Subgroups — #24920531

Order 10624.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128436	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128424	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128095	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128090	C₂	1	1	yes	maximal, characteristic
5	#19128085	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127907	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127902	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127814	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127809	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13050579	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050579	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	C₂×#13050579	C₂²	1	1	yes	characteristic
22	C₂×#13050579	C₂²	1	1	yes	characteristic
23	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
24	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
25	#14581792	C₂²	1	1	yes	characteristic
26	#14581789	C₂²	1	1	yes	characteristic
27	#14581787	—	2	1	no	—
28	#14581787	—	2	1	no	—
29	#14581781	—	2	1	no	—
30	#14581780	C₂²	1	1	yes	characteristic
31	#14581780	C₂²	1	1	yes	characteristic
32	#14581780	C₂²	1	1	yes	characteristic
33	#14581780	C₂²	1	1	yes	characteristic
34	#14581777	—	2	1	no	—
35	#14581777	—	2	1	no	—
36	#14581768	—	2	1	no	—
37	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
40	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
41	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
42	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
43	#14581754	—	2	1	no	—
44	#14581750	—	2	1	no	—
45	#14581724	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	#14581714	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₄×Q₈×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	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	#97×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	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	Dic₃₃₂	—	2	2	no	—
71	Dic₃₃₂	—	4	1	no	—
72	#13050579	—	2	2	no	—
73	#13050579	—	4	1	no	—
74	#13050579	—	4	1	no	—
75	#13050579	—	4	1	no	—
76	#13050579	—	4	1	no	—
77	D₆₆₄	—	2	2	no	—
78	D₆₆₄	—	4	1	no	—
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	—	2	1	no	—
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	—	2	1	no	—
97	C₄×Dic₈₃	—	2	1	no	—
98	C₄×Dic₈₃	—	2	1	no	—
99	C₄×Dic₈₃	—	2	1	no	—
100	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
108	C₄²×C₈₃	C₂³	1	1	yes	characteristic
109	C₄²×C₈₃	C₂³	1	1	yes	characteristic
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂²×D₈₃	—	2	1	no	—
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	4	1	no	—
115	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
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	D₃₃₂	—	2	1	no	—
130	D₃₃₂	—	2	1	no	—
131	D₃₃₂	—	2	1	no	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	4	2	no	—
134	D₃₃₂	—	8	1	no	—
135	C₄×D₈₃	—	4	2	no	—
136	C₄×D₈₃	—	8	1	no	—
137	C₄×D₈₃	—	8	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	2	1	no	—
143	Dic₁₆₆	—	2	1	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	#41	1	2	yes	—
148	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	2	1	no	—
161	C₄×C₈₃	—	2	1	no	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	—	2	1	no	—
164	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
165	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	—	4	1	no	—
171	C₂×C₈₃	#550	1	1	yes	characteristic
172	C₂×C₈₃	#575	1	1	yes	characteristic
173	C₂×C₈₃	#577	1	1	yes	characteristic
174	D₈₃	—	8	1	no	—
175	D₈₃	—	8	1	no	—
176	D₈₃	—	8	1	no	—
177	#3401	—	83	1	no	maximal, Sylow
178	C₈₃	#3401	1	1	yes	characteristic, complemented, Sylow
179	#559	—	83	1	no	—
180	#556	—	83	1	no	—
181	#497	—	83	1	no	—
182	#496	—	83	1	no	—
183	#495	—	83	1	no	—
184	#464	—	83	1	no	—
185	#464	—	83	1	no	—
186	#463	—	83	1	no	—
187	#463	—	83	1	no	—
188	#461	—	83	1	no	—
189	#460	—	83	1	no	—
190	#446	D₈₃	1	1	yes	characteristic, complemented
191	#442	—	83	1	no	—
192	#441	—	83	1	no	—
193	C₄×SD₁₆	—	83	1	no	—
194	C₂×Q₁₆	—	83	1	no	—
195	C₂×Q₁₆	—	83	1	no	—
196	C₂×SD₁₆	—	83	1	no	—
197	C₂×SD₁₆	—	83	1	no	—
198	C₂×SD₁₆	—	83	1	no	—
199	C₂×SD₁₆	—	83	1	no	—
200	C₂×SD₁₆	—	83	1	no	—
201	C₂×D₈	—	83	1	no	—
