Subgroups — #24920664

Order 10624.

1540 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13050590	C₂	1	1	yes	maximal, characteristic
2	#19128182	C₂	1	4	yes	maximal, complemented
3	#19128181	C₂	1	4	yes	maximal, complemented
4	#19128178	C₂	1	2	yes	maximal, complemented
5	#19128175	C₂	1	2	yes	maximal, complemented
6	#19128174	C₂	1	4	yes	maximal, complemented
7	C₄×#13050590	C₂	1	4	yes	maximal, complemented
8	#19128169	C₂	1	4	yes	maximal, complemented
9	C₂×#14581751	C₂	1	1	yes	maximal, characteristic, complemented
10	#117×D₈₃	C₂	1	4	yes	maximal, complemented
11	C₂×#117×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13050590	C₄	1	16	yes	complemented
13	C₂×#13050590	C₂²	1	2	yes	—
14	C₂×#13050590	C₂²	1	2	yes	—
15	C₂×#13050590	C₂²	1	2	yes	—
16	C₂×#13050590	C₂²	1	2	yes	—
17	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
18	C₂²×D₃₃₂	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊D₈₃	C₂²	1	2	yes	—
20	C₂²×C₄×D₈₃	C₂²	1	2	yes	—
21	C₄×Dic₁₆₆	C₂²	1	8	yes	complemented
22	#14581757	C₂²	1	8	yes	complemented
23	C₄×D₃₃₂	C₂²	1	8	yes	complemented
24	#14581754	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊D₈₃	C₂²	1	16	yes	complemented
26	(C₂²⋊D₈₃)⋊C₄	C₂²	1	16	yes	complemented
27	#14581751	C₂²	1	2	yes	—
28	#14581751	C₂²	1	2	yes	—
29	#14581751	C₂²	1	2	yes	complemented
30	#14581751	C₂²	1	2	yes	complemented
31	#14581750	C₂²	1	8	yes	complemented
32	#14581749	C₂²	1	8	yes	complemented
33	#14581748	C₂²	1	8	yes	—
34	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	complemented
35	C₄⋊C₄×D₈₃	C₂²	1	8	yes	complemented
36	C₂×C₂²⋊Dic₈₃	C₂²	1	2	yes	—
37	C₂²⋊C₄×D₈₃	C₂²	1	8	yes	—
38	C₂×C₄×Dic₈₃	C₂²	1	2	yes	complemented
39	C₄²×D₈₃	C₂²	1	8	yes	complemented
40	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
41	#117×C₈₃	C₂²	1	2	yes	—
42	#117×C₈₃	C₂²	1	2	yes	—
43	#117×C₈₃	C₂²	1	2	yes	complemented
44	#117×C₈₃	C₂²	1	2	yes	complemented
45	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	complemented
46	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
47	C₂×C₄²×C₈₃	C₂²	1	2	yes	complemented
48	C₂³×D₈₃	C₂³	1	2	yes	—
49	#13050590	C₂×C₄	1	16	yes	complemented
50	#13050590	C₂×C₄	1	16	yes	complemented
51	#13050590	C₂×C₄	1	16	yes	complemented
52	#13050590	C₂×C₄	1	16	yes	complemented
53	C₂×Dic₁₆₆	C₂×C₄	1	8	yes	complemented
54	C₂×Dic₁₆₆	C₂³	1	2	yes	—
55	C₂×Dic₁₆₆	C₂³	1	2	yes	—
56	C₂×D₃₃₂	C₂×C₄	1	8	yes	complemented
57	C₂×D₃₃₂	C₂³	1	2	yes	—
58	C₂×D₃₃₂	C₂³	1	2	yes	—
59	C₂×C₂²⋊D₈₃	C₂×C₄	1	16	yes	complemented
60	C₂×C₂²⋊D₈₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊D₈₃	C₂³	1	4	yes	—
62	C₂²×Dic₈₃	C₂³	1	4	yes	—
63	C₂²×Dic₈₃	C₂³	1	2	yes	—
64	C₂×C₄×D₈₃	C₂×C₄	1	16	yes	complemented
65	C₂×C₄×D₈₃	C₂³	1	16	yes	—
