Subgroups — #34187801

Order 20224.

594 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	#23764237	C₂	1	2	yes	maximal, complemented
2	#23763425	C₂	1	1	yes	maximal, characteristic, complemented
3	#23763176	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761968	C₂	1	2	yes	maximal, complemented
5	C₂×#466×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579093	—	2	1	no	—
7	C₂×#14579076	—	2	4	no	—
8	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#14579071	—	2	1	no	—
10	#19098423	C₂²	1	2	yes	—
11	C₄×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	#19098326	—	2	1	no	—
13	#19098317	—	2	1	no	—
14	#19098297	—	2	4	no	—
15	C₂²×D₈×C₇₉	C₄	1	1	yes	characteristic, complemented
16	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#121×C₇₉	C₂²	1	2	yes	—
18	#466×C₇₉	—	2	1	no	—
19	#466×C₇₉	—	2	1	no	—
20	#466×C₇₉	—	2	1	no	—
21	#466×C₇₉	—	2	1	no	—
22	C₂×#107×C₇₉	C₄	1	1	yes	characteristic, complemented
23	C₂×#102×C₇₉	C₄	1	2	yes	complemented
24	C₂²×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
27	C₂×C₂²⋊Dic₇₉	D₄	1	2	yes	—
28	C₂×C₄×Dic₇₉	—	2	1	no	—
29	#14579093	—	4	2	no	—
30	#14579076	—	4	8	no	—
31	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
32	C₂²⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
33	#14579071	—	4	2	no	—
34	C₇₉⋊#95	—	2	1	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₂²×D₄×C₇₉	—	2	2	no	—
37	C₂²×D₄×C₇₉	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₂×D₈×C₇₉	—	2	1	no	—
40	C₂×D₈×C₇₉	—	2	1	no	—
41	C₂×D₈×C₇₉	—	4	2	no	—
42	C₂²×C₈×C₇₉	—	4	2	no	—
43	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
49	#121×C₇₉	—	2	2	no	—
50	#121×C₇₉	—	2	2	no	—
51	#121×C₇₉	—	2	2	no	—
52	#121×C₇₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
55	#107×C₇₉	—	2	1	no	—
56	#107×C₇₉	—	2	1	no	—
57	#102×C₇₉	—	2	2	no	—
58	#102×C₇₉	—	2	2	no	—
59	C₂²×Dic₇₉	—	2	2	no	—
60	C₂²×Dic₇₉	—	4	2	no	—
61	C₂×C₇₉⋊C₈	—	2	2	no	—
62	C₂×C₇₉⋊C₈	—	4	1	no	—
63	C₂×C₇₉⋊C₈	—	4	1	no	—
64	C₄⋊Dic₇₉	—	4	2	no	—
65	C₄⋊Dic₇₉	—	4	2	no	—
66	C₂²⋊Dic₇₉	—	2	4	no	—
67	C₂²⋊Dic₇₉	—	4	4	no	—
68	C₄×Dic₇₉	—	4	1	no	—
69	C₄×Dic₇₉	—	4	1	no	—
70	C₂⁴×C₇₉	—	2	2	no	—
71	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×D₄×C₇₉	—	2	2	no	—
73	C₂×D₄×C₇₉	—	2	2	no	—
74	C₂×D₄×C₇₉	—	4	2	no	—
75	C₂×D₄×C₇₉	—	4	2	no	—
76	C₂×D₄×C₇₉	—	4	2	no	—
77	C₂×D₄×C₇₉	—	4	2	no	—
78	C₂×D₄×C₇₉	—	4	4	no	—
79	C₂×D₄×C₇₉	—	4	4	no	—
80	C₂²×C₄×C₇₉	—	2	2	no	—
81	C₂²×C₄×C₇₉	—	4	1	no	—
82	C₂²×C₄×C₇₉	—	4	1	no	—
83	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
89	D₈×C₇₉	—	4	2	no	—
90	D₈×C₇₉	—	4	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	4	1	no	—
95	C₂×C₈×C₇₉	—	4	1	no	—
96	C₂×C₈×C₇₉	—	4	1	no	—
97	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₇₉	—	2	2	no	—
102	C₄⋊C₄×C₇₉	—	2	2	no	—
103	C₂²⋊C₄×C₇₉	—	4	2	no	—
104	C₂²⋊C₄×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	—	4	1	no	—
106	C₂³×C₇₉	—	4	1	no	—
