Subgroups — #34196287

Order 20224.

686 conjugacy classes of subgroups.

The subgroup lattice has 344 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	#23763133	C₂	1	1	yes	maximal, characteristic
3	#23763119	C₂	1	2	yes	maximal
4	C₂×#19098319	C₂	1	2	yes	maximal
5	C₇₉×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂×#14579120	C₂²	1	2	yes	—
8	#19098590	—	2	4	no	—
9	#19098587	—	2	2	no	—
10	#19098330	—	2	2	no	—
11	#19098325	—	2	4	no	—
12	#19098319	—	2	4	no	—
13	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
16	#389×C₇₉	—	2	2	no	—
17	#385×C₇₉	—	2	2	no	—
18	#384×C₇₉	C₄	1	2	yes	complemented
19	#382×C₇₉	C₄	1	2	yes	complemented
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	2	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	—	2	2	no	—
25	C₂×#11929608	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
27	#14579120	—	2	2	no	—
28	#14579120	—	2	2	no	—
29	#14579120	—	2	2	no	—
30	#14579120	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₇₉	—	2	2	no	—
32	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄×Dic₇₉	—	2	2	no	—
34	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₇₉	—	2	1	no	—
36	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	#117×C₇₉	—	2	2	no	—
38	#117×C₇₉	—	2	2	no	—
39	#117×C₇₉	—	2	1	no	—
40	#117×C₇₉	—	2	1	no	—
41	#117×C₇₉	D₄	1	2	yes	—
42	#117×C₇₉	Q₈	1	2	yes	—
43	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
48	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
53	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₉	D₄	1	2	yes	—
55	#95×C₇₉	—	2	4	no	—
56	C₂²×Dic₇₉	—	4	2	no	—
57	#11929608	—	4	2	no	—
58	#11929608	—	4	2	no	—
59	#11929608	—	4	2	no	—
60	#11929608	—	4	2	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	—	2	4	no	—
63	C₂×C₇₉⋊C₈	—	2	4	no	—
64	C₄⋊Dic₇₉	—	2	4	no	—
65	C₄⋊Dic₇₉	—	4	2	no	—
66	C₂²⋊Dic₇₉	—	2	4	no	—
67	C₂²⋊Dic₇₉	—	4	2	no	—
68	C₄×Dic₇₉	—	2	4	no	—
69	C₄×Dic₇₉	—	4	2	no	—
70	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	—	2	2	no	—
72	C₂²×C₄×C₇₉	—	2	2	no	—
73	C₂²×C₄×C₇₉	—	4	2	no	—
74	C₂²×C₄×C₇₉	—	8	2	no	—
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	#41	1	2	yes	—
86	C₂²×C₄×C₇₉	#41	1	2	yes	—
87	C₄⋊C₄×C₇₉	—	2	2	no	—
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	—	2	4	no	—
92	C₄⋊C₄×C₇₉	—	4	1	no	—
93	C₄⋊C₄×C₇₉	—	4	1	no	—
94	C₄⋊C₄×C₇₉	—	8	4	no	—
95	C₂²⋊C₄×C₇₉	—	2	2	no	—
96	C₂²⋊C₄×C₇₉	—	2	4	no	—
97	C₂²⋊C₄×C₇₉	—	2	2	no	—
98	C₄²×C₇₉	—	2	2	no	—
99	C₄²×C₇₉	—	2	2	no	—
100	C₄²×C₇₉	—	2	2	no	—
101	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
102	C₂³×C₇₉	—	4	1	no	—
103	C₂³×C₇₉	—	4	1	no	—
104	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₇₉	#121	1	2	yes	—
106	C₂³×C₇₉	#121	1	1	yes	characteristic
107	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
108	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	—	2	2	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	4	2	no	—
116	C₂×C₄×C₇₉	—	4	4	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	—	8	4	no	—
