Subgroups — #13061785

Order 1344.

428 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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621381	C₂	1	2	yes	maximal, complemented
3	#10621380	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621380	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621377	C₂	1	2	yes	maximal
6	#10621375	C₂	1	2	yes	maximal, complemented
7	#10621375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₄×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#10621373	C₂	1	2	yes	maximal, complemented
10	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#90385	—	3	1	no	maximal, Hall
13	C₂×D₈₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₁	C₂²	1	2	yes	—
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
23	#66631	C₂²	1	1	yes	characteristic
24	#66631	C₂²	1	1	yes	characteristic
25	#66631	C₂²	1	2	yes	—
26	#66631	C₂²	1	2	yes	—
27	#66631	C₂²	1	2	yes	—
28	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₂₁	C₂²	1	2	yes	—
31	C₄⋊Dic₂₁	C₂²	1	2	yes	—
32	#66629	C₂²	1	2	yes	—
33	#66629	C₂²	1	2	yes	—
34	C₄×Dic₂₁	C₂²	1	2	yes	—
35	C₄×Dic₂₁	C₂²	1	2	yes	—
36	C₇×#125	D₃	1	1	yes	characteristic, complemented
37	#6461	—	3	2	no	—
38	#6460	—	3	1	no	—
39	#6460	—	3	1	no	—
40	#6457	—	3	2	no	—
41	#6455	—	3	2	no	—
42	#6455	—	3	1	no	—
43	D₇×C₄⋊C₄	—	3	1	no	—
44	#6453	—	3	2	no	—
45	C₄×D₂₈	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5694	—	7	1	no	maximal, Hall
48	C₂²×D₂₁	C₂³	1	1	yes	characteristic
49	C₂²×D₂₁	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
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	1	yes	characteristic
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₁	C₂³	1	2	yes	—
58	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
59	D₈₄	—	2	2	no	—
60	C₄×D₂₁	—	2	4	no	—
61	C₄×D₂₁	—	2	2	no	—
62	C₄×D₂₁	—	2	2	no	—
63	Dic₄₂	—	2	2	no	—
64	C₂×D₂₈	—	3	1	no	—
65	C₂×C₄×D₇	—	3	1	no	—
66	C₂×C₄×D₇	—	3	1	no	—
67	C₂×C₄×D₇	—	3	2	no	—
68	C₂×Dic₁₄	—	3	1	no	—
69	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
73	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
74	(C₂×D₇)⋊C₄	—	3	1	no	—
75	(C₂×D₇)⋊C₄	—	3	1	no	—
76	(C₂×D₇)⋊C₄	—	3	2	no	—
77	(C₂×D₇)⋊C₄	—	3	2	no	—
78	(C₂×D₇)⋊C₄	—	3	2	no	—
79	C₄⋊Dic₇	—	3	1	no	—
80	C₄⋊Dic₇	—	3	1	no	—
81	C₄⋊Dic₇	—	3	2	no	—
82	C₄⋊Dic₇	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	C₄×Dic₇	—	3	2	no	—
86	C₄×Dic₇	—	3	2	no	—
87	C₃×#125	D₇	1	1	yes	characteristic, complemented
88	#898	—	7	2	no	—
89	#897	—	7	1	no	—
90	#897	—	7	1	no	—
91	#894	—	7	2	no	—
92	#892	—	7	2	no	—
93	#892	—	7	1	no	—
94	D₃×C₄⋊C₄	—	7	1	no	—
95	#890	—	7	2	no	—
96	C₄×D₁₂	—	7	1	no	—
97	C₄×Dic₆	—	7	1	no	—
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×D₂₁	—	2	1	no	—
100	C₂×D₂₁	—	2	1	no	—
101	C₂×D₂₁	—	2	1	no	—
102	C₂×D₂₁	—	2	1	no	—
103	C₂×D₂₁	—	2	1	no	—
