171	C₂²×D₃	—	14	1	no	—
172	C₂²×D₃	—	14	1	no	—
173	C₂²×D₃	—	14	1	no	—
174	C₂²×D₃	—	14	1	no	—
175	C₂²×D₃	—	14	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
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	—	2	1	no	—
182	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
183	C₂×C₃×C₄	D₂₈	1	2	yes	—
184	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
185	C₂×Dic₃	—	7	2	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	—	14	1	no	—
193	C₂×Dic₃	—	14	1	no	—
194	C₃×C₇	#388	1	1	yes	characteristic, complemented, Hall
195	C₂⁴	—	21	1	no	—
196	C₂²×C₄	—	21	1	no	—
197	C₂²×C₄	—	21	1	no	—
198	C₂²×C₄	—	21	1	no	—
199	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
202	C₄⋊C₄	—	42	2	no	—
203	C₂²⋊C₄	—	21	4	no	—
204	C₂²⋊C₄	—	42	2	no	—
205	C₂²⋊C₄	—	42	2	no	—
206	C₄²	—	2	2	no	—
207	C₂×C₇	#872	1	1	yes	characteristic
208	C₂×C₇	C₄×D₁₂	1	1	yes	characteristic
209	C₂×C₇	#875	1	1	yes	characteristic
210	C₂×C₇	#876	1	1	yes	characteristic
211	C₂×C₇	C₄×#67	1	1	yes	characteristic
212	C₂×C₇	#929	1	1	yes	characteristic
213	C₂×C₇	#930	1	1	yes	characteristic
214	D₇	—	12	2	no	—
215	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
216	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
217	C₂²×C₃	#1157	1	1	yes	characteristic
218	C₂²×C₃	#1157	1	1	yes	characteristic
219	C₂²×C₃	#1157	1	1	yes	characteristic
220	C₂²×C₃	#1157	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
222	C₂×D₃	—	14	2	no	—
223	C₂×D₃	—	14	2	no	—
224	C₂×D₃	—	14	2	no	—
225	C₂×D₃	—	28	1	no	—
226	C₂×D₃	—	28	1	no	—
227	C₂×D₃	—	28	1	no	—
228	C₂×D₃	—	28	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	2	no	—
232	Dic₃	—	14	2	no	—
233	Dic₃	—	28	1	no	—
234	Dic₃	—	28	1	no	—
235	C₂³	—	21	2	no	—
236	C₂³	—	42	1	no	—
237	C₂³	—	42	1	no	—
238	C₂³	—	42	1	no	—
239	C₂³	—	42	1	no	—
240	C₂³	—	42	1	no	—
241	C₂³	—	42	1	no	—
242	C₂³	C₂²×D₂₁	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	21	2	no	—
250	C₂×C₄	—	42	1	no	—
251	C₂×C₄	—	42	1	no	—
252	C₂×C₄	—	42	1	no	—
253	C₂×C₄	—	42	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	C₄×D₂₁	1	2	yes	—
259	C₂×C₄	D₈₄	1	2	yes	—
260	C₂×C₄	#4268	1	2	yes	—
261	C₇	#4940	1	1	yes	characteristic, complemented, Sylow
262	C₂×C₃	#6435	1	1	yes	characteristic
263	C₂×C₃	C₄×D₂₈	1	1	yes	characteristic
264	C₂×C₃	#6438	1	1	yes	characteristic
265	C₂×C₃	#6439	1	1	yes	characteristic
266	C₂×C₃	C₄×C₂²⋊D₇	1	1	yes	characteristic
267	C₂×C₃	#6492	1	1	yes	characteristic
268	C₂×C₃	#6493	1	1	yes	characteristic
269	D₃	—	28	2	no	—
270	C₂²	—	42	2	no	—
271	C₂²	—	42	2	no	—
272	C₂²	—	42	2	no	—
273	C₂²	—	84	1	no	—
274	C₂²	—	84	1	no	—
275	C₂²	—	84	1	no	—
276	C₂²	—	84	1	no	—
277	C₂²	C₂×C₄×D₂₁	1	1	yes	characteristic
278	C₂²	C₂×D₈₄	1	1	yes	characteristic
279	C₂²	#66728	1	1	yes	characteristic
280	C₂²	#66728	1	1	yes	characteristic
281	C₂²	#66728	1	1	yes	characteristic
282	C₂²	#66728	1	1	yes	characteristic
283	C₂²	C₂×#4268	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	42	2	no	—
288	C₄	—	84	1	no	—
289	C₄	—	84	1	no	—
290	C₃	#89712	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	84	2	no	—
292	C₂	#10621355	1	1	yes	characteristic
293	C₂	C₄×D₈₄	1	1	yes	characteristic
294	C₂	#10621358	1	1	yes	characteristic
295	C₂	#10621359	1	1	yes	characteristic
296	C₂	C₄×#4268	1	1	yes	characteristic
297	C₂	#10621412	1	1	yes	characteristic
298	C₂	#10621413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁
commutator	C₂²×C₃×C₇	C₂²×C₄
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, cosocle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×D₂₁	C₂²
hypercenter	C₂×C₄²	D₂₁
hypocenter	C₃×C₇	#388
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄²×C₇	C₂	semidirect
5. C₂×#66629	C₂	semidirect
6. #10620670	C₂	semidirect
7. #10620670	C₂	semidirect
19. #66631	C₄	semidirect
21. C₂×C₄²×C₇	D₃	semidirect
67. C₂×C₃×C₄²	D₇	semidirect
145. C₂×C₄²	D₂₁	semidirect
194. C₃×C₇	#388	semidirect
261. C₇	#4940	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×C₃×C₄²×C₇	2
C₂×#66631	2
C₂×#66629	2
#10620670	2
#89712	3
#4940	7