Subgroups — #24919352

Order 10624.

386 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	#19128391	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128360	C₂	1	1	yes	maximal, characteristic
3	#19128056	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128015	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127942	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127930	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127865	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127859	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127698	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14581717	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	#14581785	—	2	1	no	—
22	#14581784	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581780	C₂²	1	1	yes	characteristic
25	#14581778	—	2	1	no	—
26	#14581773	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581771	C₂²	1	1	yes	characteristic
29	#14581771	C₂²	1	1	yes	characteristic
30	#14581770	—	2	1	no	—
31	#14581766	C₂²	1	1	yes	characteristic
32	#14581764	—	2	1	no	—
33	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
36	#14581751	C₂²	1	1	yes	characteristic
37	#14581748	—	2	1	no	—
38	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581726	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	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	#14581714	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	2	2	no	—
60	#13050590	—	4	1	no	—
61	C₂×Dic₁₆₆	—	2	2	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	—	4	1	no	—
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂×C₂²⋊D₈₃	—	2	1	no	—
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂²×Dic₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	Dic₃₃₂	—	2	2	no	—
74	Dic₃₃₂	—	4	1	no	—
75	#13050579	—	4	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	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	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₄×Dic₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	—	4	1	no	—
102	C₂×C₈×C₈₃	—	4	1	no	—
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	2	no	—
115	C₂×C₄×C₈₃	—	2	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	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
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	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	4	2	no	—
134	C₂×Dic₈₃	—	4	2	no	—
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	C₄×D₈₃	—	4	2	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	—	4	2	no	—
144	Dic₁₆₆	#41	1	2	yes	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₈×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	—	2	1	no	—
154	C₂²×C₈₃	#135	1	2	yes	—
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	C₂×#41	1	1	yes	characteristic
164	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₈₃	—	2	2	no	—
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	Dic₈₃	—	4	1	no	—
172	C₂×C₈₃	—	2	2	no	—
173	C₂×C₈₃	—	4	1	no	—
174	C₂×C₈₃	#546	1	1	yes	characteristic
175	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
176	C₂×C₈₃	#578	1	1	yes	characteristic
177	D₈₃	—	8	1	no	—
178	#3300	—	83	1	no	maximal, Sylow
179	C₈₃	#3300	1	1	yes	characteristic, complemented, Sylow
180	#552	—	83	1	no	—
181	#548	—	83	1	no	—
182	#488	—	83	1	no	—
183	#485	—	83	1	no	—
184	#480	—	83	1	no	—
185	#479	—	83	1	no	—
186	#467	—	83	1	no	—
187	#465	—	83	1	no	—
188	#464	—	83	1	no	—
189	#452	—	83	1	no	—
190	#451	—	83	1	no	—
191	C₄×Q₁₆	—	83	1	no	—
192	C₈×D₄	D₈₃	1	1	yes	characteristic, complemented
193	#416	—	83	1	no	—
194	C₂×#103	—	83	1	no	—
195	C₂×#41	—	83	1	no	—
196	C₂²×Q₈	—	83	1	no	—
197	C₂×Q₁₆	—	83	1	no	—
198	C₂×Q₁₆	—	83	1	no	—
199	C₂×SD₁₆	—	83	1	no	—
200	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
201	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
202	#126	—	166	1	no	—
203	#125	—	83	1	no	—
204	#125	—	166	1	no	—