66	C₂×C₄×D₈₃	C₂³	1	4	yes	—
67	C₂×C₄×D₈₃	C₂³	1	4	yes	—
68	C₄⋊Dic₈₃	C₂³	1	4	yes	—
69	C₄⋊Dic₈₃	C₂³	1	4	yes	—
70	Dic₈₃⋊C₄	C₂³	1	16	yes	—
71	C₂²⋊Dic₈₃	C₂³	1	4	yes	—
72	C₂²⋊Dic₈₃	C₂³	1	4	yes	—
73	(C₂×D₈₃)⋊C₄	C₂³	1	16	yes	—
74	C₄×Dic₈₃	C₂³	1	4	yes	—
75	C₄×Dic₈₃	C₂³	1	16	yes	—
76	C₄×Dic₈₃	C₂³	1	4	yes	—
77	C₂⁴×C₈₃	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₈₃	C₂×C₄	1	8	yes	complemented
79	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
80	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
87	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₈₃	C₂³	1	4	yes	—
89	C₂²⋊C₄×C₈₃	C₂³	1	4	yes	—
90	C₄²×C₈₃	C₂³	1	4	yes	—
91	C₄²×C₈₃	C₂³	1	4	yes	—
92	C₂³×C₈₃	—	4	2	no	—
93	C₂³×C₈₃	C₂²×C₄	1	4	yes	—
94	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
95	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
96	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
97	C₂²×D₈₃	—	4	4	no	—
98	C₂²×D₈₃	C₂²×C₄	1	8	yes	—
99	C₂²×D₈₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₈₃	—	4	4	no	—
101	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
103	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
107	C₂×C₄×C₈₃	C₂⁴	1	8	yes	—
108	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₈₃	C₂²×C₄	1	16	yes	—
113	C₂²⋊D₈₃	C₂²×C₄	1	16	yes	—
114	C₂×Dic₈₃	—	4	4	no	—
115	C₂×Dic₈₃	C₂²×C₄	1	8	yes	—
116	C₂×Dic₈₃	C₂⁴	1	8	yes	—
117	C₂×Dic₈₃	C₂⁴	1	4	yes	—
118	D₃₃₂	C₂²×C₄	1	8	yes	—
119	D₃₃₂	C₂²×C₄	1	8	yes	—
120	C₄×D₈₃	—	4	16	no	—
121	C₄×D₈₃	C₂²×C₄	1	16	yes	—
122	C₄×D₈₃	C₂²×C₄	1	16	yes	—
123	Dic₁₆₆	C₂²×C₄	1	8	yes	—
124	Dic₁₆₆	C₂²×C₄	1	8	yes	—
125	C₂²×C₈₃	—	2	2	no	—
126	C₂²×C₈₃	—	2	2	no	—
127	C₂²×C₈₃	—	2	2	no	—
128	C₂²×C₈₃	—	4	4	no	—
129	C₂²×C₈₃	C₂³×C₄	1	2	yes	—
130	C₂²×C₈₃	C₂³×C₄	1	2	yes	—
131	C₂²×C₈₃	C₂³×C₄	1	2	yes	—
132	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
133	C₂×D₈₃	—	2	8	no	—
134	C₂×D₈₃	—	4	8	no	—
135	C₂×D₈₃	C₂³×C₄	1	8	yes	—
136	C₄×C₈₃	—	2	8	no	—
137	C₄×C₈₃	C₂³×C₄	1	2	yes	—
138	C₄×C₈₃	C₂³×C₄	1	2	yes	—
139	C₄×C₈₃	C₂³×C₄	1	2	yes	—
140	C₄×C₈₃	C₂³×C₄	1	2	yes	—
141	Dic₈₃	—	2	8	no	—
142	Dic₈₃	C₂³×C₄	1	8	yes	—
143	C₂×C₈₃	—	2	2	no	—
144	C₂×C₈₃	—	2	2	no	—
145	C₂×C₈₃	—	2	2	no	—
146	C₂×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
147	C₂×C₈₃	#585	1	2	yes	—
148	D₈₃	—	2	8	no	—
149	#3428	—	83	1	no	maximal, Sylow
150	C₈₃	#3428	1	1	yes	characteristic, complemented, Sylow
151	C₂²×#41	—	83	1	no	—
152	#520	—	83	4	no	—
153	#520	—	83	4	no	—
154	#519	—	83	2	no	—
155	#518	—	83	2	no	—
156	#518	—	83	4	no	—
157	C₄×#41	—	83	4	no	—
158	C₄×#41	—	83	4	no	—
159	C₂×#117	—	83	1	no	—
160	C₂×#117	—	83	4	no	—
161	C₂×#117	D₈₃	1	1	yes	characteristic, complemented
162	C₂×#41	—	83	16	no	—
163	C₂×#41	—	83	2	no	—
