Subgroups — #13061780

Order 1344.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 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	#10621380	C₂	1	2	yes	maximal
3	#10621379	C₂	1	2	yes	maximal
4	#10621375	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
6	#10621373	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621372	C₂	1	2	yes	maximal, complemented
8	#10621371	C₂	1	1	yes	maximal, characteristic, complemented
9	#10621371	C₂	1	2	yes	maximal, complemented
10	#10621355	C₂	1	1	yes	maximal, characteristic
11	#10621352	C₂	1	1	yes	maximal, characteristic, complemented
12	#90380	—	3	1	no	maximal, Hall
13	C₂×C₄×D₂₁	C₂²	1	2	yes	—
14	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂	C₂²	1	2	yes	—
16	C₂×Dic₄₂	C₂²	1	2	yes	—
17	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
22	#66631	C₂²	1	1	yes	characteristic
23	#66631	C₂²	1	1	yes	characteristic
24	#66631	C₂²	1	2	yes	—
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₁	C₂²	1	2	yes	—
27	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₂₁	C₂²	1	2	yes	—
29	#66629	C₂²	1	1	yes	characteristic
30	#66629	C₂²	1	1	yes	characteristic
31	#66629	C₂²	1	2	yes	—
32	#66629	C₂²	1	2	yes	—
33	#66629	C₂²	1	2	yes	—
34	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁	C₂²	1	2	yes	—
36	C₇×#125	D₃	1	1	yes	characteristic, complemented
37	#6460	—	3	2	no	—
38	#6459	—	3	2	no	—
39	#6455	—	3	1	no	—
40	D₇×C₄⋊C₄	—	3	1	no	—
41	#6453	—	3	1	no	—
42	#6452	—	3	2	no	—
43	#6451	—	3	1	no	—
44	#6451	—	3	2	no	—
45	#6435	—	3	1	no	—
46	#6432	—	3	1	no	—
47	#5689	—	7	1	no	maximal, Hall
48	C₂²×D₂₁	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₁	C₂³	1	2	yes	—
57	C₂×Dic₂₁	C₂³	1	2	yes	—
58	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
59	C₄×D₂₁	—	4	2	no	—
60	C₄×D₂₁	Q₈	1	4	yes	complemented
61	Dic₄₂	—	4	2	no	—
62	Dic₄₂	—	4	2	no	—
63	C₂×C₄×D₇	—	3	1	no	—
64	C₂×C₄×D₇	—	3	2	no	—
65	C₂×Dic₁₄	—	3	2	no	—
66	C₂×Dic₁₄	—	3	2	no	—
67	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
68	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
69	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
72	(C₂×D₇)⋊C₄	—	3	1	no	—
73	(C₂×D₇)⋊C₄	—	3	1	no	—
74	(C₂×D₇)⋊C₄	—	3	2	no	—
75	C₄⋊Dic₇	—	3	1	no	—
76	C₄⋊Dic₇	—	3	2	no	—
77	C₄⋊Dic₇	—	3	1	no	—
78	C₄⋊Dic₇	—	3	2	no	—
79	Dic₇⋊C₄	—	3	1	no	—
80	Dic₇⋊C₄	—	3	1	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	2	no	—
86	C₃×#125	D₇	1	1	yes	characteristic, complemented
87	#897	—	7	2	no	—
88	#896	—	7	2	no	—
89	#892	—	7	1	no	—
90	D₃×C₄⋊C₄	—	7	1	no	—
91	#890	—	7	1	no	—
92	#889	—	7	2	no	—
93	#888	—	7	1	no	—
94	#888	—	7	2	no	—
95	#872	—	7	1	no	—
96	#869	—	7	1	no	—
97	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
98	C₂×D₂₁	—	2	1	no	—
99	C₂×D₂₁	—	2	1	no	—
100	C₂×D₂₁	C₂×Q₈	1	2	yes	—
101	C₃×C₄×C₇	—	2	1	no	—
102	C₃×C₄×C₇	—	2	2	no	—
103	C₃×C₄×C₇	—	2	2	no	—
104	C₃×C₄×C₇	—	2	1	no	—
105	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
106	Dic₂₁	—	2	1	no	—
107	Dic₂₁	—	2	2	no	—
108	Dic₂₁	—	2	2	no	—
109	Dic₂₁	—	2	1	no	—
110	Dic₂₁	C₂×Q₈	1	2	yes	—
111	#544	—	21	1	no	Sylow
112	C₂²×D₇	—	3	1	no	—
113	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
114	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
115	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂×Dic₇	—	3	1	no	—
119	C₂×Dic₇	—	3	1	no	—
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	2	no	—
122	C₂×Dic₇	—	3	2	no	—
123	C₄×D₇	—	3	4	no	—
124	C₄×D₇	—	12	2	no	—
125	Dic₁₄	—	12	2	no	—
126	Dic₁₄	—	12	2	no	—
127	C₂×C₄×D₃	—	7	1	no	—
128	C₂×C₄×D₃	—	7	2	no	—
129	C₂×Dic₆	—	7	2	no	—
130	C₂×Dic₆	—	7	2	no	—
131	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
132	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
133	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
134	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
135	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	—	7	1	no	—
137	(C₂×D₃)⋊C₄	—	7	1	no	—
138	(C₂×D₃)⋊C₄	—	7	2	no	—
139	C₄⋊Dic₃	—	7	1	no	—
140	C₄⋊Dic₃	—	7	2	no	—
