Subgroups — #24922236

Order 10624.

1424 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128537	C₂	1	2	yes	maximal, complemented
2	#19128517	C₂	1	4	yes	maximal, complemented
3	#19128511	C₂	1	4	yes	maximal, complemented
4	#19128510	C₂	1	4	yes	maximal, complemented
5	#19128505	C₂	1	4	yes	maximal, complemented
6	#19128501	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128498	C₂	1	4	yes	maximal, complemented
8	C₂×#14581812	C₂	1	2	yes	maximal, complemented
9	#135×D₈₃	C₂	1	4	yes	maximal, complemented
10	C₂×#14581800	C₂	1	1	yes	maximal, characteristic, complemented
11	#575×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14581840	C₂²	1	4	yes	complemented
13	#14581839	—	2	2	no	—
14	#14581839	C₂²	1	4	yes	complemented
15	#14581838	—	2	2	no	—
16	#14581838	C₂²	1	4	yes	complemented
17	#14581837	—	2	4	no	—
18	#14581837	C₂²	1	4	yes	complemented
19	C₂×#13050590	—	2	4	no	—
20	C₂×#13050590	C₂²	1	1	yes	characteristic
21	C₂×#13050590	C₂²	1	2	yes	complemented
22	#41×D₈₃	—	2	4	no	—
23	#41×D₈₃	C₂²	1	8	yes	complemented
24	#14581825	C₂²	1	4	yes	complemented
25	#14581824	C₂²	1	8	yes	complemented
26	#14581821	C₂²	1	8	yes	complemented
27	#14581820	C₂²	1	4	yes	complemented
28	#14581819	C₂²	1	8	yes	complemented
29	#14581816	C₂²	1	8	yes	complemented
30	#14581815	C₂²	1	4	yes	complemented
31	#14581814	C₂²	1	4	yes	complemented
32	#14581813	C₂²	1	8	yes	complemented
33	#14581812	C₂²	1	4	yes	complemented
34	#14581812	C₂²	1	4	yes	complemented
35	#14581812	C₂²	1	4	yes	complemented
36	#14581812	C₂²	1	4	yes	complemented
37	#14581811	C₂²	1	4	yes	complemented
38	C₂×#13050580	C₂²	1	2	yes	complemented
39	Q₁₆×D₈₃	C₂²	1	4	yes	complemented
40	C₂×#13050578	C₂²	1	2	yes	complemented
41	C₂×#13050577	C₂²	1	2	yes	complemented
42	SD₁₆×D₈₃	C₂²	1	8	yes	complemented
43	C₂×#13050575	C₂²	1	2	yes	complemented
44	D₈×D₈₃	C₂²	1	4	yes	complemented
45	#14581800	C₂²	1	4	yes	complemented
46	#14581800	C₂²	1	4	yes	complemented
47	C₂×#13050573	C₂²	1	2	yes	complemented
48	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₈×D₈₃	C₂²	1	2	yes	complemented
50	C₂×#41×C₈₃	C₂²	1	2	yes	complemented
51	#137×C₈₃	C₂²	1	4	yes	complemented
52	AGL(1,ℤ/8)×C₈₃	C₂²	1	4	yes	complemented
53	#135×C₈₃	C₂²	1	4	yes	complemented
54	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic, complemented
55	#13050591	—	2	4	no	—
56	#13050591	C₂³	1	8	yes	—
57	#13050590	—	2	2	no	—
58	#13050590	—	2	2	no	—
59	#13050590	—	2	8	no	—
60	#13050590	—	2	4	no	—
61	#13050590	—	2	8	no	—
62	#13050590	—	2	4	no	—
63	#13050590	D₄	1	4	yes	—
