Subgroups — #34183359

Order 20224.

528 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	#23771964	C₂	1	1	yes	maximal, characteristic
2	#23761829	C₂	1	2	yes	maximal, complemented
3	#23761714	C₂	1	1	yes	maximal, characteristic
4	#23761303	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2396	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	—	2	1	no	—
7	#19098616	C₂²	1	2	yes	—
8	#19098590	—	2	4	no	—
9	#19098587	—	2	1	no	—
10	#19098587	C₂²	1	2	yes	—
11	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
12	#19098289	—	2	1	no	—
13	#19098287	—	2	1	no	—
14	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
16	#400×C₇₉	C₄	1	1	yes	characteristic
17	#399×C₇₉	C₄	1	1	yes	characteristic
18	#398×C₇₉	—	2	1	no	—
19	#396×C₇₉	—	2	1	no	—
20	#395×C₇₉	C₄	1	1	yes	characteristic
21	#393×C₇₉	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
23	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₇₉	—	2	1	no	—
25	C₂×#11929608	—	2	2	no	—
26	C₂×#11929608	—	2	2	no	—
27	C₂×#11929608	—	4	2	no	—
28	C₂×#11929608	—	4	2	no	—
29	C₂²×C₇₉⋊C₈	—	2	2	no	—
30	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
34	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₄×C₇₉⋊C₈	—	2	2	no	—
36	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₇₉	—	2	2	no	—
39	C₄×Q₈×C₇₉	—	2	2	no	—
40	#117×C₇₉	—	2	2	no	—
41	#117×C₇₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₇₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	—	2	1	no	—
54	C₂×C₄²×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
57	#95×C₇₉	—	2	1	no	—
58	#95×C₇₉	—	2	1	no	—
59	#95×C₇₉	—	2	1	no	—
60	#95×C₇₉	—	2	1	no	—
61	#95×C₇₉	C₂×C₄	1	1	yes	characteristic
62	#95×C₇₉	C₂×C₄	1	1	yes	characteristic
63	#11929608	—	4	4	no	—
64	#11929608	—	4	2	no	—
65	#11929608	—	4	2	no	—
66	C₂×C₇₉⋊C₈	—	2	4	no	—
67	C₂×C₇₉⋊C₈	—	4	4	no	—
68	C₂×C₇₉⋊C₈	—	4	2	no	—
69	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×Q₈×C₇₉	—	2	2	no	—
71	C₂×Q₈×C₇₉	—	4	2	no	—
72	C₂²×C₄×C₇₉	—	2	1	no	—
73	C₂²×C₄×C₇₉	—	2	1	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	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	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
84	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₇₉	—	2	2	no	—
89	C₄⋊C₄×C₇₉	—	2	2	no	—
90	C₄⋊C₄×C₇₉	—	2	2	no	—
91	C₄⋊C₄×C₇₉	—	4	2	no	—
92	C₄⋊C₄×C₇₉	—	4	2	no	—
93	C₄⋊C₄×C₇₉	—	4	2	no	—
94	C₄⋊C₄×C₇₉	—	4	2	no	—
95	C₂²⋊C₄×C₇₉	—	2	2	no	—
96	C₂²⋊C₄×C₇₉	—	2	2	no	—
97	C₂²⋊C₄×C₇₉	—	4	2	no	—
98	C₂²⋊C₄×C₇₉	—	4	2	no	—
99	C₄²×C₇₉	—	2	1	no	—
100	C₄²×C₇₉	—	2	1	no	—
101	C₄²×C₇₉	—	4	1	no	—
102	C₄²×C₇₉	—	4	1	no	—
103	C₄²×C₇₉	—	4	1	no	—
104	C₄²×C₇₉	—	4	1	no	—
105	C₄²×C₇₉	C₂×D₄	1	2	yes	—
106	C₄²×C₇₉	#41	1	2	yes	—
107	C₂³×C₇₉	—	2	1	no	—
108	C₂³×C₇₉	—	2	1	no	—
109	C₂³×C₇₉	—	4	1	no	—
110	C₂³×C₇₉	—	4	1	no	—
111	C₂³×C₇₉	#103	1	2	yes	—
112	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	Q₈×C₇₉	—	2	4	no	—
114	Q₈×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	2	no	—
119	C₂×C₄×C₇₉	—	2	2	no	—
120	C₂×C₄×C₇₉	—	2	2	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	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	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	#103	1	2	yes	—
140	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#121	1	2	yes	—
143	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
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	—	8	2	no	—
150	C₂²×C₇₉	—	2	1	no	—
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	—	2	2	no	—
153	C₂²×C₇₉	—	4	2	no	—
154	C₂²×C₇₉	—	4	1	no	—
155	C₂²×C₇₉	—	4	1	no	—
156	C₂²×C₇₉	—	8	1	no	—
157	C₂²×C₇₉	#386	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
159	C₂²×C₇₉	#417	1	1	yes	characteristic
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	2	no	—
162	C₄×C₇₉	—	2	2	no	—
163	C₄×C₇₉	—	4	2	no	—
164	C₄×C₇₉	—	4	1	no	—
165	C₄×C₇₉	—	4	1	no	—
166	C₄×C₇₉	—	4	2	no	—
167	C₄×C₇₉	—	8	1	no	—
168	C₄×C₇₉	—	8	1	no	—
169	C₄×C₇₉	#450	1	1	yes	characteristic