119	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#121	1	2	yes	—
126	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
130	C₂×C₄×C₇₉	#123	1	2	yes	—
131	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#125	1	2	yes	—
133	C₂×C₄×C₇₉	#125	1	2	yes	—
134	C₂×C₄×C₇₉	#126	1	2	yes	—
135	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
137	C₂×Dic₇₉	—	4	4	no	—
138	C₂×Dic₇₉	—	8	2	no	—
139	C₂×Dic₇₉	—	8	2	no	—
140	C₇₉⋊C₈	—	4	4	no	—
141	C₂²×C₇₉	—	2	1	no	—
142	C₂²×C₇₉	—	2	2	no	—
143	C₂²×C₇₉	—	2	1	no	—
144	C₂²×C₇₉	—	2	2	no	—
145	C₂²×C₇₉	—	4	1	no	—
146	C₂²×C₇₉	—	4	2	no	—
147	C₂²×C₇₉	—	4	1	no	—
148	C₂²×C₇₉	#380	1	1	yes	characteristic
149	C₂²×C₇₉	#386	1	1	yes	characteristic
150	C₂²×C₇₉	#391	1	1	yes	characteristic
151	C₂²×C₇₉	#396	1	2	yes	—
152	C₂²×C₇₉	#398	1	2	yes	—
153	C₄×C₇₉	—	4	4	no	—
154	C₄×C₇₉	—	4	2	no	—
155	C₄×C₇₉	—	4	2	no	—
156	C₄×C₇₉	—	8	2	no	—
157	C₄×C₇₉	#387	1	2	yes	—
158	C₄×C₇₉	#389	1	2	yes	—
159	C₄×C₇₉	#394	1	1	yes	characteristic
160	C₄×C₇₉	#395	1	1	yes	characteristic
161	C₄×C₇₉	#398	1	1	yes	characteristic
162	C₄×C₇₉	#400	1	1	yes	characteristic
163	Dic₇₉	—	8	4	no	—
164	#11225	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	—	2	2	no	—
166	C₂×C₇₉	—	2	1	no	—
167	C₂×C₇₉	—	2	1	no	—
168	C₂×C₇₉	—	2	2	no	—
169	C₂×C₇₉	#1441	1	1	yes	characteristic
170	C₂×C₇₉	#1909	1	1	yes	characteristic
171	C₂×C₇₉	#1910	1	1	yes	characteristic
172	#2905	—	79	1	no	—
173	#2317	D₇₉	1	1	yes	characteristic
174	#1740	—	79	1	no	—
175	#1736	—	79	2	no	—
176	C₂×H(3,ℤ/4)	—	79	2	no	—
177	C₇₉	#11225	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	79	2	no	—
179	C₂×#117	—	79	2	no	—
180	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
181	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
182	C₂²×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
183	#424	—	158	4	no	—
184	#423	—	158	2	no	—
185	#389	—	2	2	no	—
186	#385	—	2	2	no	—
187	#384	Dic₇₉	1	2	yes	complemented
188	#382	Dic₇₉	1	2	yes	complemented
189	#342	—	158	2	no	—
190	#340	—	158	4	no	—
191	H(3,ℤ/4)	—	158	4	no	—
192	C₂³×C₄	—	2	1	no	—
193	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
194	C₂×M₁₆	—	158	2	no	—
195	C₂×M₁₆	—	158	2	no	—
196	C₂×M₁₆	—	158	2	no	—
197	C₂×M₁₆	—	158	2	no	—
198	C₂×M₁₆	—	158	2	no	—
199	C₂×M₁₆	—	158	2	no	—
200	C₂²×C₈	—	79	4	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	1	no	—
204	#117	—	2	1	no	—
205	#117	—	79	4	no	—
206	#117	—	158	2	no	—
207	#117	—	158	2	no	—
208	#117	—	158	2	no	—
209	#117	Dic₁₅₈	1	2	yes	—
210	#117	D₃₁₆	1	2	yes	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	158	2	no	—
217	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	158	2	no	—
222	C₂×C₂²⋊C₄	C₂²⋊D₇₉	1	2	yes	—
223	C₂×C₄²	—	158	2	no	—
224	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
225	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
226	C₄⋊C₈	—	158	8	no	—
227	#95	—	2	4	no	—
228	C₂⁴	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	8	2	no	—
233	C₂²×C₄	—	316	2	no	—
234	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