64	#13050590	C₂³	1	4	yes	—
65	#13050590	C₂³	1	4	yes	—
66	#13050590	C₂³	1	4	yes	—
67	#13050589	—	2	8	no	—
68	#13050589	—	2	8	no	—
69	#13050589	—	2	4	no	—
70	#13050589	C₂³	1	8	yes	—
71	C₂×Dic₁₆₆	—	2	4	no	—
72	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
73	C₂×Dic₁₆₆	C₂³	1	2	yes	—
74	Q₈×D₈₃	—	2	4	no	—
75	Q₈×D₈₃	C₂³	1	8	yes	—
76	C₂×D₃₃₂	—	2	4	no	—
77	C₂×D₃₃₂	D₄	1	1	yes	characteristic
78	C₂×D₃₃₂	C₂³	1	2	yes	—
79	C₂×C₂²⋊D₈₃	—	2	2	no	—
80	C₂×C₂²⋊D₈₃	—	2	8	no	—
81	C₂×C₂²⋊D₈₃	D₄	1	2	yes	—
82	D₄×D₈₃	—	2	8	no	—
83	D₄×D₈₃	—	2	8	no	—
84	D₄×D₈₃	—	2	4	no	—
85	D₄×D₈₃	C₂³	1	8	yes	—
86	C₂×C₄×D₈₃	—	2	2	no	—
87	C₂×C₄×D₈₃	—	2	8	no	—
88	C₂×C₄×D₈₃	C₂³	1	2	yes	—
89	Dic₃₃₂	C₂³	1	4	yes	—
90	#13050580	C₂³	1	4	yes	—
91	#13050580	C₂³	1	4	yes	—
92	#13050579	C₂³	1	8	yes	—
93	#13050578	C₂³	1	4	yes	—
94	#13050578	C₂³	1	4	yes	—
95	#13050577	C₂³	1	4	yes	—
96	#13050577	C₂³	1	4	yes	—
97	D₆₆₄	C₂³	1	4	yes	—
98	#13050575	C₂³	1	4	yes	—
99	#13050575	C₂³	1	4	yes	—
100	#13050573	C₂³	1	8	yes	—
101	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
104	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
105	C₈×D₈₃	C₂³	1	4	yes	—
106	C₈×D₈₃	C₂³	1	4	yes	—
107	#41×C₈₃	—	2	2	no	—
108	#41×C₈₃	—	2	2	no	—
109	#41×C₈₃	C₂³	1	4	yes	—
110	#41×C₈₃	C₂³	1	4	yes	—
111	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
112	C₂×D₄×C₈₃	—	2	2	no	—
113	C₂×D₄×C₈₃	C₂³	1	2	yes	—
114	C₂²×C₄×C₈₃	—	2	2	no	—
115	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
116	Q₁₆×C₈₃	C₂³	1	4	yes	—
117	SD₁₆×C₈₃	C₂³	1	8	yes	—
118	D₈×C₈₃	C₂³	1	4	yes	—
119	M₁₆×C₈₃	C₂³	1	4	yes	—
120	C₂×C₈×C₈₃	C₂³	1	2	yes	—
121	C₂³×C₈₃	—	2	2	no	—
122	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×D₈₃	—	2	2	no	—
124	C₂²×D₈₃	—	2	8	no	—
125	C₂²×D₈₃	C₂×D₄	1	2	yes	—
126	Q₈×C₈₃	—	2	2	no	—
127	Q₈×C₈₃	C₂⁴	1	4	yes	—
128	D₄×C₈₃	—	2	4	no	—
129	D₄×C₈₃	—	2	4	no	—
130	D₄×C₈₃	—	2	2	no	—
131	D₄×C₈₃	C₂⁴	1	4	yes	—
132	C₂×C₄×C₈₃	—	2	2	no	—
133	C₂×C₄×C₈₃	—	2	4	no	—
134	C₂×C₄×C₈₃	—	2	4	no	—
135	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
136	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
139	C₂²⋊D₈₃	—	2	4	no	—
140	C₂²⋊D₈₃	—	2	4	no	—
141	C₂²⋊D₈₃	—	2	8	no	—
142	C₂²⋊D₈₃	—	2	8	no	—
143	C₂²⋊D₈₃	C₂×D₄	1	8	yes	—
144	C₂×Dic₈₃	—	2	2	no	—
145	C₂×Dic₈₃	—	2	8	no	—
146	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
147	D₃₃₂	—	2	8	no	—
148	D₃₃₂	—	2	2	no	—
149	D₃₃₂	C₂×D₄	1	4	yes	—
150	D₃₃₂	C₂⁴	1	4	yes	—
151	C₄×D₈₃	—	2	8	no	—
152	C₄×D₈₃	—	2	4	no	—
153	C₄×D₈₃	—	2	4	no	—
154	C₄×D₈₃	—	2	8	no	—
155	C₄×D₈₃	C₂×D₄	1	4	yes	—