170	C₄×C₇₉	#451	1	1	yes	characteristic
171	C₄×C₇₉	#483	1	1	yes	characteristic
172	C₄×C₇₉	#484	1	1	yes	characteristic
173	#9033	—	79	1	no	maximal, Sylow
174	C₂×C₇₉	—	2	1	no	—
175	C₂×C₇₉	—	2	1	no	—
176	C₂×C₇₉	—	4	2	no	—
177	C₂×C₇₉	#1872	1	1	yes	characteristic
178	C₂×C₇₉	#1884	1	1	yes	characteristic
179	C₂×C₇₉	#1891	1	1	yes	characteristic
180	#2905	—	79	1	no	—
181	#2396	D₇₉	1	1	yes	characteristic
182	#1487	—	79	2	no	—
183	#1458	—	79	1	no	—
184	#1303	—	79	2	no	—
185	C₇₉	#9033	1	1	yes	characteristic, complemented, Sylow
186	C₂²×M₁₆	—	158	1	no	—
187	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
188	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
189	#431	—	79	2	no	—
190	#424	—	158	4	no	—
191	#423	—	79	2	no	—
192	#423	—	158	1	no	—
193	#400	Dic₇₉	1	1	yes	characteristic
194	#399	Dic₇₉	1	1	yes	characteristic
195	#398	—	2	1	no	—
196	#396	—	2	1	no	—
197	#395	Dic₇₉	1	1	yes	characteristic
198	#393	Dic₇₉	1	1	yes	characteristic
199	C₄×C₄⋊C₄	—	2	1	no	—
200	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
201	C₄×C₂²⋊C₄	—	2	1	no	—
202	Q₈⋊C₈	—	158	4	no	—
203	#324	—	158	1	no	—
204	#323	—	158	1	no	—
205	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
206	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
207	C₂×M₁₆	—	316	2	no	—
208	C₂×M₁₆	—	316	2	no	—
209	C₂×M₁₆	—	158	2	no	—
210	C₂×M₁₆	—	158	2	no	—
211	C₂²×C₈	—	158	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
231	C₄⋊C₈	—	79	4	no	—
232	C₄⋊C₈	—	158	4	no	—
233	C₄⋊C₈	—	158	2	no	—
234	C₄⋊C₈	—	158	2	no	—
235	C₂²⋊C₈	—	158	2	no	—
236	C₄×C₈	—	158	2	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#95	C₂×Dic₇₉	1	1	yes	characteristic
242	#95	C₂×Dic₇₉	1	1	yes	characteristic
243	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
257	C₂²×C₄	#11929611	1	2	yes	—
258	C₂²×C₄	#11929614	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
262	M₁₆	—	316	4	no	—
263	M₁₆	—	316	2	no	—
264	M₁₆	—	316	2	no	—
265	C₂×C₈	—	316	4	no	—
266	C₂×C₈	—	316	2	no	—
267	C₂×C₈	—	158	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	D₄×D₇₉	1	2	yes	—
286	C₄²	#11929623	1	2	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	#14579085	1	2	yes	—
292	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
293	Q₈	—	2	4	no	—
294	Q₈	—	4	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	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	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	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	#14579085	1	2	yes	—
320	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
321	C₂×C₄	#14579129	1	1	yes	characteristic
322	C₂×C₄	#14579133	1	2	yes	—
323	C₂×C₄	#14579142	1	1	yes	characteristic
324	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
325	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
326	C₂×C₄	#14579186	1	1	yes	characteristic
327	C₂×C₄	#14579191	1	1	yes	characteristic
328	C₈	—	316	4	no	—
329	C₈	—	632	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	#19098443	1	1	yes	characteristic
338	C₂²	C₂×#14579085	1	1	yes	characteristic
339	C₂²	#19098564	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	#19098713	1	1	yes	characteristic
350	C₄	#19098718	1	1	yes	characteristic
351	C₄	#19098891	1	1	yes	characteristic
352	C₄	#19098896	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	4	2	no	—
356	C₂	#23763708	1	1	yes	characteristic
357	C₂	#23763784	1	1	yes	characteristic
358	C₂	#23763819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2396	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2396	D₇₉
hypocenter	C₇₉	#9033
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2396

Complements

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

N	H ≅ G/N	type
2. #23761829	C₂	semidirect
4. #23761303	C₂	semidirect
185. C₇₉	#9033	semidirect

Maximal subgroups

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

subgroup	index
#23771964	2
#23761829	2
#23761714	2
#23761303	2
C₇₉×#2396	2
#9033	79