Subgroups — #34176726

Order 20224.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763094	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098318	C₂	1	1	yes	maximal, characteristic
3	#23761226	C₂	1	2	yes	maximal, complemented
4	#23761223	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2855	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	—	2	1	no	—
7	C₂³×C₇₉⋊C₈	—	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×C₂²⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	#391×C₇₉	C₄	1	1	yes	characteristic
15	#391×C₇₉	C₂²	1	1	yes	characteristic
16	#389×C₇₉	C₄	1	1	yes	characteristic
17	#389×C₇₉	C₂²	1	1	yes	characteristic
18	#387×C₇₉	—	2	1	no	—
19	#387×C₇₉	—	2	1	no	—
20	#385×C₇₉	—	2	1	no	—
21	#381×C₇₉	—	2	1	no	—
22	#380×C₇₉	C₄	1	1	yes	characteristic
23	C₂×#11929608	—	2	4	no	—
24	C₂×#11929608	—	4	2	no	—
25	C₂×#11929608	—	4	2	no	—
26	C₂²×C₇₉⋊C₈	—	2	4	no	—
27	C₂²×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₇₉⋊C₈	—	2	2	no	—
29	C₂²×C₇₉⋊C₈	—	4	2	no	—
30	C₂²×C₇₉⋊C₈	—	4	2	no	—
31	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²⋊Q₈×C₇₉	—	4	2	no	—
36	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
44	C₂×C₄²×C₇₉	—	4	2	no	—
45	C₂×C₄²×C₇₉	C₂×C₄	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	C₂³	1	1	yes	characteristic
49	#95×C₇₉	—	2	1	no	—
50	#95×C₇₉	—	2	1	no	—
51	#95×C₇₉	—	2	1	no	—
52	#95×C₇₉	—	2	1	no	—
53	#95×C₇₉	—	2	1	no	—
54	#95×C₇₉	—	2	1	no	—
55	#95×C₇₉	—	2	1	no	—
56	#11929608	—	4	8	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	4	8	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₂×C₇₉⋊C₈	—	4	2	no	—
62	C₂×C₇₉⋊C₈	—	4	2	no	—
63	C₂×C₇₉⋊C₈	—	4	2	no	—
64	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×Q₈×C₇₉	—	2	2	no	—
66	C₂×Q₈×C₇₉	—	4	1	no	—
67	C₂×Q₈×C₇₉	—	4	1	no	—
68	C₂²×C₄×C₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	—	2	2	no	—
70	C₂²×C₄×C₇₉	—	2	2	no	—
71	C₂²×C₄×C₇₉	—	2	2	no	—
72	C₂²×C₄×C₇₉	—	2	1	no	—
73	C₂²×C₄×C₇₉	—	2	1	no	—
74	C₂²×C₄×C₇₉	—	2	2	no	—
75	C₂²×C₄×C₇₉	—	2	2	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	—	2	1	no	—
78	C₂²×C₄×C₇₉	—	2	1	no	—
79	C₂²×C₄×C₇₉	—	2	1	no	—
80	C₂²×C₄×C₇₉	—	4	1	no	—
81	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
85	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₇₉	—	2	2	no	—
90	C₄⋊C₄×C₇₉	—	4	2	no	—
91	C₄⋊C₄×C₇₉	—	4	1	no	—
92	C₄⋊C₄×C₇₉	—	4	1	no	—
93	C₄⋊C₄×C₇₉	—	4	2	no	—
94	C₄⋊C₄×C₇₉	—	4	1	no	—
95	C₄⋊C₄×C₇₉	—	4	1	no	—
96	C₂²⋊C₄×C₇₉	—	4	2	no	—
97	C₂²⋊C₄×C₇₉	—	4	2	no	—
98	C₄²×C₇₉	—	4	1	no	—
99	C₄²×C₇₉	—	4	1	no	—
100	C₄²×C₇₉	—	4	1	no	—
101	C₄²×C₇₉	—	4	1	no	—
102	C₂³×C₇₉	—	2	1	no	—
103	C₂³×C₇₉	—	2	1	no	—
104	C₂³×C₇₉	—	2	1	no	—
105	C₂³×C₇₉	—	2	1	no	—
106	C₂³×C₇₉	#99	1	2	yes	—
107	C₂³×C₇₉	#103	1	2	yes	—
108	C₂³×C₇₉	#104	1	2	yes	—
109	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	Q₈×C₇₉	—	8	2	no	—
111	C₂×C₄×C₇₉	—	2	4	no	—
112	C₂×C₄×C₇₉	—	2	4	no	—
113	C₂×C₄×C₇₉	—	2	4	no	—
114	C₂×C₄×C₇₉	—	2	4	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	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	—	2	1	no	—
128	C₂×C₄×C₇₉	—	4	4	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	—	4	1	no	—
137	C₂×C₄×C₇₉	—	4	1	no	—
138	C₂×C₄×C₇₉	—	4	1	no	—
139	C₂×C₄×C₇₉	—	4	1	no	—
140	C₂×C₄×C₇₉	—	4	1	no	—
141	C₂×C₄×C₇₉	#103	1	2	yes	—
142	C₂×C₄×C₇₉	#104	1	2	yes	—
143	C₂×C₄×C₇₉	#131	1	2	yes	—
144	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
148	C₇₉⋊C₈	—	4	4	no	—
149	C₇₉⋊C₈	—	4	4	no	—
150	C₂²×C₇₉	—	2	2	no	—
151	C₂²×C₇₉	—	2	2	no	—
152	C₂²×C₇₉	—	2	2	no	—
153	C₂²×C₇₉	—	2	2	no	—
154	C₂²×C₇₉	—	2	2	no	—
155	C₂²×C₇₉	—	2	2	no	—
156	C₂²×C₇₉	#328	1	4	yes	—
157	C₂²×C₇₉	#408	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#99	1	1	yes	characteristic
159	C₂²×C₇₉	#413	1	1	yes	characteristic
160	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
161	C₂²×C₇₉	#417	1	1	yes	characteristic
162	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
163	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
164	C₄×C₇₉	—	2	2	no	—
165	C₄×C₇₉	—	2	2	no	—
166	C₄×C₇₉	—	2	1	no	—
167	C₄×C₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	2	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	—	4	1	no	—