202	C₂×D₈	—	83	1	no	—
203	C₄⋊Q₈	—	83	1	no	—
204	#127	—	83	1	no	—
205	#126	—	166	1	no	—
206	#126	—	166	1	no	—
207	#124	—	83	1	no	—
208	#124	—	83	1	no	—
209	#124	—	83	1	no	—
210	#124	—	83	1	no	—
211	#124	—	166	1	no	—
212	#123	—	166	1	no	—
213	#123	—	166	1	no	—
214	C₂²⋊Q₈	—	166	1	no	—
215	C₄×Q₈	—	83	1	no	—
216	C₄×Q₈	—	83	1	no	—
217	C₄×Q₈	—	83	1	no	—
218	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
219	C₄×D₄	—	83	1	no	—
220	C₄×D₄	—	83	1	no	—
221	C₄×D₄	—	83	1	no	—
222	C₄×D₄	—	166	1	no	—
223	#117	—	166	1	no	—
224	#106	—	83	1	no	—
225	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
226	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
227	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
228	#103	—	83	1	no	—
229	#103	—	83	1	no	—
230	#103	—	83	1	no	—
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#97	C₂×D₈₃	1	1	yes	characteristic
235	#97	C₂×D₈₃	1	1	yes	characteristic
236	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
237	C₂×Q₈	—	83	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	83	1	no	—
240	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
241	C₂×D₄	—	83	1	no	—
242	C₂×D₄	—	83	1	no	—
243	C₂×D₄	—	83	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂×D₄	—	166	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	Q₁₆	—	332	1	no	—
250	Q₁₆	—	166	2	no	—
251	SD₁₆	—	332	1	no	—
252	SD₁₆	—	332	1	no	—
253	SD₁₆	—	332	1	no	—
254	SD₁₆	—	332	1	no	—
255	SD₁₆	—	166	2	no	—
256	D₈	—	332	1	no	—
257	D₈	—	166	2	no	—
258	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₄⋊C₄	—	83	1	no	—
263	C₄⋊C₄	—	83	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
270	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
271	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₂²⋊C₄	—	166	1	no	—
283	C₄²	—	166	1	no	—
284	C₄²	—	166	1	no	—
285	C₄²	—	166	1	no	—
286	C₄²	C₂²×D₈₃	1	1	yes	characteristic
287	C₄²	C₂²×D₈₃	1	1	yes	characteristic
288	C₄²	C₂²×D₈₃	1	1	yes	characteristic
289	C₂³	—	166	1	no	—
290	C₂³	—	166	1	no	—
291	C₂³	—	166	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	166	1	no	—
294	Q₈	—	166	1	no	—
295	Q₈	—	166	1	no	—
296	Q₈	—	166	1	no	—
297	Q₈	—	166	1	no	—
298	Q₈	—	166	1	no	—
299	D₄	—	332	2	no	—
300	D₄	—	664	1	no	—
301	D₄	—	166	1	no	—
302	D₄	—	166	1	no	—
303	D₄	—	166	1	no	—
304	D₄	—	166	1	no	—
305	D₄	—	166	1	no	—
306	D₄	—	166	1	no	—
307	C₂×C₄	—	332	2	no	—
308	C₂×C₄	—	664	1	no	—
309	C₂×C₄	—	664	1	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	—	166	1	no	—
315	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
316	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
317	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
318	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
319	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
320	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	#13050591	1	2	yes	—
326	C₂²	—	332	1	no	—
327	C₂²	—	332	1	no	—
328	C₂²	—	332	1	no	—
329	C₂²	—	332	1	no	—
330	C₂²	—	332	1	no	—
331	C₂²	—	332	1	no	—
332	C₂²	—	332	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	332	1	no	—
343	C₄	—	332	1	no	—
344	C₄	—	332	1	no	—
345	C₄	—	332	1	no	—
346	C₄	—	332	1	no	—
347	C₄	C₂×#13050591	1	1	yes	characteristic
348	C₄	#14581838	1	1	yes	characteristic
349	C₂	—	664	1	no	—
350	C₂	—	664	1	no	—
351	C₂	—	664	1	no	—
352	C₂	#19128373	1	1	yes	characteristic
353	C₂	#19128501	1	1	yes	characteristic
354	C₂	#19128513	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#446×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128090	C₂
hypercenter	#446	D₈₃
hypocenter	C₈₃	#3401
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #19128436	C₂	semidirect
2. #19128424	C₂	semidirect
3. #19128095	C₂	semidirect
5. #19128085	C₂	semidirect
6. #19127924	C₂	semidirect
7. #19127924	C₂	semidirect
8. #19127917	C₂	semidirect
9. #19127917	C₂	semidirect
10. #19127907	C₂	semidirect
11. #19127902	C₂	semidirect
12. #19127814	C₂	semidirect
13. #19127809	C₂	semidirect
14. C₄×#13050579	C₂	semidirect
15. #446×C₈₃	C₂	semidirect
178. C₈₃	#3401	semidirect
190. #446	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128436	2
#19128424	2
#19128095	2
#19128090	2
#19128085	2
#19127924	2
#19127917	2
#19127907	2
#19127902	2
#19127814	2
#19127809	2
C₄×#13050579	2
#446×C₈₃	2
#3401	83