Subgroups — #35504359

Order 21248.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917977	C₂	1	1	yes	maximal, characteristic, complemented
2	#24908635	C₂	1	4	yes	maximal, complemented
3	#24908362	C₂	1	2	yes	maximal
4	#24907588	C₂	1	2	yes	maximal, complemented
5	#24907517	C₂	1	2	yes	maximal, complemented
6	#24907509	C₂	1	2	yes	maximal, complemented
7	#24907342	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#443×C₈₃	C₂	1	1	yes	maximal, characteristic
9	#19128258	C₂²	1	2	yes	—
10	#19128106	—	2	1	no	—
11	#19128103	—	2	1	no	—
12	#19128103	—	2	1	no	—
13	#19128099	—	2	1	no	—
14	C₂×#14581728	C₂²	1	1	yes	characteristic
15	#19127739	C₂²	1	2	yes	—
16	#19127739	C₂²	1	1	yes	characteristic
17	#19127739	C₂²	1	2	yes	—
18	#19127718	C₂²	1	4	yes	—
19	#19127675	C₂²	1	4	yes	—
20	#19127656	C₂²	1	2	yes	—
21	#19127473	C₂²	1	4	yes	—
22	#19127461	C₂²	1	2	yes	—
23	#19127458	C₂²	1	4	yes	—
24	C₂×#135×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×#131×C₈₃	C₂²	1	2	yes	—
26	#443×C₈₃	—	2	1	no	—
27	#443×C₈₃	—	2	1	no	—
28	#443×C₈₃	—	2	1	no	—
29	#443×C₈₃	—	2	1	no	—
30	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×#104×C₈₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
33	C₂×#13050574	C₂³	1	2	yes	—
34	#14581792	—	2	2	no	—
35	#14581784	—	2	2	no	—
36	#14581771	—	4	2	no	—
37	C₄×Dic₁₆₆	—	4	2	no	—
38	#14581751	C₂³	1	2	yes	—
39	#14581751	C₂³	1	4	yes	—
40	#14581728	—	2	1	no	—
41	#14581728	—	2	1	no	—
42	#14581726	—	2	2	no	—
43	#14581726	—	2	2	no	—
44	#14581726	—	2	1	no	—
45	#14581724	—	2	2	no	—
46	#14581724	—	2	2	no	—
47	#14581724	—	2	1	no	—
48	#14581719	—	4	4	no	—
49	#14581707	—	4	4	no	—
50	#14581702	—	2	2	no	—
51	C₈×Dic₈₃	—	2	2	no	—
52	C₂×#41×C₈₃	C₂³	1	2	yes	—
53	#135×C₈₃	—	2	1	no	—
54	#135×C₈₃	—	2	1	no	—
55	#135×C₈₃	—	2	1	no	—
56	#135×C₈₃	—	2	1	no	—
57	C₂×Q₁₆×C₈₃	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₈₃	Q₈	1	2	yes	complemented
59	C₂×D₈×C₈₃	Q₈	1	1	yes	characteristic, complemented
60	#131×C₈₃	—	2	2	no	—
61	#131×C₈₃	—	2	2	no	—
62	#131×C₈₃	—	2	2	no	—
63	#131×C₈₃	—	2	2	no	—
64	C₂×M₁₆×C₈₃	—	2	2	no	—
65	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₈₃	—	2	2	no	—
67	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
70	#108×C₈₃	—	2	1	no	—
71	#108×C₈₃	—	2	1	no	—
72	#104×C₈₃	—	2	2	no	—
73	#104×C₈₃	—	2	2	no	—
74	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
78	C₂×Dic₁₆₆	—	4	2	no	—
79	#13050574	—	2	2	no	—
80	#13050574	—	2	2	no	—
81	C₂×C₈₃⋊C₈	—	2	2	no	—
82	C₄⋊Dic₈₃	—	2	4	no	—
83	C₄⋊Dic₈₃	—	2	2	no	—
84	C₄⋊Dic₈₃	—	4	2	no	—
85	Dic₈₃⋊C₄	—	4	4	no	—
86	C₂²⋊Dic₈₃	—	2	4	no	—
87	C₂²⋊Dic₈₃	—	2	2	no	—
88	C₄×Dic₈₃	—	2	2	no	—
89	C₄×Dic₈₃	—	2	4	no	—
90	#41×C₈₃	—	2	2	no	—
91	#41×C₈₃	—	2	2	no	—
92	#41×C₈₃	—	2	2	no	—
93	#41×C₈₃	—	2	2	no	—
94	C₂×Q₈×C₈₃	C₂×Q₈	1	2	yes	—
95	C₂×D₄×C₈₃	—	2	2	no	—
96	C₂×D₄×C₈₃	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	C₂²×C₄×C₈₃	—	2	2	no	—
99	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
100	Q₁₆×C₈₃	—	2	1	no	—
101	Q₁₆×C₈₃	—	2	1	no	—
102	SD₁₆×C₈₃	—	2	2	no	—
103	SD₁₆×C₈₃	—	2	2	no	—
104	D₈×C₈₃	—	2	1	no	—
105	D₈×C₈₃	—	2	1	no	—
106	M₁₆×C₈₃	—	2	2	no	—
107	M₁₆×C₈₃	—	2	2	no	—
108	M₁₆×C₈₃	—	4	2	no	—
109	M₁₆×C₈₃	—	4	2	no	—
110	C₂×C₈×C₈₃	—	2	2	no	—
111	C₂×C₈×C₈₃	—	4	2	no	—
112	C₂×C₈×C₈₃	—	4	2	no	—