172	C₄×C₇₉	—	8	1	no	—
173	C₄×C₇₉	—	8	1	no	—
174	C₄×C₇₉	—	8	1	no	—
175	C₄×C₇₉	—	8	1	no	—
176	#7681	—	79	1	no	maximal, Sylow
177	C₂×C₇₉	—	2	4	no	—
178	C₂×C₇₉	#1468	1	1	yes	characteristic
179	C₂×C₇₉	#1483	1	1	yes	characteristic
180	C₂×C₇₉	#1493	1	1	yes	characteristic
181	C₂×C₇₉	C₂×#328	1	1	yes	characteristic
182	C₂×C₇₉	#1504	1	1	yes	characteristic
183	C₂×C₇₉	#1524	1	1	yes	characteristic
184	C₂×C₇₉	#1550	1	1	yes	characteristic
185	#2855	D₇₉	1	1	yes	characteristic
186	#1728	—	79	1	no	—
187	C₂×#336	—	79	1	no	—
188	#1272	—	79	2	no	—
189	#1271	—	79	2	no	—
190	C₇₉	#7681	1	1	yes	characteristic, complemented, Sylow
191	C₂²×M₁₆	—	158	1	no	—
192	C₂³×C₈	—	158	1	no	—
193	C₂×C₂²⋊Q₈	Dic₇₉	1	1	yes	characteristic
194	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
195	#391	Dic₇₉	1	1	yes	characteristic
196	#391	C₂×D₇₉	1	1	yes	characteristic
197	#389	Dic₇₉	1	1	yes	characteristic
198	#389	C₂×D₇₉	1	1	yes	characteristic
199	#387	—	2	1	no	—
200	#387	—	2	1	no	—
201	#385	—	2	1	no	—
202	#381	—	2	1	no	—
203	#380	Dic₇₉	1	1	yes	characteristic
204	#336	—	79	2	no	—
205	#336	—	79	2	no	—
206	#336	—	158	1	no	—
207	#336	—	158	1	no	—
208	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
209	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
210	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
211	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
212	C₂×M₁₆	—	316	2	no	—
213	C₂×M₁₆	—	316	2	no	—
214	C₂×M₁₆	—	158	4	no	—
215	C₂²×C₈	—	316	2	no	—
216	C₂²×C₈	—	316	2	no	—
217	C₂²×C₈	—	158	4	no	—
218	C₂²×C₈	—	158	2	no	—
219	C₂²×C₈	—	158	2	no	—
220	C₂²⋊Q₈	—	4	2	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	—	2	1	no	—
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₄²	—	4	2	no	—
230	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
261	C₂²×C₄	#11929611	1	2	yes	—
262	C₂²×C₄	#11929614	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
266	M₁₆	—	316	8	no	—
267	C₂×C₈	—	316	8	no	—
268	C₂×C₈	—	316	2	no	—
269	C₂×C₈	—	316	2	no	—
270	C₂×C₈	—	316	2	no	—
271	C₂×C₈	—	316	2	no	—
272	C₂×C₈	—	158	4	no	—
273	C₂×C₈	—	158	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	#14579076	1	2	yes	—
292	C₂³	#14579085	1	2	yes	—
293	C₂³	#14579088	1	2	yes	—
294	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
295	Q₈	—	8	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	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	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	#14579085	1	2	yes	—
327	C₂×C₄	#14579088	1	2	yes	—
328	C₂×C₄	#14579168	1	2	yes	—
329	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
330	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
331	C₂×C₄	#14579186	1	1	yes	characteristic
332	C₂×C₄	#14579191	1	1	yes	characteristic
333	C₈	—	316	4	no	—
334	C₈	—	316	4	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	#19098300	1	4	yes	—
342	C₂²	#19098530	1	1	yes	characteristic
343	C₂²	C₂×#14579076	1	1	yes	characteristic
344	C₂²	#19098547	1	1	yes	characteristic
345	C₂²	C₂×#14579085	1	1	yes	characteristic
346	C₂²	#19098564	1	1	yes	characteristic
347	C₂²	C₂×#14579088	1	1	yes	characteristic
348	C₂²	C₂×#14579088	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	8	1	no	—
358	C₄	—	8	1	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₂	—	2	4	no	—
362	C₂	#23761748	1	1	yes	characteristic
363	C₂	#23761813	1	1	yes	characteristic
364	C₂	#23761855	1	1	yes	characteristic
365	C₂	C₂×#19098300	1	1	yes	characteristic
366	C₂	#23761904	1	1	yes	characteristic
367	C₂	#23761988	1	1	yes	characteristic
368	C₂	#23762090	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×#2855	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2855	D₇₉
hypocenter	C₇₉	#7681
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2855

Complements

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

N	H ≅ G/N	type
3. #23761226	C₂	semidirect
4. #23761223	C₂	semidirect
190. C₇₉	#7681	semidirect

Maximal subgroups

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

subgroup	index
#23763094	2
C₂×#19098318	2
#23761226	2
#23761223	2
C₇₉×#2855	2
#7681	79