156	C₄×D₈₃	C₂⁴	1	4	yes	—
157	Dic₁₆₆	—	2	2	no	—
158	Dic₁₆₆	—	2	8	no	—
159	Dic₁₆₆	C₂×D₄	1	4	yes	—
160	Dic₁₆₆	C₂⁴	1	4	yes	—
161	C₈×C₈₃	C₂⁴	1	4	yes	—
162	C₈₃⋊C₈	C₂⁴	1	1	yes	characteristic
163	C₈₃⋊C₈	C₂⁴	1	1	yes	characteristic
164	C₈₃⋊C₈	C₂⁴	1	2	yes	—
165	C₂²×C₈₃	—	2	4	no	—
166	C₂²×C₈₃	—	4	1	no	—
167	C₂²×C₈₃	—	8	2	no	—
168	C₂²×C₈₃	C₂²×D₄	1	2	yes	—
169	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₈₃	—	2	4	no	—
171	C₂×D₈₃	—	4	2	no	—
172	C₂×D₈₃	—	8	8	no	—
173	C₂×D₈₃	—	8	2	no	—
174	C₂×D₈₃	C₂²×D₄	1	4	yes	—
175	C₄×C₈₃	—	2	4	no	—
176	C₄×C₈₃	C₂²×D₄	1	2	yes	—
177	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
178	C₄×C₈₃	C₂⁵	1	1	yes	characteristic
179	Dic₈₃	—	2	4	no	—
180	Dic₈₃	C₂²×D₄	1	4	yes	—
181	C₂×C₈₃	—	2	1	no	—
182	C₂×C₈₃	—	2	2	no	—
183	C₂×C₈₃	—	4	4	no	—
184	C₂×C₈₃	C₂³×D₄	1	1	yes	characteristic
185	D₈₃	—	2	4	no	—
186	D₈₃	—	4	4	no	—
187	#3584	—	83	1	no	maximal, Sylow
188	C₈₃	#3584	1	1	yes	characteristic, complemented, Sylow
189	#585	—	83	2	no	—
190	#578	—	83	4	no	—
191	#577	—	83	4	no	—
192	#577	—	83	4	no	—
193	#576	—	83	4	no	—
194	#575	—	83	4	no	—
195	#575	—	83	1	no	—
196	#575	D₈₃	1	1	yes	characteristic, complemented
197	C₂×#135	—	83	2	no	—
198	C₂×#135	—	83	4	no	—
199	C₂×#131	—	83	1	no	—
200	2₋¹⁺⁴	—	83	4	no	—
201	2₋¹⁺⁴	—	83	4	no	—
202	2₋¹⁺⁴	—	166	2	no	—
203	H(5,𝔽₂)	—	83	4	no	—
204	H(5,𝔽₂)	—	83	4	no	—
205	H(5,𝔽₂)	—	166	4	no	—
206	H(5,𝔽₂)	—	166	2	no	—
207	C₂×#41	—	83	1	no	—
208	C₂×#41	—	83	8	no	—
209	C₂×#41	—	83	2	no	—
210	C₂×#41	—	166	4	no	—
211	C₂×#41	—	166	4	no	—
212	C₂×#41	C₂×D₈₃	1	2	yes	complemented
213	#137	—	83	8	no	—
214	#137	—	83	8	no	—
215	#137	—	83	4	no	—
216	#137	C₂×D₈₃	1	4	yes	complemented
217	AGL(1,ℤ/8)	—	83	8	no	—
218	AGL(1,ℤ/8)	—	83	8	no	—
219	AGL(1,ℤ/8)	—	83	4	no	—
220	AGL(1,ℤ/8)	C₂×D₈₃	1	4	yes	complemented
221	#135	—	83	4	no	—
222	#135	—	83	4	no	—
223	#135	—	83	4	no	—
224	#135	—	83	4	no	—
225	#135	—	83	4	no	—
226	#135	—	83	8	no	—
227	#135	—	83	4	no	—
228	#135	—	83	4	no	—
229	#135	C₂×D₈₃	1	4	yes	complemented
230	C₂×Q₁₆	—	83	2	no	—
231	C₂×Q₁₆	—	83	4	no	—
232	C₂×SD₁₆	—	83	2	no	—
233	C₂×SD₁₆	—	83	2	no	—
234	C₂×SD₁₆	—	83	8	no	—
235	C₂×D₈	—	83	2	no	—
236	C₂×D₈	—	83	4	no	—
237	#131	—	83	4	no	—
238	#131	—	83	4	no	—
239	C₂×M₁₆	—	83	2	no	—
240	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic, complemented
241	C₂²×C₈	—	83	1	no	—
242	C₂²×C₈	—	83	2	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	83	4	no	—
246	#41	—	83	4	no	—
247	#41	—	83	8	no	—
248	#41	—	83	8	no	—
249	#41	—	83	4	no	—
250	#41	—	83	4	no	—
251	#41	—	166	8	no	—
