Subgroups — #24918125

Order 10624.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128313	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128107	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128096	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128055	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128053	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127957	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127945	C₂	1	1	yes	maximal, characteristic
8	#19127940	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127928	C₂	1	1	yes	maximal, characteristic
10	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13050580	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13050578	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127657	C₂	1	1	yes	maximal, characteristic, complemented
15	#542×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂×#13050578	C₂²	1	1	yes	characteristic
20	C₂×#13050574	C₂²	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
22	#14581792	C₂²	1	1	yes	characteristic
23	#14581785	—	2	1	no	—
24	#14581785	—	2	1	no	—
25	#14581784	C₂²	1	1	yes	characteristic
26	#14581782	—	2	1	no	—
27	#14581782	—	2	1	no	—
28	#14581771	C₂²	1	1	yes	characteristic
29	#14581771	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	#14581725	C₂²	1	1	yes	characteristic
33	#14581725	C₂²	1	1	yes	characteristic
34	#14581723	C₂²	1	1	yes	characteristic
35	#14581723	C₂²	1	1	yes	characteristic
36	#14581716	C₂²	1	1	yes	characteristic
37	#14581716	C₂²	1	1	yes	characteristic
38	#14581715	C₂²	1	1	yes	characteristic
39	#14581715	C₂²	1	1	yes	characteristic
40	#14581715	C₂²	1	1	yes	characteristic
41	#14581715	C₂²	1	1	yes	characteristic
42	#14581713	C₂²	1	1	yes	characteristic
43	#14581713	C₂²	1	1	yes	characteristic
44	#14581703	C₂²	1	1	yes	characteristic
45	#14581703	C₂²	1	1	yes	characteristic
46	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
47	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
49	#126×C₈₃	—	2	1	no	—
50	#124×C₈₃	—	2	1	no	—
51	#124×C₈₃	C₂²	1	1	yes	characteristic
52	#123×C₈₃	—	2	1	no	—
53	C₂²⋊Q₈×C₈₃	—	2	1	no	—
54	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
58	#117×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	#13050580	—	2	2	no	—
62	#13050580	—	2	2	no	—
63	#13050578	—	2	2	no	—
64	#13050578	—	2	2	no	—
65	#13050574	—	2	2	no	—
66	C₂×C₈₃⋊C₈	—	2	2	no	—
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₈₃	—	2	1	no	—
82	C₂²⋊Dic₈₃	—	2	1	no	—
83	C₄×Dic₈₃	—	2	1	no	—
84	C₄×Dic₈₃	—	2	1	no	—
85	C₂×Q₈×C₈₃	—	2	1	no	—
86	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
87	C₂×D₄×C₈₃	—	2	1	no	—
88	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₈₃	—	2	1	no	—
90	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₈₃	—	2	1	no	—
92	C₄⋊C₄×C₈₃	—	2	1	no	—
93	C₄⋊C₄×C₈₃	—	2	1	no	—
94	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²⋊C₄×C₈₃	—	2	1	no	—
99	C₂²⋊C₄×C₈₃	—	2	1	no	—
100	C₂²⋊C₄×C₈₃	—	2	1	no	—
101	C₂²⋊C₄×C₈₃	—	2	1	no	—
102	C₂²⋊C₄×C₈₃	—	2	1	no	—
103	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
105	C₄²×C₈₃	—	2	1	no	—
106	C₄²×C₈₃	C₂³	1	1	yes	characteristic
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	8	1	no	—
113	D₄×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	1	no	—
115	D₄×C₈₃	—	8	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	8	1	no	—
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	Dic₁₆₆	—	2	1	no	—
133	Dic₁₆₆	—	2	1	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	C₈₃⋊C₈	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₈₃⋊C₈	#41	1	2	yes	—
139	C₈₃⋊C₈	#41	1	2	yes	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	—	4	1	no	—
145	C₂²×C₈₃	—	4	1	no	—
146	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	C₂×C₈₃	—	4	1	no	—
162	C₂×C₈₃	—	8	1	no	—
163	C₂×C₈₃	#532	1	1	yes	characteristic
164	C₂×C₈₃	#577	1	1	yes	characteristic
