Subgroups — #24917694

Order 10624.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128322	C₂	1	1	yes	maximal, characteristic
2	#19128178	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128026	C₂	1	1	yes	maximal, characteristic
5	#19127997	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127987	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127859	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127842	C₂	1	1	yes	maximal, characteristic
11	#19127809	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13050579	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
19	C₂×#13050579	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	#14581792	C₂²	1	1	yes	characteristic
22	#14581787	—	2	1	no	—
23	#14581780	C₂²	1	1	yes	characteristic
24	#14581778	—	2	1	no	—
25	#14581774	—	2	1	no	—
26	#14581771	C₂²	1	1	yes	characteristic
27	#14581770	—	2	1	no	—
28	#14581766	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
34	#14581757	—	2	1	no	—
35	#14581757	—	2	1	no	—
36	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
37	#14581751	C₂²	1	1	yes	characteristic
38	#14581748	—	2	1	no	—
39	#14581748	—	2	1	no	—
40	#14581726	C₂²	1	1	yes	characteristic
41	#14581726	C₂²	1	1	yes	characteristic
42	#14581724	C₂²	1	1	yes	characteristic
43	#14581724	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	C₂²	1	1	yes	characteristic
48	#14581717	C₂²	1	1	yes	characteristic
49	#14581717	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	#117×C₈₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#97×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×Dic₁₆₆	—	2	2	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	Dic₃₃₂	—	2	2	no	—
71	Dic₃₃₂	—	2	2	no	—
72	#13050579	—	2	2	no	—
73	#13050579	—	2	2	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
106	C₄²×C₈₃	C₂³	1	1	yes	characteristic
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₃	—	8	1	no	—
119	C₂×Dic₈₃	—	2	4	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	D₃₃₂	—	2	1	no	—
129	C₄×D₈₃	—	8	1	no	—
130	Dic₁₆₆	—	2	4	no	—
131	Dic₁₆₆	—	2	1	no	—
132	Dic₁₆₆	—	2	1	no	—
133	Dic₁₆₆	—	2	1	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	8	1	no	—
136	Dic₁₆₆	#41	1	2	yes	—
137	Dic₁₆₆	#41	1	2	yes	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₈₃	—	4	1	no	—
147	C₂×D₈₃	—	4	1	no	—
148	C₂×D₈₃	—	4	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
156	Dic₈₃	—	2	2	no	—
157	Dic₈₃	—	2	2	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	C₂×C₈₃	—	4	1	no	—
164	C₂×C₈₃	#525	1	1	yes	characteristic
165	C₂×C₈₃	#577	1	1	yes	characteristic
166	C₂×C₈₃	#578	1	1	yes	characteristic
167	D₈₃	—	8	1	no	—
168	#3121	—	83	1	no	maximal, Sylow
169	C₈₃	#3121	1	1	yes	characteristic, complemented, Sylow
170	#542	—	83	1	no	—
171	#519	—	83	1	no	—
172	#485	—	83	1	no	—
173	#482	—	83	1	no	—
174	#477	—	83	1	no	—
175	#475	—	83	1	no	—
176	#464	—	83	1	no	—
177	#464	—	83	1	no	—
178	#451	—	83	1	no	—
179	#448	—	83	1	no	—
180	#441	—	83	1	no	—
181	#440	—	83	1	no	—
182	C₄×Q₁₆	—	83	1	no	—
183	C₄×SD₁₆	—	83	1	no	—
184	#433	D₈₃	1	1	yes	characteristic, complemented
185	C₂²×Q₈	—	83	1	no	—
186	C₂×Q₁₆	—	83	1	no	—
187	C₂×Q₁₆	—	83	1	no	—
188	C₂×SD₁₆	—	83	1	no	—
189	C₂×SD₁₆	—	83	1	no	—
190	C₄⋊Q₈	—	83	1	no	—
191	#126	—	166	1	no	—
192	#124	—	83	1	no	—
193	#124	—	166	1	no	—
194	#123	—	166	1	no	—
195	C₂²⋊Q₈	—	83	1	no	—
196	C₂²⋊Q₈	—	166	1	no	—
197	#121	—	83	1	no	—
198	C₄×Q₈	—	83	1	no	—
199	C₄×Q₈	—	83	1	no	—
200	C₄×Q₈	—	83	1	no	—
