Subgroups — #21437041

Order 7552.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838130	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837812	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837808	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837801	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837706	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837706	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837702	C₂	1	1	yes	maximal, characteristic
8	#16837702	C₂	1	1	yes	maximal, characteristic
9	#16837692	C₂	1	1	yes	maximal, characteristic
10	#16837684	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837533	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837502	C₂	1	1	yes	maximal, characteristic
13	#16837502	C₂	1	1	yes	maximal, characteristic
14	C₄×#11745670	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#554	C₂	1	1	yes	maximal, characteristic
16	C₂×#11745670	C₂²	1	1	yes	characteristic
17	#14273727	C₂²	1	1	yes	characteristic
18	#14273725	—	2	1	no	—
19	#14273719	C₂²	1	1	yes	characteristic
20	#14273719	C₂²	1	1	yes	characteristic
21	#14273717	—	2	1	no	—
22	#14273717	—	2	1	no	—
23	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
26	#14273692	—	2	1	no	—
27	#14273660	C₂²	1	1	yes	characteristic
28	#14273660	C₂²	1	1	yes	characteristic
29	#14273660	C₂²	1	1	yes	characteristic
30	#14273660	C₂²	1	1	yes	characteristic
31	#14273658	C₂²	1	1	yes	characteristic
32	#14273658	C₂²	1	1	yes	characteristic
33	#14273658	C₂²	1	1	yes	characteristic
34	#14273658	C₂²	1	1	yes	characteristic
35	#14273658	C₂²	1	1	yes	characteristic
36	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
37	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
39	#14273650	C₂²	1	1	yes	characteristic
40	#14273650	C₂²	1	1	yes	characteristic
41	#14273650	C₂²	1	1	yes	characteristic
42	#14273648	C₂²	1	1	yes	characteristic
43	#14273648	C₂²	1	1	yes	characteristic
44	#14273648	C₂²	1	1	yes	characteristic
45	#14273638	C₂²	1	1	yes	characteristic
46	#14273638	C₂²	1	1	yes	characteristic
47	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
49	#125×C₅₉	C₂²	1	1	yes	characteristic
50	#125×C₅₉	C₂²	1	1	yes	characteristic
51	C₂²⋊Q₈×C₅₉	—	2	1	no	—
52	C₂²⋊Q₈×C₅₉	—	2	1	no	—
53	C₂²⋊Q₈×C₅₉	—	2	1	no	—
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂²×C₄×C₅₉	—	2	1	no	—
64	C₂²×C₄×C₅₉	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₄⋊C₄×C₅₉	—	2	1	no	—
69	C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₄⋊C₄×C₅₉	—	2	1	no	—
71	C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₅₉	—	2	1	no	—
79	C₂²⋊C₄×C₅₉	—	2	1	no	—
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	C₂³	1	1	yes	characteristic
82	C₄²×C₅₉	C₂³	1	1	yes	characteristic
83	C₄²×C₅₉	C₂³	1	1	yes	characteristic
84	#11745670	Q₈	1	4	yes	complemented
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂³×C₅₉	—	2	1	no	—
104	Q₈×C₅₉	—	4	1	no	—
105	Q₈×C₅₉	—	8	1	no	—
106	D₄×C₅₉	—	2	1	no	—
107	D₄×C₅₉	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	4	2	no	—
112	C₂×C₄×C₅₉	—	8	1	no	—
113	C₂×C₄×C₅₉	—	8	1	no	—
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	Dic₁₁₈	—	2	1	no	—
126	Dic₁₁₈	—	4	2	no	—
127	Dic₁₁₈	C₂×Q₈	1	2	yes	—
128	C₅₉⋊C₈	—	2	1	no	—
129	C₅₉⋊C₈	—	2	1	no	—
130	C₅₉⋊C₈	—	2	1	no	—
131	C₅₉⋊C₈	C₂×Q₈	1	2	yes	—
132	C₂²×C₅₉	—	2	2	no	—
133	C₂²×C₅₉	—	4	1	no	—
134	C₂²×C₅₉	—	4	1	no	—
135	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	2	1	no	—
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	4	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
146	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₅₉	—	2	2	no	—
148	Dic₅₉	—	4	1	no	—
149	Dic₅₉	—	4	1	no	—
150	Dic₅₉	—	4	1	no	—
151	#3388	—	59	1	no	maximal, Sylow
152	C₂×C₅₉	—	4	2	no	—
