Subgroups — #34253969

Order 20224.

734 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	#23772366	C₂	1	1	yes	maximal, characteristic
2	#23772356	C₂	1	1	yes	maximal, characteristic
3	#23772045	C₂	1	1	yes	maximal, characteristic
4	#23761701	C₂	1	2	yes	maximal, complemented
5	#23761694	C₂	1	2	yes	maximal, complemented
6	#23761691	C₂	1	2	yes	maximal, complemented
7	#23761687	C₂	1	1	yes	maximal, characteristic, complemented
8	#23761684	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×#11929607	C₂	1	2	yes	maximal, complemented
10	C₂×#19098284	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11929624	C₂²	1	1	yes	characteristic
13	#19098693	C₄	1	2	yes	—
14	#19098687	C₄	1	2	yes	—
15	#19098642	C₄	1	2	yes	—
16	#19098634	C₄	1	4	yes	—
17	#19098628	C₄	1	2	yes	—
18	(C₂²⋊D₇₉)⋊C₈	C₄	1	4	yes	—
19	#19098625	C₄	1	4	yes	—
20	#19098616	C₂²	1	1	yes	characteristic
21	#19098614	C₂²	1	2	yes	—
22	#19098614	C₂²	1	2	yes	—
23	#19098614	C₂²	1	2	yes	—
24	#19098614	C₂²	1	2	yes	—
25	#19098591	C₄	1	2	yes	—
26	#19098531	C₄	1	2	yes	—
27	C₄×#11929607	C₂²	1	2	yes	—
28	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₇₉	C₂²	1	2	yes	—
31	#19098285	C₂²	1	2	yes	—
32	#19098285	C₂²	1	2	yes	—
33	#19098285	C₂²	1	2	yes	—
34	#19098284	C₂²	1	1	yes	characteristic
35	#19098284	C₂²	1	1	yes	characteristic
36	#19098284	C₂²	1	1	yes	characteristic
37	#19098284	C₂²	1	1	yes	characteristic
38	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
39	#431×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
42	#322×C₇₉	C₂²	1	2	yes	—
43	C₈²×C₇₉	C₂²	1	2	yes	—
44	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
45	C₂×#11929607	C₂×C₄	1	4	yes	—
46	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇₉	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
51	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
52	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
53	C₄×C₂²⋊D₇₉	C₂×C₄	1	2	yes	—
54	C₄×C₂²⋊D₇₉	C₂×C₄	1	2	yes	—
55	#14579120	C₂×C₄	1	1	yes	characteristic
56	#14579118	C₂×C₄	1	2	yes	—
57	C₄²×D₇₉	C₂³	1	2	yes	—
58	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
59	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
60	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
61	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
62	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
63	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
65	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
66	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
67	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
68	#14579071	C₂×C₄	1	4	yes	—
69	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
70	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
76	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
77	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
78	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
81	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
82	#97×C₇₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₇₉	C₂³	1	2	yes	—
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₉	C₂³	1	2	yes	—
87	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
88	#11929624	—	2	4	no	—
89	C₂×Dic₁₅₈	C₄²	1	1	yes	characteristic
90	C₂×D₃₁₆	C₄²	1	1	yes	characteristic
91	C₂×C₂²⋊D₇₉	C₄²	1	2	yes	—
92	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
95	#11929607	—	2	8	no	—
96	C₂×C₇₉⋊C₈	—	2	4	no	—
97	C₂×C₇₉⋊C₈	—	4	2	no	—
98	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
103	C₈×D₇₉	—	4	4	no	—
104	C₄⋊Dic₇₉	C₄²	1	1	yes	characteristic