252	#41	—	166	8	no	—
253	#41	—	166	4	no	—
254	#41	—	166	4	no	—
255	#41	—	166	8	no	—
256	#41	—	166	8	no	—
257	#41	—	166	4	no	—
258	#41	—	166	4	no	—
259	#41	—	166	2	no	—
260	#41	—	166	2	no	—
261	#41	C₂²×D₈₃	1	4	yes	—
262	#41	C₂²×D₈₃	1	4	yes	—
263	C₂×Q₈	—	83	1	no	—
264	C₂×Q₈	—	83	8	no	—
265	C₂×Q₈	—	83	2	no	—
266	C₂×Q₈	—	166	4	no	—
267	C₂×Q₈	—	166	4	no	—
268	C₂×Q₈	C₂²×D₈₃	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	83	2	no	—
271	C₂×D₄	—	83	8	no	—
272	C₂×D₄	—	83	1	no	—
273	C₂×D₄	—	83	2	no	—
274	C₂×D₄	—	166	4	no	—
275	C₂×D₄	—	166	8	no	—
276	C₂×D₄	—	166	4	no	—
277	C₂×D₄	—	166	8	no	—
278	C₂×D₄	—	166	8	no	—
279	C₂×D₄	—	166	2	no	—
280	C₂×D₄	C₂²×D₈₃	1	2	yes	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	83	2	no	—
283	C₂²×C₄	—	166	8	no	—
284	C₂²×C₄	—	166	2	no	—
285	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
286	Q₁₆	—	83	4	no	—
287	Q₁₆	—	83	4	no	—
288	Q₁₆	—	83	4	no	—
289	Q₁₆	C₂²×D₈₃	1	4	yes	—
290	SD₁₆	—	83	4	no	—
291	SD₁₆	—	83	4	no	—
292	SD₁₆	—	83	4	no	—
293	SD₁₆	—	83	4	no	—
294	SD₁₆	—	83	8	no	—
295	SD₁₆	C₂²×D₈₃	1	8	yes	—
296	D₈	—	83	4	no	—
297	D₈	—	83	4	no	—
298	D₈	—	83	4	no	—
299	D₈	C₂²×D₈₃	1	4	yes	—
300	M₁₆	—	83	8	no	—
301	M₁₆	C₂²×D₈₃	1	4	yes	—
302	C₂×C₈	—	83	1	no	—
303	C₂×C₈	—	83	1	no	—
304	C₂×C₈	—	83	2	no	—
305	C₂×C₈	—	83	2	no	—
306	C₂×C₈	—	83	4	no	—
307	C₂×C₈	—	83	4	no	—
308	C₂×C₈	C₂²×D₈₃	1	2	yes	—
309	C₂³	—	2	2	no	—
310	C₂³	—	83	2	no	—
311	C₂³	—	166	8	no	—
312	C₂³	—	166	2	no	—
313	C₂³	D₄×D₈₃	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	83	4	no	—
316	Q₈	—	83	4	no	—
317	Q₈	—	166	8	no	—
318	Q₈	—	166	2	no	—
319	Q₈	C₂³×D₈₃	1	4	yes	—
320	D₄	—	2	4	no	—
321	D₄	—	2	4	no	—
322	D₄	—	2	2	no	—
323	D₄	—	83	8	no	—
324	D₄	—	83	4	no	—
325	D₄	—	83	4	no	—
326	D₄	—	166	8	no	—
327	D₄	—	166	8	no	—
328	D₄	—	166	8	no	—
329	D₄	—	166	4	no	—
330	D₄	—	166	4	no	—
331	D₄	—	166	2	no	—
332	D₄	C₂³×D₈₃	1	4	yes	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	83	2	no	—
337	C₂×C₄	—	83	4	no	—
338	C₂×C₄	—	83	4	no	—
339	C₂×C₄	—	166	8	no	—
340	C₂×C₄	—	166	8	no	—
341	C₂×C₄	—	166	8	no	—
342	C₂×C₄	—	166	2	no	—
343	C₂×C₄	—	166	4	no	—
344	C₂×C₄	—	166	4	no	—
345	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
346	C₂×C₄	D₄×D₈₃	1	2	yes	—
347	C₂×C₄	C₂³×D₈₃	1	2	yes	—
348	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
349	C₈	—	83	1	no	—
350	C₈	—	83	1	no	—
351	C₈	—	83	2	no	—
352	C₈	C₂³×D₈₃	1	4	yes	—
353	C₂²	—	2	4	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	332	2	no	—
357	C₂²	—	83	4	no	—
358	C₂²	—	664	2	no	—
359	C₂²	—	664	8	no	—
360	C₂²	—	166	4	no	—
