Subgroups — #24918868

Order 10624.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 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, complemented
2	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127991	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127981	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127952	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127945	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127935	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127928	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127916	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127916	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127898	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127893	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127728	C₂	1	1	yes	maximal, characteristic, complemented
15	#540×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050578	C₂²	1	1	yes	characteristic
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂×#13050575	C₂²	1	1	yes	characteristic
19	C₂×#13050575	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	#14581787	—	2	1	no	—
24	#14581780	C₂²	1	1	yes	characteristic
25	#14581778	—	2	1	no	—
26	#14581774	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581770	—	2	1	no	—
29	#14581766	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
31	C₄×D₃₃₂	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	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
37	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₈₃⋊C₈)	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	#14581713	C₂²	1	1	yes	characteristic
45	#14581713	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#127×C₈₃	C₂²	1	1	yes	characteristic
49	#125×C₈₃	C₂²	1	1	yes	characteristic
50	#123×C₈₃	—	2	1	no	—
51	#123×C₈₃	—	2	1	no	—
52	#121×C₈₃	—	2	1	no	—
53	#121×C₈₃	—	2	1	no	—
54	#121×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×D₄×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	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	—	2	1	no	—
64	#13050578	—	4	1	no	—
65	#13050578	—	4	1	no	—
66	#13050575	—	4	1	no	—
67	#13050575	—	4	1	no	—
68	#13050574	—	4	1	no	—
69	C₂×C₈₃⋊C₈	—	4	1	no	—
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	C₂²⋊Dic₈₃	—	2	1	no	—
80	(C₂×D₈₃)⋊C₄	—	2	1	no	—
81	(C₂×D₈₃)⋊C₄	—	2	1	no	—
82	(C₂×D₈₃)⋊C₄	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	C₄×Dic₈₃	—	2	1	no	—
85	C₂×D₄×C₈₃	—	2	1	no	—
86	C₂×D₄×C₈₃	—	2	1	no	—
87	C₂×D₄×C₈₃	—	2	1	no	—
88	C₂×D₄×C₈₃	—	2	1	no	—
89	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₈₃	—	2	1	no	—
92	C₂²×C₄×C₈₃	—	2	1	no	—
93	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	—	2	1	no	—
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
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	—	2	1	no	—
109	C₂³×C₈₃	—	2	1	no	—
110	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₈₃	—	2	1	no	—
112	D₄×C₈₃	—	2	1	no	—
113	D₄×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	1	no	—
115	D₄×C₈₃	—	2	1	no	—
116	D₄×C₈₃	—	8	1	no	—
117	D₄×C₈₃	—	8	1	no	—
118	D₄×C₈₃	—	8	1	no	—
119	D₄×C₈₃	—	8	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	8	1	no	—
125	C₂×C₄×C₈₃	—	8	1	no	—
126	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₃	—	8	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	D₃₃₂	—	2	1	no	—
138	C₄×D₈₃	—	8	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	8	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	—	4	1	no	—
152	C₂²×C₈₃	—	4	1	no	—
153	C₂²×C₈₃	—	4	1	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₂×D₈₃	—	4	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	—	4	1	no	—
165	C₄×C₈₃	—	4	1	no	—
166	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
168	Dic₈₃	—	4	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	—	4	1	no	—
171	C₂×C₈₃	—	4	1	no	—
172	C₂×C₈₃	—	8	1	no	—
173	C₂×C₈₃	—	8	1	no	—
174	C₂×C₈₃	#537	1	1	yes	characteristic
175	C₂×C₈₃	#576	1	1	yes	characteristic
176	C₂×C₈₃	#577	1	1	yes	characteristic
177	D₈₃	—	8	1	no	—
178	#3257	—	83	1	no	maximal, Sylow
179	C₈₃	#3257	1	1	yes	characteristic, complemented, Sylow
