Subgroups — #21433878

Order 7552.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11745685	C₂	1	1	yes	maximal, characteristic
2	#16837706	C₂	1	2	yes	maximal, complemented
3	#16837699	C₂	1	2	yes	maximal, complemented
4	#16837694	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837689	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837684	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837679	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837438	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837396	C₂	1	2	yes	maximal
11	C₅₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11745685	C₂²	1	1	yes	characteristic
13	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
14	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
16	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
17	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₅₉	—	2	1	no	—
19	C₄×C₂²⋊D₅₉	—	2	1	no	—
20	#14273686	C₂²	1	1	yes	characteristic
21	#14273684	—	2	1	no	—
22	C₄²×D₅₉	—	2	1	no	—
23	#14273660	C₂²	1	2	yes	—
24	#14273660	C₂²	1	2	yes	—
25	#14273658	C₂²	1	2	yes	—
26	#14273658	C₂²	1	2	yes	—
27	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
30	#14273650	C₂²	1	2	yes	—
31	#14273650	C₂²	1	2	yes	—
32	#14273647	C₂²	1	2	yes	—
33	#14273647	C₂²	1	2	yes	—
34	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
36	#125×C₅₉	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₅₉	—	2	1	no	—
38	C₂²⋊Q₈×C₅₉	—	2	1	no	—
39	C₄×Q₈×C₅₉	—	2	1	no	—
40	C₄×Q₈×C₅₉	—	2	1	no	—
41	#117×C₅₉	C₂²	1	2	yes	—
42	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₅₉	—	2	1	no	—
44	C₂×Q₈×C₅₉	—	2	1	no	—
45	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅₉	—	2	1	no	—
49	#11745685	—	2	2	no	—
50	#11745685	Q₈	1	4	yes	complemented
51	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₅₉	—	2	1	no	—
53	C₂×C₄×D₅₉	—	2	1	no	—
54	C₂×C₄×D₅₉	—	2	1	no	—
55	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	—	2	2	no	—
57	C₄⋊C₄×C₅₉	—	2	2	no	—
58	C₄⋊C₄×C₅₉	—	2	1	no	—
59	C₄⋊C₄×C₅₉	—	2	1	no	—
60	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
61	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₅₉	—	2	2	no	—
63	C₄²×C₅₉	—	2	2	no	—
64	C₄²×C₅₉	D₄	1	1	yes	characteristic
65	C₄²×C₅₉	D₄	1	1	yes	characteristic
66	C₄²×C₅₉	C₂³	1	1	yes	characteristic
67	C₄²×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²⋊Dic₅₉	—	2	1	no	—
69	(C₂×D₅₉)⋊C₄	—	2	1	no	—
70	(C₂×D₅₉)⋊C₄	—	2	1	no	—
71	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
73	Dic₅₉⋊C₄	—	2	1	no	—
74	Dic₅₉⋊C₄	—	2	1	no	—
75	C₄×Dic₅₉	—	2	1	no	—
76	C₄×Dic₅₉	—	2	1	no	—
77	C₄×Dic₅₉	—	2	1	no	—
78	C₂×C₅₉⋊C₈	—	2	4	no	—
79	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
80	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
81	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₅₉	—	2	1	no	—
83	Q₈×C₅₉	—	4	2	no	—
84	Q₈×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	—	2	2	no	—
86	C₂×C₄×C₅₉	—	2	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	4	1	no	—
89	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₅₉	#41	1	2	yes	—
96	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₅₉	—	2	4	no	—
98	C₂×Dic₅₉	—	2	1	no	—
99	C₂×Dic₅₉	—	2	1	no	—
100	C₂×Dic₅₉	—	2	1	no	—
101	D₂₃₆	—	2	1	no	—
102	D₂₃₆	C₂×Q₈	1	2	yes	—
103	C₄×D₅₉	—	2	4	no	—
104	C₄×D₅₉	—	4	2	no	—
105	C₄×D₅₉	—	4	2	no	—
106	Dic₁₁₈	—	2	1	no	—
107	Dic₁₁₈	C₂×Q₈	1	2	yes	—
108	C₅₉⋊C₈	—	2	2	no	—
109	C₅₉⋊C₈	—	2	2	no	—
110	C₂²×C₅₉	—	2	1	no	—
111	C₂²×C₅₉	—	2	1	no	—
112	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₅₉	—	2	2	no	—
115	C₂×D₅₉	—	4	1	no	—
116	C₂×D₅₉	—	4	1	no	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	4	2	no	—
121	C₄×C₅₉	—	4	2	no	—
122	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₅₉	#135	1	2	yes	—
124	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
126	Dic₅₉	—	2	2	no	—