107	C₂³×C₇₉	—	4	2	no	—
108	C₂³×C₇₉	—	4	4	no	—
109	C₂³×C₇₉	—	8	2	no	—
110	C₂³×C₇₉	—	8	2	no	—
111	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₂³×C₇₉	#121	1	1	yes	characteristic
113	C₂³×C₇₉	#127	1	1	yes	characteristic
114	C₂³×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
115	D₄×C₇₉	—	4	2	no	—
116	D₄×C₇₉	—	4	2	no	—
117	D₄×C₇₉	—	8	2	no	—
118	D₄×C₇₉	—	8	4	no	—
119	D₄×C₇₉	—	8	2	no	—
120	D₄×C₇₉	—	8	2	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	1	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	4	4	no	—
132	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	8	2	no	—
140	C₂×Dic₇₉	—	8	2	no	—
141	C₂×Dic₇₉	—	8	2	no	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	—	2	1	no	—
144	C₈×C₇₉	—	2	1	no	—
145	C₈×C₇₉	#99	1	2	yes	complemented
146	C₇₉⋊C₈	—	4	2	no	—
147	C₂²×C₇₉	—	2	1	no	—
148	C₂²×C₇₉	—	2	1	no	—
149	C₂²×C₇₉	—	2	1	no	—
150	C₂²×C₇₉	—	2	1	no	—
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	—	2	1	no	—
153	C₂²×C₇₉	—	4	4	no	—
154	C₂²×C₇₉	—	8	2	no	—
155	C₂²×C₇₉	—	8	2	no	—
156	C₂²×C₇₉	—	8	2	no	—
157	C₂²×C₇₉	—	16	4	no	—
158	C₂²×C₇₉	#390	1	1	yes	characteristic
159	C₂²×C₇₉	#442	1	2	yes	—
160	C₂²×C₇₉	#469	1	2	yes	—
161	C₂²×C₇₉	#496	1	2	yes	—
162	C₄×C₇₉	—	2	1	no	—
163	C₄×C₇₉	—	2	1	no	—
164	C₄×C₇₉	—	2	1	no	—
165	C₄×C₇₉	—	8	2	no	—
166	C₄×C₇₉	C₂×#99	1	1	yes	characteristic
167	C₄×C₇₉	#410	1	1	yes	characteristic
168	Dic₇₉	—	8	2	no	—
169	Dic₇₉	—	16	2	no	—
170	#9785	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	2	2	no	—
172	C₂×C₇₉	—	2	2	no	—
173	C₂×C₇₉	—	2	2	no	—
174	C₂×C₇₉	—	16	2	no	—
175	C₂×C₇₉	#1964	1	1	yes	characteristic
176	C₂×C₇₉	#1972	1	1	yes	characteristic
177	C₂×C₇₉	#1976	1	1	yes	characteristic
178	C₂×#466	D₇₉	1	1	yes	characteristic
179	#1964	—	79	2	no	—
180	#1807	—	79	1	no	—
181	#1752	—	79	1	no	—
182	#1519	—	79	2	no	—
183	C₇₉	#9785	1	1	yes	characteristic, complemented, Sylow
184	C₂²×D₈	Dic₇₉	1	1	yes	characteristic, complemented
185	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
186	C₂×#121	C₂×D₇₉	1	2	yes	—
187	#466	—	2	1	no	—
188	#466	—	2	1	no	—
189	#466	—	2	1	no	—
190	#466	—	2	1	no	—
191	C₂×#107	Dic₇₉	1	1	yes	characteristic, complemented
192	C₂×#106	—	158	1	no	—
193	C₂×#102	Dic₇₉	1	2	yes	complemented
194	C₂×#99	—	158	4	no	—
195	C₂×C₂²⋊C₈	—	79	1	no	—
196	C₂×#97	—	158	1	no	—
197	#380	—	79	2	no	—
198	C₄×C₂²⋊C₄	—	79	1	no	—
199	#340	—	158	1	no	—
200	#336	—	158	1	no	—
201	#327	—	158	4	no	—
202	C₂²×D₄	—	2	2	no	—
203	C₂²×D₄	C₂×Dic₇₉	1	2	yes	—
204	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	4	2	no	—
208	C₂²×C₈	—	4	2	no	—
209	C₂²×C₈	—	158	1	no	—
210	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	2	2	no	—
220	C₂×C₄⋊C₄	—	158	2	no	—
221	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	—	79	2	no	—
224	C₂×C₂²⋊C₄	—	158	2	no	—
225	C₂×C₄²	—	158	1	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#106	—	316	2	no	—
229	#102	—	2	2	no	—