235	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₇₉	1	2	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
240	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
241	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
242	C₂²×C₄	#11929623	1	2	yes	—
243	C₂²×C₄	#11929624	1	1	yes	characteristic
244	C₂²×C₄	#11929624	1	1	yes	characteristic
245	C₂²×C₄	#11929625	1	2	yes	—
246	M₁₆	—	316	2	no	—
247	M₁₆	—	316	2	no	—
248	M₁₆	—	316	2	no	—
249	M₁₆	—	316	2	no	—
250	C₂×C₈	—	158	4	no	—
251	C₂×C₈	—	158	4	no	—
252	C₂×C₈	—	158	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	8	4	no	—
261	C₄⋊C₄	—	316	2	no	—
262	C₄⋊C₄	—	158	4	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	316	2	no	—
267	C₂²⋊C₄	—	158	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	316	2	no	—
272	C₄²	—	158	4	no	—
273	C₄²	C₄⋊Dic₇₉	1	4	yes	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
277	C₂³	#14579133	1	2	yes	—
278	C₂³	#14579135	1	1	yes	characteristic
279	C₂³	#14579137	1	2	yes	—
280	C₂³	#14579140	1	1	yes	characteristic
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	316	4	no	—
292	C₂×C₄	—	632	2	no	—
293	C₂×C₄	—	632	2	no	—
294	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
295	C₂×C₄	#14579120	1	1	yes	characteristic
296	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
297	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
298	C₂×C₄	#14579129	1	1	yes	characteristic
299	C₂×C₄	#14579135	1	1	yes	characteristic
300	C₂×C₄	#14579136	1	2	yes	—
301	C₂×C₄	#14579139	1	2	yes	—
302	C₂×C₄	#14579140	1	1	yes	characteristic
303	C₂×C₄	#14579141	1	2	yes	—
304	C₂×C₄	#14579142	1	1	yes	characteristic
305	C₂×C₄	#14579146	1	2	yes	—
306	C₂×C₄	#14579150	1	1	yes	characteristic
307	C₂×C₄	#14579151	1	2	yes	—
308	C₂×C₄	#14579152	1	2	yes	—
309	C₂×C₄	#14579156	1	2	yes	—
310	C₂×C₄	#14579159	1	1	yes	characteristic
311	C₂×C₄	#14579160	1	2	yes	—
312	C₈	—	316	4	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	#19098423	1	1	yes	characteristic
321	C₂²	#19098443	1	1	yes	characteristic
322	C₂²	#19098470	1	1	yes	characteristic
323	C₂²	#19098485	1	2	yes	—
324	C₂²	#19098496	1	2	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	632	4	no	—
330	C₄	#19098449	1	2	yes	—
331	C₄	#19098462	1	2	yes	—
332	C₄	#19098480	1	1	yes	characteristic
333	C₄	#19098483	1	1	yes	characteristic
334	C₄	#19098493	1	1	yes	characteristic
335	C₄	#19098502	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	#23761653	1	1	yes	characteristic
341	C₂	#23763923	1	1	yes	characteristic
342	C₂	#23763928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098443
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#2317	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#2317	D₇₉
hypocenter	C₇₉	#11225
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
18. #384×C₇₉	C₄	semidirect
19. #382×C₇₉	C₄	semidirect
177. C₇₉	#11225	semidirect
187. #384	Dic₇₉	semidirect
188. #382	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23771964	2
#23763133	2
#23763119	2
C₂×#19098319	2
C₇₉×#2317	2
#11225	79