164	C₂×#41	—	83	2	no	—
165	C₂×#41	—	83	2	no	—
166	C₂×#41	—	83	2	no	—
167	C₂²×Q₈	—	83	1	no	—
168	C₂²×D₄	—	83	2	no	—
169	C₂²×D₄	—	83	1	no	—
170	C₂³×C₄	—	83	2	no	—
171	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
172	C₄×Q₈	—	83	8	no	—
173	C₄×Q₈	—	83	8	no	—
174	C₄×D₄	—	83	16	no	—
175	C₄×D₄	—	83	8	no	—
176	C₄×D₄	—	83	16	no	—
177	C₄×D₄	—	83	8	no	—
178	#117	—	83	2	no	—
179	#117	—	83	2	no	—
180	#117	—	83	2	no	—
181	#117	—	83	2	no	—
182	#117	—	83	8	no	—
183	#117	—	83	8	no	—
184	#117	—	83	8	no	—
185	#117	C₂×D₈₃	1	2	yes	—
186	#117	C₂×D₈₃	1	2	yes	—
187	#117	C₂×D₈₃	1	2	yes	complemented
188	#117	C₂×D₈₃	1	2	yes	complemented
189	C₂×C₄⋊C₄	—	83	2	no	—
190	C₂×C₄⋊C₄	—	83	8	no	—
191	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	complemented
192	C₂×C₂²⋊C₄	—	83	2	no	—
193	C₂×C₂²⋊C₄	—	83	8	no	—
194	C₂×C₂²⋊C₄	C₂×D₈₃	1	2	yes	—
195	C₂×C₄²	—	83	2	no	—
196	C₂×C₄²	—	83	8	no	—
197	C₂×C₄²	C₂×D₈₃	1	2	yes	complemented
198	C₂⁴	—	83	2	no	—
199	C₂⁴	C₂²×D₈₃	1	1	yes	characteristic
200	#41	—	83	16	no	—
201	#41	—	83	16	no	—
202	#41	—	83	16	no	—
203	#41	—	83	16	no	—
204	C₂×Q₈	—	83	8	no	—
205	C₂×Q₈	—	83	2	no	—
206	C₂×Q₈	—	83	2	no	—
207	C₂×D₄	—	83	16	no	—
208	C₂×D₄	—	83	4	no	—
209	C₂×D₄	—	83	4	no	—
210	C₂×D₄	—	83	8	no	—
211	C₂×D₄	—	83	2	no	—
212	C₂×D₄	—	83	2	no	—
213	C₂²×C₄	—	83	4	no	—
214	C₂²×C₄	—	83	16	no	—
215	C₂²×C₄	—	83	2	no	—
216	C₂²×C₄	—	83	16	no	—
217	C₂²×C₄	—	83	4	no	—
218	C₂²×C₄	—	83	4	no	—
219	C₂²×C₄	C₄×D₈₃	1	8	yes	complemented
220	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
221	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
227	C₄⋊C₄	—	83	4	no	—
228	C₄⋊C₄	—	83	16	no	—
229	C₄⋊C₄	—	83	4	no	—
230	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
231	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
232	C₂²⋊C₄	—	83	4	no	—
233	C₂²⋊C₄	—	83	4	no	—
234	C₂²⋊C₄	—	83	16	no	—
235	C₂²⋊C₄	C₂²×D₈₃	1	4	yes	—
236	C₂²⋊C₄	C₂²×D₈₃	1	4	yes	—
237	C₄²	—	83	16	no	—
238	C₄²	—	83	4	no	—
239	C₄²	—	83	4	no	—
240	C₄²	C₂²×D₈₃	1	4	yes	—
241	C₄²	C₂²×D₈₃	1	4	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	—	332	4	no	—
244	C₂³	—	83	8	no	—
245	C₂³	—	83	4	no	—
246	C₂³	C₂×C₄×D₈₃	1	4	yes	—
247	C₂³	C₂³×D₈₃	1	1	yes	characteristic
248	C₂³	C₂³×D₈₃	1	1	yes	characteristic
249	C₂³	C₂³×D₈₃	1	1	yes	characteristic
250	Q₈	—	83	8	no	—
251	Q₈	—	83	8	no	—
252	D₄	—	83	16	no	—
253	D₄	—	83	16	no	—
254	D₄	—	83	8	no	—
255	D₄	—	83	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	332	16	no	—
258	C₂×C₄	—	332	4	no	—
259	C₂×C₄	—	83	8	no	—
260	C₂×C₄	—	83	4	no	—
261	C₂×C₄	—	83	16	no	—
262	C₂×C₄	—	83	16	no	—
263	C₂×C₄	—	83	8	no	—
264	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
265	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