205	#124	—	83	1	no	—
206	#124	—	166	1	no	—
207	#123	—	166	1	no	—
208	C₂²⋊Q₈	—	83	1	no	—
209	C₂²⋊Q₈	—	83	1	no	—
210	C₂²⋊Q₈	—	83	1	no	—
211	C₂²⋊Q₈	—	166	1	no	—
212	#121	—	83	1	no	—
213	#121	—	166	1	no	—
214	C₄×Q₈	—	83	1	no	—
215	C₄×Q₈	—	83	1	no	—
216	C₄×D₄	—	166	1	no	—
217	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
218	#117	—	83	1	no	—
219	#117	—	166	1	no	—
220	C₂×C₄⋊C₄	—	83	1	no	—
221	#107	—	83	1	no	—
222	#107	—	83	1	no	—
223	#106	—	83	1	no	—
224	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	—	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	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
236	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
237	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
238	#41	—	332	1	no	—
239	#41	—	166	2	no	—
240	C₂×Q₈	—	332	1	no	—
241	C₂×Q₈	—	83	1	no	—
242	C₂×Q₈	—	83	1	no	—
243	C₂×Q₈	—	83	1	no	—
244	C₂×Q₈	—	166	2	no	—
245	C₂×Q₈	—	166	1	no	—
246	C₂×D₄	—	83	1	no	—
247	C₂×D₄	—	166	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂×D₄	D₃₃₂	1	1	yes	characteristic
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	—	166	1	no	—
252	C₂²×C₄	—	166	1	no	—
253	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
255	Q₁₆	—	332	1	no	—
256	Q₁₆	—	166	2	no	—
257	SD₁₆	—	332	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	—	166	1	no	—
274	C₄⋊C₄	—	166	1	no	—
275	C₄⋊C₄	—	166	1	no	—
276	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
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	C₂²⋊C₄	—	166	1	no	—
284	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
285	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
286	C₄²	—	166	1	no	—
287	C₄²	—	166	1	no	—
288	C₄²	—	166	1	no	—
289	C₄²	C₂²×D₈₃	1	1	yes	characteristic
290	C₂³	—	166	1	no	—
291	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
292	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
293	Q₈	—	332	2	no	—
294	Q₈	—	83	2	no	—
295	Q₈	—	166	1	no	—
296	Q₈	—	166	1	no	—
297	Q₈	—	166	1	no	—
298	Q₈	—	166	1	no	—
299	Q₈	—	166	1	no	—
300	D₄	—	2	2	no	—
301	D₄	—	332	2	no	—
302	D₄	—	332	2	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	—	332	2	no	—
310	C₂×C₄	—	332	2	no	—
311	C₂×C₄	—	332	2	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	—	166	1	no	—
315	C₂×C₄	—	166	1	no	—
316	C₂×C₄	—	166	1	no	—
317	C₂×C₄	—	166	1	no	—
318	C₂×C₄	—	166	1	no	—
319	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
320	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
321	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
322	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	—	332	1	no	—
336	C₂²	#14581814	1	2	yes	—
337	C₂²	C₂²×D₃₃₂	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	—	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	—	332	1	no	—
348	C₄	—	166	2	no	—
349	C₄	#41×D₈₃	1	1	yes	characteristic
350	C₄	#14581837	1	1	yes	characteristic
351	C₂	—	2	2	no	—
352	C₂	—	4	1	no	—
353	C₂	—	664	1	no	—
354	C₂	#19128346	1	1	yes	characteristic
355	C₂	C₂×#14581814	1	1	yes	characteristic
356	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	C₈×D₄×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128360	C₂
hypercenter	C₈×D₄	D₈₃
hypocenter	C₈₃	#3300
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #19128391	C₂	semidirect
3. #19128056	C₂	semidirect
4. #19128043	C₂	semidirect
5. #19128015	C₂	semidirect
6. #19128009	C₂	semidirect
7. #19127942	C₂	semidirect
8. #19127930	C₂	semidirect
9. #19127924	C₂	semidirect
10. #19127865	C₂	semidirect
11. #19127859	C₂	semidirect
12. C₄×Dic₃₃₂	C₂	semidirect
13. #19127698	C₂	semidirect
14. C₂×#14581717	C₂	semidirect
15. C₈×D₄×C₈₃	C₂	semidirect
179. C₈₃	#3300	semidirect
192. C₈×D₄	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128391	2
#19128360	2
#19128056	2
#19128043	2
#19128015	2
#19128009	2
#19127942	2
#19127930	2
#19127924	2
#19127865	2
#19127859	2
C₄×Dic₃₃₂	2
#19127698	2
C₂×#14581717	2
C₈×D₄×C₈₃	2
#3300	83