165	C₂×C₈₃	#578	1	1	yes	characteristic
166	#3178	—	83	1	no	maximal, Sylow
167	C₈₃	#3178	1	1	yes	characteristic, complemented, Sylow
168	#542	D₈₃	1	1	yes	characteristic
169	#541	—	83	1	no	—
170	#501	—	83	1	no	—
171	#497	—	83	1	no	—
172	#488	—	83	1	no	—
173	#487	—	83	1	no	—
174	#470	—	83	1	no	—
175	#468	—	83	1	no	—
176	#467	—	83	1	no	—
177	#465	—	83	1	no	—
178	#441	—	83	1	no	—
179	#440	—	83	1	no	—
180	C₄×Q₁₆	—	83	1	no	—
181	C₄×SD₁₆	—	83	1	no	—
182	#405	—	83	1	no	—
183	C₂×Q₁₆	—	83	1	no	—
184	C₂×Q₁₆	—	83	1	no	—
185	C₂×SD₁₆	—	83	1	no	—
186	C₂×SD₁₆	—	83	1	no	—
187	C₂×M₁₆	—	83	1	no	—
188	C₂²×C₈	—	83	1	no	—
189	C₄⋊Q₈	—	83	1	no	—
190	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
191	#126	—	2	1	no	—
192	#126	—	166	1	no	—
193	#126	—	166	1	no	—
194	#125	—	83	1	no	—
195	#125	—	166	1	no	—
196	#125	—	166	1	no	—
197	#124	—	2	1	no	—
198	#124	C₂×D₈₃	1	1	yes	characteristic
199	#123	—	2	1	no	—
200	C₂²⋊Q₈	—	2	1	no	—
201	C₂²⋊Q₈	—	83	1	no	—
202	C₂²⋊Q₈	—	83	1	no	—
203	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
204	#121	C₂×D₈₃	1	1	yes	characteristic
205	C₄×Q₈	—	83	1	no	—
206	C₄×Q₈	—	83	1	no	—
207	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
208	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
209	#117	C₂×D₈₃	1	1	yes	characteristic
210	#107	—	83	1	no	—
211	#107	—	83	1	no	—
212	#106	—	83	1	no	—
213	#106	—	83	1	no	—
214	#103	—	83	1	no	—
215	#103	—	83	1	no	—
216	#103	—	83	1	no	—
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#102	—	83	1	no	—
221	#102	—	83	1	no	—
222	#97	—	83	1	no	—
223	#97	—	83	1	no	—
224	C₄×C₈	—	83	1	no	—
225	C₄×C₈	—	83	1	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	83	1	no	—
228	C₂×Q₈	—	83	1	no	—
229	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
234	Q₁₆	—	166	2	no	—
235	Q₁₆	—	166	2	no	—
236	SD₁₆	—	166	2	no	—
237	SD₁₆	—	166	2	no	—
238	M₁₆	—	166	2	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	—	83	1	no	—
242	C₂×C₈	—	83	1	no	—
243	C₂×C₈	—	166	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₂²⋊C₄	—	166	1	no	—
268	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
269	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
270	C₄²	—	2	1	no	—
271	C₄²	—	166	1	no	—
272	C₄²	—	166	1	no	—
273	C₄²	C₂²×D₈₃	1	1	yes	characteristic
274	C₄²	C₂²×D₈₃	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	8	1	no	—
280	Q₈	—	166	1	no	—
281	Q₈	—	166	1	no	—
282	Q₈	—	166	1	no	—
283	Q₈	—	166	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	166	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
303	C₈	—	83	2	no	—
304	C₈	—	83	2	no	—
305	C₈	—	166	1	no	—
306	C₈	—	166	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	332	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	332	1	no	—
326	C₄	#41×D₈₃	1	1	yes	characteristic
327	C₄	#41×D₈₃	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	#19128247	1	1	yes	characteristic
331	C₂	#19128512	1	1	yes	characteristic
332	C₂	#19128516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#542×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#121×C₈₃	C₂²
hypercenter	#542	D₈₃
hypocenter	C₈₃	#3178
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #542

Complements

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

N	H ≅ G/N	type
1. #19128313	C₂	semidirect
2. #19128107	C₂	semidirect
3. #19128096	C₂	semidirect
4. #19128055	C₂	semidirect
5. #19128053	C₂	semidirect
6. #19127957	C₂	semidirect
8. #19127940	C₂	semidirect
10. #19127806	C₂	semidirect
11. #19127801	C₂	semidirect
12. C₄×#13050580	C₂	semidirect
13. C₄×#13050578	C₂	semidirect
14. #19127657	C₂	semidirect
167. C₈₃	#3178	semidirect

Maximal subgroups

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

subgroup	index
#19128313	2
#19128107	2
#19128096	2
#19128055	2
#19128053	2
#19127957	2
#19127945	2
#19127940	2
#19127928	2
#19127806	2
#19127801	2
C₄×#13050580	2
C₄×#13050578	2
#19127657	2
#542×C₈₃	2
#3178	83