Subgroups — #34176554

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763543	C₂	1	1	yes	maximal, characteristic, complemented
2	#23763517	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761223	C₂	1	2	yes	maximal, complemented
4	#23761221	C₂	1	2	yes	maximal
5	C₇₉×#2293	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	#19098318	—	2	1	no	—
8	#19098318	—	2	1	no	—
9	#19098318	—	2	1	no	—
10	#19098318	C₂²	1	2	yes	—
11	#19098318	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₇₉	C₄	1	1	yes	characteristic
13	C₂×#117×C₇₉	—	2	1	no	—
14	#390×C₇₉	C₄	1	1	yes	characteristic
15	#390×C₇₉	C₂²	1	1	yes	characteristic
16	#389×C₇₉	C₄	1	1	yes	characteristic
17	#389×C₇₉	C₂²	1	1	yes	characteristic
18	#388×C₇₉	—	2	1	no	—
19	#388×C₇₉	—	2	1	no	—
20	C₄×C₄⋊C₄×C₇₉	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₇₉	—	2	1	no	—
22	C₄³×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
26	C₂²×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₇₉	—	2	1	no	—
28	C₄×D₄×C₇₉	—	2	2	no	—
29	C₄×D₄×C₇₉	—	2	2	no	—
30	#117×C₇₉	—	4	2	no	—
31	#117×C₇₉	—	4	2	no	—
32	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₂×C₄²×C₇₉	—	2	1	no	—
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	#95×C₇₉	—	2	1	no	—
51	#95×C₇₉	—	2	1	no	—
52	#95×C₇₉	—	2	1	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	2	no	—
59	C₂⁴×C₇₉	—	2	1	no	—
60	C₂×D₄×C₇₉	—	2	1	no	—
61	C₂×D₄×C₇₉	—	4	1	no	—
62	C₂×D₄×C₇₉	—	4	1	no	—
63	C₂×D₄×C₇₉	M₁₆	1	2	yes	—
64	C₂²×C₄×C₇₉	—	2	4	no	—
65	C₂²×C₄×C₇₉	—	2	1	no	—
66	C₂²×C₄×C₇₉	—	2	1	no	—
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	—	4	2	no	—
69	C₂²×C₄×C₇₉	—	4	2	no	—
70	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₇₉	—	2	1	no	—
78	C₄⋊C₄×C₇₉	—	4	1	no	—
79	C₄⋊C₄×C₇₉	—	4	1	no	—
80	C₄⋊C₄×C₇₉	—	4	2	no	—
81	C₄⋊C₄×C₇₉	—	4	1	no	—
82	C₄⋊C₄×C₇₉	—	4	1	no	—
83	C₄⋊C₄×C₇₉	M₁₆	1	2	yes	—
84	C₂²⋊C₄×C₇₉	—	2	4	no	—
85	C₂²⋊C₄×C₇₉	—	4	2	no	—
86	C₂²⋊C₄×C₇₉	—	4	2	no	—
87	C₂²⋊C₄×C₇₉	—	4	1	no	—
88	C₂²⋊C₄×C₇₉	—	4	1	no	—
89	C₂²⋊C₄×C₇₉	—	4	2	no	—
90	C₄²×C₇₉	—	2	1	no	—
91	C₄²×C₇₉	—	2	1	no	—
92	C₄²×C₇₉	—	2	2	no	—
93	C₄²×C₇₉	—	4	1	no	—
94	C₄²×C₇₉	—	4	1	no	—
95	C₄²×C₇₉	—	4	1	no	—
96	C₄²×C₇₉	—	4	1	no	—
97	C₄²×C₇₉	—	4	2	no	—
98	C₄²×C₇₉	D₈	1	2	yes	—
99	C₄²×C₇₉	SD₁₆	1	2	yes	—
100	C₂³×C₇₉	—	2	2	no	—
101	C₂³×C₇₉	—	4	1	no	—
102	C₂³×C₇₉	—	4	1	no	—
103	C₂³×C₇₉	—	4	1	no	—
104	C₂³×C₇₉	—	4	1	no	—
105	C₂³×C₇₉	—	4	1	no	—
106	C₂³×C₇₉	—	4	1	no	—
107	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	D₄×C₇₉	—	4	2	no	—
109	D₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	2	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	8	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	1	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	—	4	1	no	—
134	C₂×C₄×C₇₉	—	4	1	no	—
135	C₂×C₄×C₇₉	—	4	1	no	—
136	C₂×C₄×C₇₉	#102	1	2	yes	—
137	C₂×C₄×C₇₉	#102	1	2	yes	—
138	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#135	1	2	yes	—
143	C₇₉⋊C₈	—	8	2	no	—
144	C₇₉⋊C₈	—	8	2	no	—
145	C₂²×C₇₉	—	4	2	no	—
146	C₂²×C₇₉	—	4	2	no	—
147	C₂²×C₇₉	—	4	2	no	—
148	C₂²×C₇₉	—	8	1	no	—
149	C₂²×C₇₉	—	8	1	no	—
150	C₂²×C₇₉	—	8	1	no	—
151	C₂²×C₇₉	—	8	1	no	—
152	C₂²×C₇₉	#407	1	1	yes	characteristic
153	C₂²×C₇₉	#410	1	1	yes	characteristic
154	C₂²×C₇₉	#413	1	1	yes	characteristic
155	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
156	C₂²×C₇₉	#416	1	1	yes	characteristic
157	C₂²×C₇₉	#421	1	1	yes	characteristic
158	C₂²×C₇₉	#421	1	1	yes	characteristic
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	2	1	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	2	2	no	—
163	C₄×C₇₉	—	2	1	no	—
164	C₄×C₇₉	—	2	1	no	—
165	C₄×C₇₉	—	2	2	no	—