361	C₂²	C₂×D₄×D₈₃	1	2	yes	—
362	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
363	C₄	—	2	4	no	—
364	C₄	—	83	4	no	—
365	C₄	—	166	4	no	—
366	C₄	C₂×D₄×D₈₃	1	2	yes	—
367	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
368	C₄	C₂⁴×D₈₃	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	2	1	no	—
371	C₂	—	4	4	no	—
372	C₂	—	332	4	no	—
373	C₂	—	166	4	no	—
374	C₂	C₂²×D₄×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₈₃
commutator, cosocle	C₄×C₈₃	C₂⁵
Frattini	C₄	C₂⁴×D₈₃
Fitting	#575×C₈₃	C₂
socle	C₂×C₈₃	C₂³×D₄
hypercenter	#575	D₈₃
hypocenter	C₈₃	#3584
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

Complements

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

N	H ≅ G/N	type
1. #19128537	C₂	semidirect
2. #19128517	C₂	semidirect
3. #19128511	C₂	semidirect
4. #19128510	C₂	semidirect
5. #19128505	C₂	semidirect
6. #19128501	C₂	semidirect
7. #19128498	C₂	semidirect
8. C₂×#14581812	C₂	semidirect
9. #135×D₈₃	C₂	semidirect
10. C₂×#14581800	C₂	semidirect
11. #575×C₈₃	C₂	semidirect
12. #14581840	C₂²	semidirect
14. #14581839	C₂²	semidirect
16. #14581838	C₂²	semidirect
18. #14581837	C₂²	semidirect
21. C₂×#13050590	C₂²	semidirect
23. #41×D₈₃	C₂²	semidirect
24. #14581825	C₂²	semidirect
25. #14581824	C₂²	semidirect
26. #14581821	C₂²	semidirect
27. #14581820	C₂²	semidirect
28. #14581819	C₂²	semidirect
29. #14581816	C₂²	semidirect
30. #14581815	C₂²	semidirect
31. #14581814	C₂²	semidirect
32. #14581813	C₂²	semidirect
33. #14581812	C₂²	semidirect
34. #14581812	C₂²	semidirect
35. #14581812	C₂²	semidirect
36. #14581812	C₂²	semidirect
37. #14581811	C₂²	semidirect
38. C₂×#13050580	C₂²	semidirect
39. Q₁₆×D₈₃	C₂²	semidirect
40. C₂×#13050578	C₂²	semidirect
41. C₂×#13050577	C₂²	semidirect
42. SD₁₆×D₈₃	C₂²	semidirect
43. C₂×#13050575	C₂²	semidirect
44. D₈×D₈₃	C₂²	semidirect
45. #14581800	C₂²	semidirect
46. #14581800	C₂²	semidirect
47. C₂×#13050573	C₂²	semidirect
48. C₂²×C₈₃⋊C₈	C₂²	semidirect
49. C₂×C₈×D₈₃	C₂²	semidirect
50. C₂×#41×C₈₃	C₂²	semidirect
51. #137×C₈₃	C₂²	semidirect
52. AGL(1,ℤ/8)×C₈₃	C₂²	semidirect
53. #135×C₈₃	C₂²	semidirect
54. C₂×M₁₆×C₈₃	C₂²	semidirect
188. C₈₃	#3584	semidirect
196. #575	D₈₃	semidirect
212. C₂×#41	C₂×D₈₃	semidirect
216. #137	C₂×D₈₃	semidirect
220. AGL(1,ℤ/8)	C₂×D₈₃	semidirect
229. #135	C₂×D₈₃	semidirect
240. C₂×M₁₆	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128537	2
#19128517	2
#19128511	2
#19128510	2
#19128505	2
#19128501	2
#19128498	2
C₂×#14581812	2
#135×D₈₃	2
C₂×#14581800	2
#575×C₈₃	2
#3584	83