127	Dic₅₉	—	4	1	no	—
128	Dic₅₉	—	4	1	no	—
129	#3076	—	59	1	no	maximal, Sylow
130	C₂×C₅₉	—	2	2	no	—
131	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
133	C₂×C₅₉	#575	1	1	yes	characteristic
134	D₅₉	—	4	2	no	—
135	#533	D₅₉	1	1	yes	characteristic, complemented
136	C₄×#41	—	59	1	no	—
137	#479	—	59	2	no	—
138	#478	—	59	2	no	—
139	#477	—	59	1	no	—
140	#476	—	59	1	no	—
141	#475	—	59	1	no	—
142	#474	—	59	1	no	—
143	#427	—	59	2	no	—
144	C₂×C₄⋊C₈	—	59	1	no	—
145	#416	—	59	2	no	—
146	C₅₉	#3076	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	59	1	no	—
148	C₂²×C₈	—	59	2	no	—
149	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
150	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
151	#125	C₂×D₅₉	1	2	yes	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	—	59	1	no	—
157	C₄×Q₈	—	59	1	no	—
158	C₄×D₄	—	59	1	no	—
159	C₄×D₄	—	59	1	no	—
160	C₄×D₄	—	118	1	no	—
161	C₄×D₄	—	118	1	no	—
162	#117	—	59	1	no	—
163	#117	—	118	1	no	—
164	#117	C₂×D₅₉	1	2	yes	—
165	C₂×C₄²	—	118	1	no	—
166	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
167	#107	—	59	2	no	—
168	#107	—	59	2	no	—
169	#106	—	59	2	no	—
170	#106	—	59	2	no	—
171	C₄⋊C₈	—	59	1	no	—
172	C₄⋊C₈	—	59	1	no	—
173	C₄⋊C₈	—	59	2	no	—
174	#103	—	59	2	no	—
175	#103	—	59	2	no	—
176	#102	—	59	2	no	—
177	#102	—	59	2	no	—
178	#41	—	59	4	no	—
179	#41	—	118	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	59	1	no	—
183	C₂×D₄	—	59	1	no	—
184	C₂×D₄	—	118	1	no	—
185	C₂²×C₄	—	118	1	no	—
186	C₂²×C₄	—	118	1	no	—
187	C₂²×C₄	—	118	1	no	—
188	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
191	C₂×C₈	—	59	2	no	—
192	C₂×C₈	—	59	2	no	—
193	C₂×C₈	—	118	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	59	1	no	—
199	C₄⋊C₄	—	59	1	no	—
200	C₄⋊C₄	—	118	1	no	—
201	C₄⋊C₄	—	118	1	no	—
202	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
203	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	118	1	no	—
206	C₂²⋊C₄	—	118	1	no	—
207	C₂²⋊C₄	—	118	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	118	1	no	—
210	C₄²	—	118	1	no	—
211	C₄²	—	118	1	no	—
212	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
213	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
214	C₄²	C₂²×D₅₉	1	1	yes	characteristic
215	C₄²	C₂²×D₅₉	1	1	yes	characteristic
216	C₂³	—	118	1	no	—
217	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	59	2	no	—
221	Q₈	—	118	1	no	—
222	D₄	—	59	2	no	—
223	D₄	—	118	4	no	—
224	D₄	—	118	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	118	1	no	—
230	C₂×C₄	—	118	4	no	—
231	C₂×C₄	—	118	1	no	—
232	C₂×C₄	—	118	1	no	—
233	C₂×C₄	—	236	2	no	—
234	C₂×C₄	—	236	2	no	—
235	C₂×C₄	Q₈×D₅₉	1	2	yes	—
236	C₂×C₄	#11745689	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
243	C₈	—	118	2	no	—
244	C₈	—	118	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	118	2	no	—
248	C₂²	—	236	1	no	—
249	C₂²	—	236	1	no	—
250	C₂²	#14273703	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	118	2	no	—
258	C₄	—	236	1	no	—
259	C₄	—	236	1	no	—
260	C₄	#14273703	1	2	yes	—
261	C₄	#14273747	1	2	yes	—
262	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
263	C₄	C₂×#11745689	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	236	2	no	—
266	C₂	C₂×#14273703	1	1	yes	characteristic
267	C₂	C₂×#14273747	1	1	yes	characteristic
268	C₂	#16838200	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#533	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#533	D₅₉
hypocenter	C₅₉	#3076
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #16837706	C₂	semidirect
3. #16837699	C₂	semidirect
4. #16837694	C₂	semidirect
5. #16837689	C₂	semidirect
6. #16837684	C₂	semidirect
7. #16837679	C₂	semidirect
8. #16837438	C₂	semidirect
9. C₂×C₄⋊(C₅₉⋊C₈)	C₂	semidirect
11. C₅₉×#533	C₂	semidirect
50. #11745685	Q₈	semidirect
135. #533	D₅₉	semidirect
146. C₅₉	#3076	semidirect