166	C₄×C₇₉	—	4	2	no	—
167	C₄×C₇₉	—	4	1	no	—
168	C₄×C₇₉	—	4	1	no	—
169	C₄×C₇₉	—	8	1	no	—
170	C₄×C₇₉	—	8	1	no	—
171	#7642	—	79	1	no	maximal, Sylow
172	C₂×C₇₉	—	8	2	no	—
173	C₂×C₇₉	#1464	1	1	yes	characteristic
174	C₂×C₇₉	#1486	1	1	yes	characteristic
175	C₂×C₇₉	#1488	1	1	yes	characteristic
176	C₂×C₇₉	#1517	1	1	yes	characteristic
177	C₂×C₇₉	#1521	1	1	yes	characteristic
178	C₂×C₇₉	#1535	1	1	yes	characteristic
179	C₂×C₇₉	#1540	1	1	yes	characteristic
180	#2293	D₇₉	1	1	yes	characteristic
181	#1839	—	79	1	no	—
182	#1831	—	79	1	no	—
183	#1271	—	79	2	no	—
184	#1270	—	79	2	no	—
185	C₇₉	#7642	1	1	yes	characteristic, complemented, Sylow
186	C₂×C₄×D₄	Dic₇₉	1	1	yes	characteristic
187	C₂×#117	—	2	1	no	—
188	C₂×C₄⋊C₈	—	158	1	no	—
189	#390	Dic₇₉	1	1	yes	characteristic
190	#390	C₂×D₇₉	1	1	yes	characteristic
191	#389	Dic₇₉	1	1	yes	characteristic
192	#389	C₂×D₇₉	1	1	yes	characteristic
193	#388	—	2	1	no	—
194	#388	—	2	1	no	—
195	C₄×C₄⋊C₄	Dic₇₉	1	1	yes	characteristic
196	C₄×C₂²⋊C₄	—	2	1	no	—
197	C₄³	C₂×D₇₉	1	1	yes	characteristic
198	#336	—	79	2	no	—
199	#336	—	79	2	no	—
200	#336	—	158	1	no	—
201	#336	—	158	1	no	—
202	#336	—	158	1	no	—
203	C₂²×D₄	C₂×Dic₇₉	1	1	yes	characteristic
204	C₂³×C₄	—	2	1	no	—
205	C₂²×C₈	—	158	2	no	—
206	C₂²×C₈	—	158	2	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	#117	—	4	2	no	—
210	#117	—	4	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
229	C₄⋊C₈	—	316	2	no	—
230	#95	—	2	1	no	—
231	#95	—	2	1	no	—
232	#95	—	2	1	no	—
233	C₂⁴	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	#11929608	1	2	yes	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
251	C₂×C₈	—	316	2	no	—
252	C₂×C₈	—	316	2	no	—
253	C₂×C₈	—	316	2	no	—
254	C₂×C₈	—	316	2	no	—
255	C₂×C₈	—	316	2	no	—
256	C₂×C₈	—	316	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	#11929608	1	2	yes	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	2	no	—
278	C₄²	#11929609	1	2	yes	—
279	C₄²	#11929612	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	8	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	#14579082	1	2	yes	—
317	C₂×C₄	#14579082	1	2	yes	—
318	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
319	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
320	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
321	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
322	C₂×C₄	#14579181	1	2	yes	—
323	C₈	—	632	2	no	—
324	C₈	—	632	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	—	8	1	no	—
332	C₂²	#19098527	1	1	yes	characteristic
333	C₂²	#19098537	1	1	yes	characteristic
334	C₂²	#19098547	1	1	yes	characteristic
335	C₂²	C₂×#14579082	1	1	yes	characteristic
336	C₂²	#19098559	1	1	yes	characteristic
337	C₂²	#19098580	1	1	yes	characteristic
338	C₂²	#19098580	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₂	—	8	2	no	—
352	C₂	#23761734	1	1	yes	characteristic
353	C₂	#23761825	1	1	yes	characteristic
354	C₂	#23761834	1	1	yes	characteristic
355	C₂	#23761960	1	1	yes	characteristic
356	C₂	#23761977	1	1	yes	characteristic
357	C₂	#23762031	1	1	yes	characteristic
358	C₂	#23762049	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2293	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2293	D₇₉
hypocenter	C₇₉	#7642
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2293

Complements

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

N	H ≅ G/N	type
1. #23763543	C₂	semidirect
2. #23763517	C₂	semidirect
3. #23761223	C₂	semidirect
185. C₇₉	#7642	semidirect

Maximal subgroups

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

subgroup	index
#23763543	2
#23763517	2
#23761223	2
#23761221	2
C₇₉×#2293	2
#7642	79