180	#542	—	83	1	no	—
181	#540	D₈₃	1	1	yes	characteristic, complemented
182	#479	—	83	1	no	—
183	#478	—	83	1	no	—
184	#476	—	83	1	no	—
185	#474	—	83	1	no	—
186	#469	—	83	1	no	—
187	#468	—	83	1	no	—
188	#466	—	83	1	no	—
189	#465	—	83	1	no	—
190	#463	—	83	1	no	—
191	#463	—	83	1	no	—
192	#460	—	83	1	no	—
193	#459	—	83	1	no	—
194	#424	—	83	1	no	—
195	C₂×SD₁₆	—	83	1	no	—
196	C₂×SD₁₆	—	83	1	no	—
197	C₂×D₈	—	83	1	no	—
198	C₂×D₈	—	83	1	no	—
199	C₂×M₁₆	—	83	1	no	—
200	C₂²×C₈	—	83	1	no	—
201	C₄⋊Q₈	—	83	1	no	—
202	#127	C₂×D₈₃	1	1	yes	characteristic
203	#126	—	166	1	no	—
204	#125	C₂×D₈₃	1	1	yes	characteristic
205	#124	—	83	1	no	—
206	#124	—	166	1	no	—
207	#123	—	2	1	no	—
208	#123	—	2	1	no	—
209	#123	—	166	1	no	—
210	C₂²⋊Q₈	—	83	1	no	—
211	C₂²⋊Q₈	—	166	1	no	—
212	#121	—	2	1	no	—
213	#121	—	2	1	no	—
214	#121	—	83	1	no	—
215	#121	C₂×D₈₃	1	1	yes	characteristic
216	#121	C₂×D₈₃	1	1	yes	characteristic
217	C₄×Q₈	—	83	1	no	—
218	C₄×D₄	—	83	1	no	—
219	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
220	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
221	#117	C₂×D₈₃	1	1	yes	characteristic
222	#107	—	83	1	no	—
223	#107	—	83	1	no	—
224	#106	—	83	1	no	—
225	#106	—	83	1	no	—
226	C₄⋊C₈	—	83	1	no	—
227	C₄⋊C₈	—	83	1	no	—
228	C₄⋊C₈	—	83	1	no	—
229	C₄⋊C₈	—	83	1	no	—
230	#103	—	83	1	no	—
231	#103	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	166	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	83	1	no	—
245	C₂×D₄	—	166	1	no	—
246	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
252	SD₁₆	—	332	1	no	—
253	SD₁₆	—	332	1	no	—
254	D₈	—	332	1	no	—
255	D₈	—	332	1	no	—
256	M₁₆	—	332	1	no	—
257	C₂×C₈	—	332	1	no	—
258	C₂×C₈	—	83	1	no	—
259	C₂×C₈	—	83	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	—	83	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	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	C₂²⋊D₈₃	1	1	yes	characteristic
283	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
284	C₄²	—	166	1	no	—
285	C₄²	C₂²×D₈₃	1	1	yes	characteristic
286	C₄²	C₂²×D₈₃	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	166	1	no	—
290	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
291	Q₈	—	664	1	no	—
292	Q₈	—	166	1	no	—
293	Q₈	—	166	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	D₄	—	8	1	no	—
302	D₄	—	664	1	no	—
303	D₄	—	166	1	no	—
304	D₄	—	166	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	664	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	—	166	1	no	—
315	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
318	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
321	C₈	—	166	1	no	—
322	C₈	—	166	1	no	—
323	C₈	—	166	1	no	—
324	C₈	—	166	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	—	332	1	no	—
336	C₂²	—	332	1	no	—
337	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	332	1	no	—
346	C₄	—	332	1	no	—
347	C₄	—	332	1	no	—
348	C₄	#14581838	1	1	yes	characteristic
349	C₄	#14581839	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	8	1	no	—
352	C₂	—	8	1	no	—
353	C₂	—	664	1	no	—
354	C₂	#19128279	1	1	yes	characteristic
355	C₂	#19128504	1	1	yes	characteristic
356	C₂	#19128512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#540×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#540	D₈₃
hypocenter	C₈₃	#3257
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #540

Complements

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

N	H ≅ G/N	type
1. #19128322	C₂	semidirect
2. #19128005	C₂	semidirect
3. #19127998	C₂	semidirect
4. #19127991	C₂	semidirect
5. #19127981	C₂	semidirect
6. #19127952	C₂	semidirect
7. #19127945	C₂	semidirect
8. #19127935	C₂	semidirect
9. #19127928	C₂	semidirect
10. #19127916	C₂	semidirect
11. #19127916	C₂	semidirect
12. #19127898	C₂	semidirect
13. #19127893	C₂	semidirect
14. #19127728	C₂	semidirect
15. #540×C₈₃	C₂	semidirect
179. C₈₃	#3257	semidirect
181. #540	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
#19128005	2
#19127998	2
#19127991	2
#19127981	2
#19127952	2
#19127945	2
#19127935	2
#19127928	2
#19127916	2
#19127898	2
#19127893	2
#19127728	2
#540×C₈₃	2
#3257	83