201	C₄×Q₈	—	83	1	no	—
202	C₄×Q₈	—	83	1	no	—
203	C₄×Q₈	—	166	1	no	—
204	C₄×Q₈	—	166	1	no	—
205	C₄×D₄	—	83	1	no	—
206	#117	—	83	1	no	—
207	#117	—	166	1	no	—
208	#117	—	166	1	no	—
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	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
216	#103	—	83	1	no	—
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	—	83	1	no	—
222	#102	—	83	1	no	—
223	#102	—	83	1	no	—
224	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
226	#97	C₂×D₈₃	1	1	yes	characteristic
227	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
228	C₂×Q₈	—	83	1	no	—
229	C₂×Q₈	—	83	1	no	—
230	C₂×Q₈	—	83	1	no	—
231	C₂×Q₈	—	166	2	no	—
232	C₂×Q₈	—	166	2	no	—
233	C₂×Q₈	—	166	1	no	—
234	C₂×Q₈	—	166	1	no	—
235	C₂×D₄	—	83	1	no	—
236	C₂×D₄	—	166	1	no	—
237	C₂²×C₄	—	166	1	no	—
238	C₂²×C₄	—	166	1	no	—
239	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
240	Q₁₆	—	166	2	no	—
241	Q₁₆	—	166	2	no	—
242	SD₁₆	—	166	2	no	—
243	SD₁₆	—	166	2	no	—
244	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	83	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	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
260	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
268	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
269	C₄²	—	166	1	no	—
270	C₄²	—	166	1	no	—
271	C₄²	—	166	1	no	—
272	C₄²	—	166	1	no	—
273	C₄²	—	166	1	no	—
274	C₄²	—	166	1	no	—
275	C₄²	C₂²×D₈₃	1	1	yes	characteristic
276	C₄²	C₂²×D₈₃	1	1	yes	characteristic
277	C₂³	—	166	1	no	—
278	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
279	Q₈	—	83	2	no	—
280	Q₈	—	83	2	no	—
281	Q₈	—	664	1	no	—
282	Q₈	—	166	4	no	—
283	Q₈	—	166	1	no	—
284	Q₈	—	166	1	no	—
285	Q₈	—	166	1	no	—
286	Q₈	—	166	1	no	—
287	D₄	—	664	1	no	—
288	D₄	—	166	1	no	—
289	D₄	—	166	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	664	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	—	166	1	no	—
298	C₂×C₄	—	166	4	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	1	no	—
301	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
303	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
304	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
305	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	332	1	no	—
315	C₂²	—	332	1	no	—
316	C₂²	—	332	1	no	—
317	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	332	1	no	—
326	C₄	—	332	1	no	—
327	C₄	—	332	1	no	—
328	C₄	—	166	2	no	—
329	C₄	—	166	2	no	—
330	C₄	#41×D₈₃	1	1	yes	characteristic
331	C₄	#41×D₈₃	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	664	1	no	—
334	C₂	#19128213	1	1	yes	characteristic
335	C₂	#19128513	1	1	yes	characteristic
336	C₂	#19128518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#433×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581766	C₂²
hypercenter	#433	D₈₃
hypocenter	C₈₃	#3121
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
2. #19128178	C₂	semidirect
3. #19128043	C₂	semidirect
5. #19127997	C₂	semidirect
6. #19127987	C₂	semidirect
7. #19127924	C₂	semidirect
8. #19127924	C₂	semidirect
9. #19127859	C₂	semidirect
11. #19127809	C₂	semidirect
12. #19127803	C₂	semidirect
13. C₄×Dic₃₃₂	C₂	semidirect
14. C₄×#13050579	C₂	semidirect
15. #433×C₈₃	C₂	semidirect
169. C₈₃	#3121	semidirect
184. #433	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128322	2
#19128178	2
#19128043	2
#19128026	2
#19127997	2
#19127987	2
#19127924	2
#19127859	2
#19127842	2
#19127809	2
#19127803	2
C₄×Dic₃₃₂	2
C₄×#13050579	2
#433×C₈₃	2
#3121	83