141	C₄⋊Dic₃	—	7	1	no	—
142	C₄⋊Dic₃	—	7	2	no	—
143	Dic₃⋊C₄	—	7	1	no	—
144	Dic₃⋊C₄	—	7	1	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	C₄×Dic₃	—	7	1	no	—
149	C₄×Dic₃	—	7	2	no	—
150	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
153	D₂₁	—	2	2	no	—
154	C₄⋊Q₈	—	21	2	no	—
155	C₄⋊Q₈	—	21	1	no	—
156	C₄⋊Q₈	—	21	1	no	—
157	#125	—	21	2	no	—
158	#125	—	21	1	no	—
159	#125	D₂₁	1	1	yes	characteristic, complemented
160	C₂²⋊Q₈	—	21	2	no	—
161	C₂²⋊Q₈	—	21	2	no	—
162	#117	—	21	1	no	—
163	#117	—	21	1	no	—
164	C₂×C₄⋊C₄	—	21	1	no	—
165	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
166	C₂×D₇	—	3	2	no	—
167	C₂×D₇	—	6	1	no	—
168	C₂×D₇	—	6	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	2	2	no	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	D₃×Q₈	1	2	yes	—
174	Dic₇	—	3	2	no	—
175	Dic₇	—	6	1	no	—
176	Dic₇	—	6	2	no	—
177	Dic₇	—	6	2	no	—
178	Dic₇	—	6	1	no	—
179	C₂²×D₃	—	7	1	no	—
180	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
182	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
185	C₂×Dic₃	—	7	1	no	—
186	C₂×Dic₃	—	7	1	no	—
187	C₂×Dic₃	—	7	1	no	—
188	C₂×Dic₃	—	7	2	no	—
189	C₂×Dic₃	—	7	2	no	—
190	C₄×D₃	—	7	4	no	—
191	C₄×D₃	—	28	2	no	—
192	Dic₆	—	28	2	no	—
193	Dic₆	—	28	2	no	—
194	C₃×C₇	#544	1	1	yes	characteristic, complemented, Hall
195	C₂×Q₈	—	21	2	no	—
196	C₂×Q₈	—	21	2	no	—
197	C₂²×C₄	—	21	1	no	—
198	C₂²×C₄	—	21	2	no	—
199	C₄⋊C₄	—	21	2	no	—
200	C₄⋊C₄	—	21	1	no	—
201	C₄⋊C₄	—	21	1	no	—
202	C₄⋊C₄	—	21	2	no	—
203	C₄⋊C₄	—	21	2	no	—
204	C₄⋊C₄	—	21	1	no	—
205	C₄⋊C₄	—	21	1	no	—
206	C₄⋊C₄	—	21	2	no	—
207	C₄⋊C₄	—	21	2	no	—
208	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
211	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
212	C₂²⋊C₄	—	21	1	no	—
213	C₂²⋊C₄	—	21	2	no	—
214	C₂²⋊C₄	—	21	1	no	—
215	C₄²	—	21	2	no	—
216	C₄²	—	21	1	no	—
217	C₄²	C₂×D₂₁	1	1	yes	characteristic
218	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
219	C₂×C₇	#1009	1	1	yes	characteristic
220	C₂×C₇	#1010	1	1	yes	characteristic
221	D₇	—	6	2	no	—
222	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
223	C₂×D₃	—	7	2	no	—
224	C₂×D₃	—	14	1	no	—
225	C₂×D₃	—	14	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	Q₈×D₇	1	2	yes	—
231	Dic₃	—	7	2	no	—
232	Dic₃	—	14	1	no	—
233	Dic₃	—	14	2	no	—
234	Dic₃	—	14	2	no	—
235	Dic₃	—	14	1	no	—
236	C₂³	—	21	1	no	—
237	Q₈	—	84	2	no	—
238	Q₈	—	84	2	no	—
239	C₂×C₄	—	21	1	no	—
240	C₂×C₄	—	21	4	no	—
241	C₂×C₄	—	21	1	no	—
242	C₂×C₄	—	21	1	no	—
243	C₂×C₄	—	21	2	no	—
244	C₂×C₄	—	21	2	no	—
245	C₂×C₄	—	84	2	no	—
246	C₂×C₄	C₂²×D₂₁	1	2	yes	—
247	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₂₁	1	2	yes	—
250	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
251	C₇	#5689	1	1	yes	characteristic, complemented, Sylow
252	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
253	C₂×C₃	#6553	1	1	yes	characteristic
254	C₂×C₃	#6554	1	1	yes	characteristic
255	D₃	—	14	2	no	—
256	C₂²	—	21	2	no	—
257	C₂²	—	42	1	no	—
258	C₂²	—	42	1	no	—
259	C₂²	C₂³×D₂₁	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	21	2	no	—
265	C₄	—	42	2	no	—
266	C₄	—	42	1	no	—
267	C₄	—	42	2	no	—
268	C₄	—	42	1	no	—
269	C₄	Q₈×D₂₁	1	2	yes	—
270	C₃	#90380	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	42	2	no	—
272	C₂	C₂×Q₈×D₂₁	1	1	yes	characteristic
273	C₂	#10621662	1	1	yes	characteristic
274	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	C₂²×D₂₁	C₂³
hypercenter	#125	D₂₁
hypocenter	C₃×C₇	#544
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
6. #10621373	C₂	semidirect
7. #10621372	C₂	semidirect
8. #10621371	C₂	semidirect
9. #10621371	C₂	semidirect
11. #10621352	C₂	semidirect
36. C₇×#125	D₃	semidirect
60. C₄×D₂₁	Q₈	semidirect
86. C₃×#125	D₇	semidirect
159. #125	D₂₁	semidirect
194. C₃×C₇	#544	semidirect
251. C₇	#5689	semidirect
270. C₃	#90380	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
#10621380	2
#10621379	2
#10621375	2
C₄⋊C₄×D₂₁	2
#10621373	2
#10621372	2
#10621371	2
#10621355	2
#10621352	2
#90380	3
#5689	7