113	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
120	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
121	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
122	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
123	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
124	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
125	C₄²×C₈₃	—	2	1	no	—
126	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₈₃	—	2	2	no	—
129	C₂³×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₈₃	—	2	2	no	—
131	Q₈×C₈₃	—	2	2	no	—
132	D₄×C₈₃	—	2	2	no	—
133	D₄×C₈₃	—	2	2	no	—
134	D₄×C₈₃	—	4	2	no	—
135	D₄×C₈₃	—	4	2	no	—
136	C₂×C₄×C₈₃	—	2	1	no	—
137	C₂×C₄×C₈₃	—	2	1	no	—
138	C₂×C₄×C₈₃	—	2	2	no	—
139	C₂×C₄×C₈₃	—	4	2	no	—
140	C₂×C₄×C₈₃	—	4	2	no	—
141	C₂×C₄×C₈₃	—	4	2	no	—
142	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₈₃	—	4	4	no	—
149	C₂×Dic₈₃	—	4	2	no	—
150	Dic₁₆₆	—	8	4	no	—
151	C₈×C₈₃	—	4	2	no	—
152	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₈₃	#124	1	1	yes	characteristic
157	C₈×C₈₃	#124	1	1	yes	characteristic
158	C₈×C₈₃	#124	1	1	yes	characteristic
159	C₈×C₈₃	#124	1	1	yes	characteristic
160	C₈₃⋊C₈	—	4	2	no	—
161	C₂²×C₈₃	—	4	1	no	—
162	C₂²×C₈₃	—	4	2	no	—
163	C₂²×C₈₃	—	8	2	no	—
164	C₂²×C₈₃	#535	1	1	yes	characteristic
165	C₂²×C₈₃	#554	1	1	yes	characteristic
166	C₂²×C₈₃	#555	1	1	yes	characteristic
167	C₄×C₈₃	—	2	2	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	—	4	2	no	—
170	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
172	C₄×C₈₃	#554	1	1	yes	characteristic
173	C₄×C₈₃	#555	1	1	yes	characteristic
174	Dic₈₃	—	8	4	no	—
175	Dic₈₃	—	8	2	no	—
176	#29154	—	83	1	no	maximal, Sylow
177	C₂×C₈₃	—	2	1	no	—
178	C₂×C₈₃	—	2	1	no	—
179	C₂×C₈₃	—	2	1	no	—
180	C₂×C₈₃	—	8	2	no	—
181	C₂×C₈₃	#2563	1	1	yes	characteristic
182	#3160	—	83	1	no	—
183	C₂×#443	D₈₃	1	1	yes	characteristic
184	#2031	—	83	4	no	—
185	#1976	—	83	2	no	—
186	#1833	—	83	2	no	—
187	#1814	—	83	2	no	—
188	#1812	—	83	2	no	—
189	#1772	—	83	1	no	—
190	C₈₃	#29154	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	C₂×D₈₃	1	1	yes	characteristic
192	C₂×#131	C₂×D₈₃	1	2	yes	—
193	#533	—	83	2	no	—
194	#500	—	166	1	no	—
195	#499	—	166	1	no	—
196	#499	—	166	1	no	—
197	#498	—	166	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	#443	—	2	1	no	—
202	C₂×#108	—	83	1	no	—
203	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
204	#427	—	83	1	no	—
205	#427	—	83	2	no	—
206	#427	—	83	2	no	—
207	#421	—	83	4	no	—
208	C₂×#104	C₂×D₈₃	1	2	yes	—
209	#410	—	83	4	no	—
210	#405	—	83	2	no	—
211	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
212	#344	—	83	4	no	—
213	#339	—	83	2	no	—
214	H(3,ℤ/4)	—	83	4	no	—
215	C₂×#41	C₂²×D₈₃	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Dic₁₆₆	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Dic₁₆₆	1	2	yes	complemented
222	C₂×D₈	Dic₁₆₆	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	83	2	no	—
229	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
233	C₄⋊Q₈	—	166	2	no	—
234	#125	—	166	2	no	—
235	C₂²⋊Q₈	—	332	2	no	—
236	C₄×Q₈	—	332	2	no	—
237	#117	—	83	4	no	—
238	#117	—	83	2	no	—
239	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	166	1	no	—
243	#108	—	166	1	no	—
244	#107	—	166	2	no	—
245	#107	—	166	2	no	—
246	#107	—	166	1	no	—
247	#106	—	166	2	no	—