230	#102	—	2	2	no	—
231	#99	—	316	8	no	—
232	C₂²⋊C₈	—	79	2	no	—
233	C₂²⋊C₈	—	158	1	no	—
234	#97	—	316	2	no	—
235	#95	—	158	2	no	—
236	#95	—	158	1	no	—
237	C₂⁴	—	2	2	no	—
238	C₂⁴	D₄×D₇₉	1	1	yes	characteristic
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	316	2	no	—
251	C₂²×C₄	—	158	2	no	—
252	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
256	C₂²×C₄	#11929623	1	1	yes	characteristic
257	C₂²×C₄	#11929623	1	1	yes	characteristic
258	D₈	—	4	2	no	—
259	D₈	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	316	1	no	—
267	C₂×C₈	—	316	1	no	—
268	C₂×C₈	—	158	2	no	—
269	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
272	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	316	2	no	—
276	C₄⋊C₄	—	316	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	316	4	no	—
280	C₂²⋊C₄	—	158	4	no	—
281	C₄²	—	316	1	no	—
282	C₄²	—	316	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	4	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
290	C₂³	#14579132	1	1	yes	characteristic
291	C₂³	#14579157	1	1	yes	characteristic
292	C₂³	#14579185	1	4	yes	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	316	2	no	—
311	C₂×C₄	—	316	2	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	632	2	no	—
314	C₂×C₄	—	632	2	no	—
315	C₂×C₄	—	632	2	no	—
316	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
317	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
318	C₂×C₄	#14579132	1	1	yes	characteristic
319	C₂×C₄	#14579148	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	316	2	no	—
324	C₈	#14579076	1	2	yes	complemented
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	16	4	no	—
336	C₂²	#19098466	1	1	yes	characteristic
337	C₂²	#19098675	1	2	yes	—
338	C₂²	#19098813	1	2	yes	—
339	C₂²	#19098951	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	632	2	no	—
345	C₄	—	1264	2	no	—
346	C₄	C₂×#14579076	1	1	yes	characteristic
347	C₄	#19098537	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	#23764237	1	1	yes	characteristic
353	C₂	#23764277	1	1	yes	characteristic
354	C₂	#23764299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098466
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting, prime radical	C₂×#466×C₇₉	C₂
socle	C₂²×C₇₉	#390
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#466	D₇₉
hypocenter	C₇₉	#9785
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. #23764237	C₂	semidirect
2. #23763425	C₂	semidirect
3. #23763176	C₂	semidirect
4. #23761968	C₂	semidirect
15. C₂²×D₈×C₇₉	C₄	semidirect
22. C₂×#107×C₇₉	C₄	semidirect
23. C₂×#102×C₇₉	C₄	semidirect
145. C₈×C₇₉	#99	semidirect
183. C₇₉	#9785	semidirect
184. C₂²×D₈	Dic₇₉	semidirect
191. C₂×#107	Dic₇₉	semidirect
193. C₂×#102	Dic₇₉	semidirect
324. C₈	#14579076	semidirect

Maximal subgroups

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

subgroup	index
#23764237	2
#23763425	2
#23763176	2
#23761968	2
C₂×#466×C₇₉	2
#9785	79