266	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
267	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
268	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
269	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
270	C₂×C₄	C₂³×D₈₃	1	8	yes	—
271	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
272	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	332	8	no	—
280	C₂²	—	83	8	no	—
281	C₂²	—	166	8	no	—
282	C₂²	C₂²×C₄×D₈₃	1	2	yes	—
283	C₂²	C₂²×C₄×D₈₃	1	2	yes	—
284	C₂²	C₂²×C₄×D₈₃	1	2	yes	—
285	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	83	8	no	—
288	C₄	—	166	8	no	—
289	C₄	C₂²×C₄×D₈₃	1	2	yes	—
290	C₄	C₂²×C₄×D₈₃	1	2	yes	—
291	C₄	C₂²×C₄×D₈₃	1	2	yes	—
292	C₄	C₂²×C₄×D₈₃	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	166	8	no	—
297	C₂	C₂³×C₄×D₈₃	1	1	yes	characteristic
298	C₂	#19128537	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₈₃
commutator	C₂×C₈₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₈₃
Fitting	C₂×#117×C₈₃	C₂
socle, cosocle	C₂²×C₈₃	C₂⁵
prime radical	C₂²×#13050590	C₂
hypercenter	C₂×#117	D₈₃
hypocenter	C₈₃	#3428
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #19128182	C₂	semidirect
3. #19128181	C₂	semidirect
4. #19128178	C₂	semidirect
5. #19128175	C₂	semidirect
6. #19128174	C₂	semidirect
7. C₄×#13050590	C₂	semidirect
8. #19128169	C₂	semidirect
9. C₂×#14581751	C₂	semidirect
10. #117×D₈₃	C₂	semidirect
11. C₂×#117×C₈₃	C₂	semidirect
12. C₂×#13050590	C₄	semidirect
21. C₄×Dic₁₆₆	C₂²	semidirect
22. #14581757	C₂²	semidirect
23. C₄×D₃₃₂	C₂²	semidirect
24. #14581754	C₂²	semidirect
25. C₄×C₂²⋊D₈₃	C₂²	semidirect
26. (C₂²⋊D₈₃)⋊C₄	C₂²	semidirect
29. #14581751	C₂²	semidirect
30. #14581751	C₂²	semidirect
31. #14581750	C₂²	semidirect
32. #14581749	C₂²	semidirect
34. C₂×C₄⋊Dic₈₃	C₂²	semidirect
35. C₄⋊C₄×D₈₃	C₂²	semidirect
38. C₂×C₄×Dic₈₃	C₂²	semidirect
39. C₄²×D₈₃	C₂²	semidirect
43. #117×C₈₃	C₂²	semidirect
44. #117×C₈₃	C₂²	semidirect
45. C₂×C₄⋊C₄×C₈₃	C₂²	semidirect
47. C₂×C₄²×C₈₃	C₂²	semidirect
49. #13050590	C₂×C₄	semidirect
50. #13050590	C₂×C₄	semidirect
51. #13050590	C₂×C₄	semidirect
52. #13050590	C₂×C₄	semidirect
53. C₂×Dic₁₆₆	C₂×C₄	semidirect
56. C₂×D₃₃₂	C₂×C₄	semidirect
59. C₂×C₂²⋊D₈₃	C₂×C₄	semidirect
64. C₂×C₄×D₈₃	C₂×C₄	semidirect
78. C₂²×C₄×C₈₃	C₂×C₄	semidirect
150. C₈₃	#3428	semidirect
161. C₂×#117	D₈₃	semidirect
187. #117	C₂×D₈₃	semidirect
188. #117	C₂×D₈₃	semidirect
191. C₂×C₄⋊C₄	C₂×D₈₃	semidirect
197. C₂×C₄²	C₂×D₈₃	semidirect
219. C₂²×C₄	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13050590	2
#19128182	2
#19128181	2
#19128178	2
#19128175	2
#19128174	2
C₄×#13050590	2
#19128169	2
C₂×#14581751	2
#117×D₈₃	2
C₂×#117×C₈₃	2
#3428	83