105	C₄⋊Dic₇₉	C₄²	1	1	yes	characteristic
106	Dic₇₉⋊C₄	C₄²	1	2	yes	—
107	Dic₇₉⋊C₄	C₄²	1	2	yes	—
108	C₂²⋊Dic₇₉	C₄²	1	2	yes	—
109	(C₂×D₇₉)⋊C₄	C₄²	1	2	yes	—
110	(C₂×D₇₉)⋊C₄	C₄²	1	2	yes	—
111	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
112	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
113	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
117	M₁₆×C₇₉	—	2	4	no	—
118	C₂×C₈×C₇₉	—	2	4	no	—
119	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
122	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
123	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
128	C₂²×D₇₉	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#131	1	4	yes	—
135	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₇₉	—	2	4	no	—
139	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
140	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
141	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
142	D₃₁₆	—	2	2	no	—
143	C₄×D₇₉	—	2	4	no	—
144	C₄×D₇₉	—	2	4	no	—
145	C₄×D₇₉	—	2	4	no	—
146	Dic₁₅₈	—	2	2	no	—
147	C₈×C₇₉	—	2	2	no	—
148	C₈×C₇₉	—	2	4	no	—
149	C₈×C₇₉	—	2	4	no	—
150	C₈×C₇₉	#131	1	4	yes	—
151	C₇₉⋊C₈	—	2	2	no	—
152	C₇₉⋊C₈	—	2	2	no	—
153	C₇₉⋊C₈	—	2	4	no	—
154	C₇₉⋊C₈	—	2	2	no	—
155	C₇₉⋊C₈	—	2	2	no	—
156	C₂²×C₇₉	—	2	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	#405	1	2	yes	—
159	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
160	C₂×D₇₉	—	2	2	no	—
161	C₂×D₇₉	—	2	2	no	—
162	C₂×D₇₉	—	2	2	no	—
163	C₄×C₇₉	—	2	1	no	—
164	C₄×C₇₉	—	2	1	no	—
165	C₄×C₇₉	#405	1	2	yes	—
166	C₄×C₇₉	C₂×#131	1	2	yes	—
167	C₄×C₇₉	C₂×#131	1	2	yes	—
168	C₄×C₇₉	#568	1	2	yes	—
169	Dic₇₉	—	2	2	no	—
170	Dic₇₉	—	2	2	no	—
171	Dic₇₉	—	2	2	no	—
172	#17331	—	79	1	no	maximal, Sylow
173	C₂×C₇₉	—	2	2	no	—
174	C₂×C₇₉	C₂×#405	1	1	yes	characteristic
175	C₂×C₇₉	C₄×#131	1	1	yes	characteristic
176	C₂×C₇₉	#2875	1	1	yes	characteristic
177	D₇₉	—	4	2	no	—
178	#2965	—	79	1	no	—
179	C₈×#41	—	79	1	no	—
180	#2922	—	79	1	no	—
181	#1454	—	79	2	no	—
182	#1453	—	79	2	no	—
183	#1452	—	79	2	no	—
184	#1451	—	79	1	no	—
185	#1450	—	79	1	no	—
186	C₈×M₁₆	—	79	2	no	—
187	C₈×M₁₆	D₇₉	1	1	yes	characteristic, complemented
188	C₂×#322	—	79	1	no	—
189	C₇₉	#17331	1	1	yes	characteristic, complemented, Sylow
190	C₄×#41	—	79	1	no	—
191	#446	—	79	2	no	—
192	C₈×Q₈	—	79	2	no	—
193	#436	—	79	2	no	—
194	#435	—	79	4	no	—
195	C₈×D₄	—	79	2	no	—
196	C₈×D₄	—	79	4	no	—
197	#433	—	79	4	no	—
198	#431	—	79	1	no	—
199	#431	—	79	2	no	—
200	#431	—	79	2	no	—
201	#431	—	79	2	no	—
202	#431	—	79	2	no	—
203	#431	C₂×D₇₉	1	1	yes	characteristic
204	#424	—	79	2	no	—
205	#408	—	79	2	no	—
206	C₄×M₁₆	—	79	2	no	—
207	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
208	C₂×C₄×C₈	—	79	1	no	—
209	C₂×C₄×C₈	—	79	1	no	—
210	C₂×C₄×C₈	—	79	2	no	—
211	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
212	#322	—	79	1	no	—
213	#322	—	79	1	no	—
214	#322	—	79	1	no	—
215	#322	—	79	1	no	—
216	#322	—	79	2	no	—
217	#322	—	79	2	no	—
218	#322	—	79	2	no	—
219	#322	C₂×D₇₉	1	2	yes	—
220	C₈²	—	79	2	no	—
221	C₈²	C₂×D₇₉	1	2	yes	—
222	C₂×#41	—	79	1	no	—
223	C₂×M₁₆	—	79	4	no	—
224	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
225	C₂²×C₈	—	79	2	no	—
226	C₂²×C₈	—	79	4	no	—
227	C₂²×C₈	—	79	2	no	—
228	C₂²×C₈	C₄×D₇₉	1	2	yes	—
229	C₄×Q₈	—	79	1	no	—
230	C₄×Q₈	—	79	1	no	—
231	C₄×D₄	—	79	2	no	—
232	C₄×D₄	—	79	2	no	—
233	C₄×D₄	—	79	1	no	—
234	C₄×D₄	—	79	1	no	—
235	#117	—	79	1	no	—
236	#117	—	79	2	no	—
237	C₂×C₄²	—	79	2	no	—
238	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
239	C₄⋊C₈	—	79	4	no	—
240	C₄⋊C₈	—	79	4	no	—
241	C₄⋊C₈	—	79	4	no	—
242	C₄⋊C₈	—	79	4	no	—