104	C₂×D₂₁	#41	1	2	yes	—
105	C₃×C₄×C₇	—	2	2	no	—
106	C₃×C₄×C₇	—	2	1	no	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	—	2	2	no	—
109	C₃×C₄×C₇	#41	1	2	yes	—
110	Dic₂₁	—	2	2	no	—
111	Dic₂₁	—	2	1	no	—
112	Dic₂₁	—	2	1	no	—
113	Dic₂₁	—	2	2	no	—
114	#552	—	21	1	no	Sylow
115	C₂²×D₇	—	3	1	no	—
116	C₂²×D₇	—	3	1	no	—
117	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
119	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
121	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	—	3	1	no	—
124	C₂×Dic₇	—	3	2	no	—
125	C₂×Dic₇	—	3	1	no	—
126	D₂₈	—	6	2	no	—
127	C₄×D₇	—	6	4	no	—
128	C₄×D₇	—	6	2	no	—
129	C₄×D₇	—	6	2	no	—
130	Dic₁₄	—	6	2	no	—
131	C₂×D₁₂	—	7	1	no	—
132	C₂×C₄×D₃	—	7	1	no	—
133	C₂×C₄×D₃	—	7	1	no	—
134	C₂×C₄×D₃	—	7	2	no	—
135	C₂×Dic₆	—	7	1	no	—
136	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
137	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
138	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
139	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
140	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
141	(C₂×D₃)⋊C₄	—	7	1	no	—
142	(C₂×D₃)⋊C₄	—	7	1	no	—
143	(C₂×D₃)⋊C₄	—	7	2	no	—
144	(C₂×D₃)⋊C₄	—	7	2	no	—
145	(C₂×D₃)⋊C₄	—	7	2	no	—
146	C₄⋊Dic₃	—	7	1	no	—
147	C₄⋊Dic₃	—	7	1	no	—
148	C₄⋊Dic₃	—	7	2	no	—
149	C₄⋊Dic₃	—	7	2	no	—
150	Dic₃⋊C₄	—	7	2	no	—
151	Dic₃⋊C₄	—	7	2	no	—
152	C₄×Dic₃	—	7	2	no	—
153	C₄×Dic₃	—	7	2	no	—
154	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
157	D₂₁	—	2	2	no	—
158	D₂₁	—	4	1	no	—
159	#126	—	21	2	no	—
160	#125	—	21	2	no	—
161	#125	D₂₁	1	1	yes	characteristic, complemented
162	#123	—	21	2	no	—
163	C₂²⋊Q₈	—	21	1	no	—
164	C₂²⋊Q₈	—	21	1	no	—
165	C₄×Q₈	—	21	1	no	—
166	C₄×D₄	—	21	1	no	—
167	#117	—	21	1	no	—
168	#117	—	21	2	no	—
169	C₂×C₄⋊C₄	—	21	1	no	—
170	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
171	C₂×D₇	—	3	2	no	—
172	C₂×D₇	—	6	1	no	—
173	C₂×D₇	—	6	1	no	—
174	C₂×D₇	—	6	1	no	—
175	C₂×D₇	—	6	1	no	—
176	C₂×D₇	—	6	1	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	#239	1	2	yes	—
182	Dic₇	—	6	2	no	—
183	Dic₇	—	6	1	no	—
184	Dic₇	—	6	1	no	—
185	Dic₇	—	6	2	no	—
186	C₂²×D₃	—	7	1	no	—
187	C₂²×D₃	—	7	1	no	—
188	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
190	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
192	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
193	C₂×Dic₃	—	7	2	no	—
194	C₂×Dic₃	—	7	1	no	—
195	C₂×Dic₃	—	7	2	no	—
196	C₂×Dic₃	—	7	1	no	—
197	D₁₂	—	14	2	no	—
198	C₄×D₃	—	14	4	no	—
199	C₄×D₃	—	14	2	no	—
200	C₄×D₃	—	14	2	no	—
201	Dic₆	—	14	2	no	—
202	C₃×C₇	#552	1	1	yes	characteristic, complemented, Hall
203	C₂×Q₈	—	21	1	no	—
204	C₂×D₄	—	21	1	no	—
205	C₂²×C₄	—	21	2	no	—
206	C₂²×C₄	—	21	1	no	—
207	C₂²×C₄	—	21	1	no	—
208	C₄⋊C₄	—	21	2	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	1	no	—
211	C₄⋊C₄	—	21	2	no	—