153	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
154	C₂×C₅₉	#575	1	1	yes	characteristic
155	C₂×C₅₉	#577	1	1	yes	characteristic
156	#557	—	59	1	no	—
157	#554	D₅₉	1	1	yes	characteristic
158	#501	—	59	1	no	—
159	#501	—	59	1	no	—
160	#499	—	59	1	no	—
161	#479	—	59	1	no	—
162	#479	—	59	1	no	—
163	#478	—	59	1	no	—
164	#478	—	59	1	no	—
165	#476	—	59	1	no	—
166	#475	—	59	1	no	—
167	#446	—	59	1	no	—
168	#440	—	59	1	no	—
169	#440	—	59	1	no	—
170	C₄×SD₁₆	—	59	1	no	—
171	C₅₉	#3388	1	1	yes	characteristic, complemented, Sylow
172	C₂×SD₁₆	—	59	1	no	—
173	C₄⋊Q₈	—	59	1	no	—
174	C₄⋊Q₈	—	118	1	no	—
175	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
176	#125	—	59	1	no	—
177	#125	—	59	1	no	—
178	#125	—	118	1	no	—
179	#125	—	118	1	no	—
180	#125	C₂×D₅₉	1	1	yes	characteristic
181	#125	C₂×D₅₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₄×Q₈	—	59	1	no	—
186	C₄×Q₈	—	59	1	no	—
187	C₄×Q₈	—	59	1	no	—
188	C₄×Q₈	—	118	1	no	—
189	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
190	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
191	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
192	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	2	1	no	—
194	#107	—	59	1	no	—
195	#107	—	59	1	no	—
196	#107	—	59	1	no	—
197	#107	—	59	1	no	—
198	#106	—	59	1	no	—
199	#106	—	59	1	no	—
200	#106	—	59	1	no	—
201	#106	—	59	1	no	—
202	#106	—	59	1	no	—
203	C₄⋊C₈	—	59	1	no	—
204	C₄⋊C₈	—	59	1	no	—
205	C₄⋊C₈	—	59	1	no	—
206	#103	—	59	1	no	—
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	#102	—	59	1	no	—
210	#102	—	59	1	no	—
211	#102	—	59	1	no	—
212	#97	—	59	1	no	—
213	#97	—	59	1	no	—
214	C₄×C₈	—	59	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	59	1	no	—
217	C₂×Q₈	—	118	1	no	—
218	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	SD₁₆	—	59	4	no	—
224	C₂×C₈	—	59	1	no	—
225	C₂×C₈	—	59	1	no	—
226	C₂×C₈	—	59	1	no	—
227	C₂×C₈	—	59	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	59	1	no	—
234	C₄⋊C₄	—	59	1	no	—
235	C₄⋊C₄	—	59	1	no	—
236	C₄⋊C₄	—	118	1	no	—
237	C₄⋊C₄	—	118	1	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	1	no	—
241	C₄⋊C₄	—	118	1	no	—
242	C₄⋊C₄	—	118	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
246	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	118	1	no	—
254	C₄²	—	118	1	no	—
255	C₄²	—	118	1	no	—
256	C₄²	C₂²×D₅₉	1	1	yes	characteristic
257	C₄²	C₂²×D₅₉	1	1	yes	characteristic
258	C₄²	C₂²×D₅₉	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	59	2	no	—
263	Q₈	—	118	1	no	—
264	Q₈	—	236	2	no	—
265	D₄	—	2	1	no	—
266	D₄	Q₈×D₅₉	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	118	1	no	—
274	C₂×C₄	—	118	1	no	—
275	C₂×C₄	—	118	1	no	—
276	C₂×C₄	—	118	1	no	—
277	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
284	C₈	—	59	2	no	—
285	C₈	—	118	1	no	—
286	C₈	—	118	1	no	—
287	C₈	—	118	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	118	2	no	—
302	C₄	—	236	1	no	—
303	C₄	—	236	1	no	—
304	C₄	—	236	1	no	—
305	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
306	C₄	#14273775	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	Q₈×C₂²⋊D₅₉	1	1	yes	characteristic
309	C₂	#16838200	1	1	yes	characteristic
310	C₂	#16838213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#554	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#554	D₅₉
hypocenter	C₅₉	#3388
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #554

Complements

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

N	H ≅ G/N	type
1. #16838130	C₂	semidirect
2. #16837812	C₂	semidirect
3. #16837808	C₂	semidirect
4. #16837801	C₂	semidirect
5. #16837706	C₂	semidirect
6. #16837706	C₂	semidirect
10. #16837684	C₂	semidirect
11. #16837533	C₂	semidirect
84. #11745670	Q₈	semidirect
171. C₅₉	#3388	semidirect