243	C₄⋊C₈	—	79	2	no	—
244	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
245	C₂²⋊C₈	—	79	2	no	—
246	C₂²⋊C₈	—	79	4	no	—
247	C₂²⋊C₈	—	79	4	no	—
248	C₂²⋊C₈	—	79	4	no	—
249	C₂²⋊C₈	—	79	4	no	—
250	C₂²⋊C₈	C₄×D₇₉	1	2	yes	—
251	#97	—	79	4	no	—
252	#97	C₄×D₇₉	1	2	yes	—
253	C₄×C₈	—	79	4	no	—
254	C₄×C₈	—	79	2	no	—
255	C₄×C₈	—	79	2	no	—
256	C₄×C₈	—	79	1	no	—
257	C₄×C₈	—	79	2	no	—
258	C₄×C₈	—	79	1	no	—
259	C₄×C₈	—	79	1	no	—
260	C₄×C₈	—	79	1	no	—
261	C₄×C₈	C₄×D₇₉	1	2	yes	—
262	C₄×C₈	C₂²×D₇₉	1	2	yes	—
263	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₉	1	2	yes	—
265	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
266	#41	—	158	4	no	—
267	C₂×Q₈	—	79	1	no	—
268	C₂×D₄	—	79	2	no	—
269	C₂×D₄	—	79	1	no	—
270	C₂²×C₄	—	79	2	no	—
271	C₂²×C₄	—	79	2	no	—
272	C₂²×C₄	—	79	2	no	—
273	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	158	8	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	316	4	no	—
280	C₂×C₈	—	316	2	no	—
281	C₂×C₈	—	79	2	no	—
282	C₂×C₈	—	79	2	no	—
283	C₂×C₈	—	79	2	no	—
284	C₂×C₈	—	79	2	no	—
285	C₂×C₈	—	79	4	no	—
286	C₂×C₈	—	158	4	no	—
287	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
288	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
290	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
291	C₄⋊C₄	—	79	2	no	—
292	C₄⋊C₄	—	79	2	no	—
293	C₄⋊C₄	—	79	1	no	—
294	C₄⋊C₄	—	79	1	no	—
295	C₂²⋊C₄	—	79	2	no	—
296	C₂²⋊C₄	—	79	2	no	—
297	C₂²⋊C₄	—	79	2	no	—
298	C₄²	—	79	2	no	—
299	C₄²	—	79	2	no	—
300	C₄²	—	79	2	no	—
301	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
304	C₄²	C₂³×D₇₉	1	1	yes	characteristic
305	C₂³	—	79	2	no	—
306	C₂³	C₄²×D₇₉	1	1	yes	characteristic
307	Q₈	—	158	2	no	—
308	D₄	—	158	4	no	—
309	D₄	—	158	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	79	2	no	—
313	C₂×C₄	—	79	2	no	—
314	C₂×C₄	—	79	2	no	—
315	C₂×C₄	—	158	4	no	—
316	C₂×C₄	—	158	4	no	—
317	C₂×C₄	—	158	4	no	—
318	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
319	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
320	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
321	C₂×C₄	#14579167	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	4	no	—
328	C₈	—	158	2	no	—
329	C₈	—	158	2	no	—
330	C₈	—	158	4	no	—
331	C₈	—	158	2	no	—
332	C₈	—	158	2	no	—
333	C₈	#14579169	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	158	2	no	—
337	C₂²	—	158	2	no	—
338	C₂²	—	158	2	no	—
339	C₂²	#19098517	1	2	yes	—
340	C₂²	C₂×C₄²×D₇₉	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	158	2	no	—
344	C₄	—	158	2	no	—
345	C₄	—	158	2	no	—
346	C₄	#19098517	1	2	yes	—
347	C₄	C₂×#14579167	1	2	yes	—
348	C₄	C₂×#14579169	1	2	yes	—
349	C₄	#19099328	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	316	2	no	—
352	C₂	C₂×#19098517	1	1	yes	characteristic
353	C₂	C₄×#14579169	1	1	yes	characteristic
354	C₂	#23771827	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂²×C₄²
Fitting	C₈×M₁₆×C₇₉	C₂
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₈×M₁₆	D₇₉
hypocenter	C₇₉	#17331
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #23761701	C₂	semidirect
5. #23761694	C₂	semidirect
6. #23761691	C₂	semidirect
7. #23761687	C₂	semidirect
8. #23761684	C₂	semidirect
9. C₈×#11929607	C₂	semidirect
10. C₂×#19098284	C₂	semidirect
11. C₈×M₁₆×C₇₉	C₂	semidirect
187. C₈×M₁₆	D₇₉	semidirect
189. C₇₉	#17331	semidirect

Maximal subgroups

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

subgroup	index
#23772366	2
#23772356	2
#23772045	2
#23761701	2
#23761694	2
#23761691	2
#23761687	2
#23761684	2
C₈×#11929607	2
C₂×#19098284	2
C₈×M₁₆×C₇₉	2
#17331	79