248	#106	—	166	2	no	—
249	#106	—	166	1	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	332	4	no	—
253	#99	—	332	4	no	—
254	#97	—	166	2	no	—
255	C₄×C₈	—	166	2	no	—
256	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
257	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	332	2	no	—
265	C₂×Q₈	C₂×Dic₁₆₆	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Dic₁₆₆	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	166	2	no	—
282	M₁₆	—	166	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	166	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
291	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
292	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
293	C₂×C₈	#13050589	1	1	yes	characteristic
294	C₂×C₈	#13050589	1	1	yes	characteristic
295	C₂×C₈	#13050589	1	1	yes	characteristic
296	C₂×C₈	#13050589	1	1	yes	characteristic
297	C₂×C₈	#13050590	1	1	yes	characteristic
298	C₂×C₈	#13050590	1	1	yes	characteristic
299	C₄⋊C₄	—	332	4	no	—
300	C₄⋊C₄	—	332	2	no	—
301	C₄⋊C₄	—	166	4	no	—
302	C₄⋊C₄	—	166	2	no	—
303	C₂²⋊C₄	—	166	2	no	—
304	C₂²⋊C₄	—	166	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	166	2	no	—
307	C₄²	—	166	4	no	—
308	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	#14581838	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	664	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	332	2	no	—
326	C₂×C₄	—	332	4	no	—
327	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
329	C₂×C₄	C₂×#13050589	1	1	yes	characteristic
330	C₂×C₄	C₂×#13050589	1	1	yes	characteristic
331	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
332	C₂×C₄	#14581838	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	332	2	no	—
335	C₈	#14581771	1	1	yes	characteristic
336	C₈	#14581771	1	1	yes	characteristic
337	C₈	#14581771	1	1	yes	characteristic
338	C₈	#14581771	1	1	yes	characteristic
339	C₈	#14581779	1	1	yes	characteristic
340	C₈	#14581779	1	1	yes	characteristic
341	C₈	#14581779	1	1	yes	characteristic
342	C₈	#14581779	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#19128265	1	1	yes	characteristic
347	C₂²	#19128408	1	1	yes	characteristic
348	C₂²	#19128416	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	664	4	no	—
353	C₄	—	664	2	no	—
354	C₄	C₂×#14581771	1	1	yes	characteristic
355	C₄	C₂×#14581779	1	1	yes	characteristic
356	C₄	#19128408	1	1	yes	characteristic
357	C₄	#19128416	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	#24912636	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19128416
commutator, cosocle	C₂²×C₄×C₈₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₂×#443×C₈₃	C₂
socle	C₂×C₈₃	#2563
prime radical	C₂×#135×C₈₃	C₂²
hypercenter	C₂×#443	D₈₃
hypocenter	C₈₃	#29154
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #24917977	C₂	semidirect
2. #24908635	C₂	semidirect
4. #24907588	C₂	semidirect
5. #24907517	C₂	semidirect
6. #24907509	C₂	semidirect
7. #24907342	C₂	semidirect
57. C₂×Q₁₆×C₈₃	Q₈	semidirect
58. C₂×SD₁₆×C₈₃	Q₈	semidirect
59. C₂×D₈×C₈₃	Q₈	semidirect
190. C₈₃	#29154	semidirect
220. C₂×Q₁₆	Dic₁₆₆	semidirect
221. C₂×SD₁₆	Dic₁₆₆	semidirect
222. C₂×D₈	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24917977	2
#24908635	2
#24908362	2
#24907588	2
#24907517	2
#24907509	2
#24907342	2
C₂×#443×C₈₃	2
#29154	83