212	C₄⋊C₄	—	21	1	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
215	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
216	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
217	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
218	C₂²⋊C₄	—	21	1	no	—
219	C₂²⋊C₄	—	21	2	no	—
220	C₂²⋊C₄	—	21	2	no	—
221	C₂²⋊C₄	—	21	1	no	—
222	C₂²⋊C₄	—	21	2	no	—
223	C₄²	—	21	2	no	—
224	C₄²	—	21	2	no	—
225	C₄²	C₂×D₂₁	1	1	yes	characteristic
226	C₂×C₇	C₂×#239	1	1	yes	characteristic
227	C₂×C₇	D₃×#41	1	1	yes	characteristic
228	C₂×C₇	#1010	1	1	yes	characteristic
229	D₇	—	6	2	no	—
230	D₇	—	12	1	no	—
231	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
232	C₂×D₃	—	7	2	no	—
233	C₂×D₃	—	14	1	no	—
234	C₂×D₃	—	14	1	no	—
235	C₂×D₃	—	14	1	no	—
236	C₂×D₃	—	14	1	no	—
237	C₂×D₃	—	14	1	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	#1161	1	2	yes	—
243	Dic₃	—	14	2	no	—
244	Dic₃	—	14	1	no	—
245	Dic₃	—	14	1	no	—
246	Dic₃	—	14	2	no	—
247	C₂³	—	21	1	no	—
248	C₂³	—	21	1	no	—
249	Q₈	—	42	2	no	—
250	D₄	—	42	2	no	—
251	C₂×C₄	—	21	1	no	—
252	C₂×C₄	—	21	2	no	—
253	C₂×C₄	—	21	2	no	—
254	C₂×C₄	—	21	1	no	—
255	C₂×C₄	—	42	4	no	—
256	C₂×C₄	—	42	2	no	—
257	C₂×C₄	—	42	2	no	—
258	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₂₁	1	2	yes	—
260	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₂₁	1	2	yes	—
262	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
263	C₇	#5694	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	C₂×#1161	1	1	yes	characteristic
265	C₂×C₃	D₇×#41	1	1	yes	characteristic
266	C₂×C₃	#6554	1	1	yes	characteristic
267	D₃	—	14	2	no	—
268	D₃	—	28	1	no	—
269	C₂²	—	21	2	no	—
270	C₂²	—	42	1	no	—
271	C₂²	—	42	1	no	—
272	C₂²	—	42	1	no	—
273	C₂²	—	42	1	no	—
274	C₂²	—	42	1	no	—
275	C₂²	C₂³×D₂₁	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	42	1	no	—
281	C₄	—	42	1	no	—
282	C₄	—	42	2	no	—
283	C₄	—	42	2	no	—
284	C₄	#66732	1	2	yes	—
285	C₃	#90385	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	42	2	no	—
287	C₂	—	84	1	no	—
288	C₂	C₂×#66732	1	1	yes	characteristic
289	C₂	#41×D₂₁	1	1	yes	characteristic
290	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×#125	C₂
prime radical	C₂×D₈₄	C₂²
hypercenter	#125	D₂₁
hypocenter	C₃×C₇	#552
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#125	C₂	semidirect
2. #10621381	C₂	semidirect
3. #10621380	C₂	semidirect
4. #10621380	C₂	semidirect
6. #10621375	C₂	semidirect
7. #10621375	C₂	semidirect
8. C₄⋊C₄×D₂₁	C₂	semidirect
9. #10621373	C₂	semidirect
11. C₄×Dic₄₂	C₂	semidirect
36. C₇×#125	D₃	semidirect
87. C₃×#125	D₇	semidirect
161. #125	D₂₁	semidirect
202. C₃×C₇	#552	semidirect
263. C₇	#5694	semidirect
285. C₃	#90385	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#125	2
#10621381	2
#10621380	2
#10621377	2
#10621375	2
C₄⋊C₄×D₂₁	2
#10621373	2
C₄×D₈₄	2
C₄×Dic₄₂	2
#90385	3
#5694	7