Subgroups — #34205142

Order 20224.

456 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	#23776502	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772301	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098348	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765014	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761490	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#2117	C₂	1	1	yes	maximal, characteristic, complemented
8	#19099361	C₄	1	1	yes	characteristic
9	C₂×#14579167	C₂²	1	1	yes	characteristic
10	#19098981	—	2	2	no	—
11	#19098979	—	2	2	no	—
12	C₂×#14579101	C₂²	1	1	yes	characteristic
13	M₃₂×D₇₉	—	2	2	no	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	#19098683	—	2	2	no	—
16	#19098678	—	2	2	no	—
17	#19098611	C₄	1	1	yes	characteristic
18	#19098580	C₄	1	1	yes	characteristic
19	#19098579	C₄	1	1	yes	characteristic
20	#19098518	C₂²	1	1	yes	characteristic
21	#19098348	—	2	2	no	—
22	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
23	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
26	#350×C₇₉	—	2	2	no	—
27	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
28	#14579192	C₈	1	2	yes	—
29	#14579191	—	2	1	no	—
30	#14579187	C₈	1	2	yes	—
31	#14579186	—	2	1	no	—
32	#14579181	—	2	2	no	—
33	#14579167	—	2	1	no	—
34	#14579167	—	2	1	no	—
35	#14579167	D₄	1	2	yes	—
36	#14579167	D₄	1	2	yes	—
37	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
38	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
39	C₂×#11929607	—	2	1	no	—
40	C₂×C₈×D₇₉	—	2	1	no	—
41	#14579120	C₂×C₄	1	1	yes	characteristic
42	#14579101	—	2	1	no	—
43	#14579101	—	2	1	no	—
44	#14579100	—	2	4	no	—
45	C₂×C₇₉⋊C₁₆	—	2	1	no	—
46	C₂×C₇₉⋊C₁₆	—	2	1	no	—
47	C₂×C₇₉⋊C₁₆	—	2	1	no	—
48	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
49	C₁₆×D₇₉	—	2	4	no	—
50	#14579096	—	2	2	no	—
51	#14579088	—	2	2	no	—
52	#14579087	—	2	2	no	—
53	#14579071	—	2	1	no	—
54	C₈×Dic₇₉	—	2	1	no	—
55	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
56	#131×C₇₉	—	2	1	no	—
57	#131×C₇₉	—	2	1	no	—
58	#131×C₇₉	—	2	1	no	—
59	#131×C₇₉	—	2	1	no	—
60	C₂×M₁₆×C₇₉	—	2	1	no	—
61	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	—	2	1	no	—
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	M₃₂×C₇₉	—	2	1	no	—
65	M₃₂×C₇₉	—	2	1	no	—
66	M₃₂×C₇₉	—	2	1	no	—
67	M₃₂×C₇₉	D₄	1	2	yes	—
68	C₂×C₁₆×C₇₉	—	2	1	no	—
69	C₂×C₁₆×C₇₉	—	2	1	no	—
70	#11929624	—	2	1	no	—
71	#11929624	—	2	1	no	—
72	#11929624	C₂²⋊C₄	1	2	yes	—
73	#11929624	C₂×C₈	1	2	yes	—
74	C₂×Dic₁₅₈	M₁₆	1	1	yes	characteristic
75	C₂×D₃₁₆	M₁₆	1	1	yes	characteristic
76	C₂×C₂²⋊D₇₉	—	2	1	no	—
77	C₂×C₄×D₇₉	—	2	1	no	—
78	#11929614	—	2	2	no	—
79	#11929612	—	2	2	no	—
80	#11929611	—	2	2	no	—
81	#11929609	—	2	2	no	—
82	#11929608	—	2	1	no	—
83	#11929608	—	2	1	no	—
84	#11929608	C₂²⋊C₄	1	2	yes	—
85	#11929608	C₂×C₈	1	2	yes	—
86	#11929607	—	2	2	no	—
87	#11929607	—	2	2	no	—
88	C₂×C₇₉⋊C₈	—	2	1	no	—
89	C₂×C₇₉⋊C₈	—	2	1	no	—
90	C₈×D₇₉	—	2	2	no	—
91	C₈×D₇₉	—	2	2	no	—
92	C₄⋊Dic₇₉	—	2	1	no	—
93	C₂²⋊Dic₇₉	—	2	1	no	—
94	C₄×Dic₇₉	—	2	1	no	—
95	C₇₉⋊C₁₆	—	2	2	no	—
96	C₇₉⋊C₁₆	—	4	1	no	—
97	#41×C₇₉	—	2	1	no	—
98	#41×C₇₉	—	2	1	no	—
99	#41×C₇₉	—	2	1	no	—
100	#41×C₇₉	—	2	1	no	—
101	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
102	C₂×D₄×C₇₉	—	2	1	no	—
103	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	—	2	1	no	—
105	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₇₉	—	2	1	no	—
107	M₁₆×C₇₉	—	2	1	no	—
108	M₁₆×C₇₉	—	4	1	no	—
109	M₁₆×C₇₉	—	4	1	no	—
110	C₂×C₈×C₇₉	—	2	1	no	—
111	C₂×C₈×C₇₉	—	4	1	no	—
112	C₂×C₈×C₇₉	—	4	1	no	—
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
119	C₁₆×C₇₉	—	2	2	no	—
120	C₁₆×C₇₉	—	4	1	no	—
121	C₂³×C₇₉	—	2	1	no	—
122	C₂³×C₇₉	#131	1	1	yes	characteristic
123	C₂²×D₇₉	—	2	1	no	—
124	Q₈×C₇₉	—	2	1	no	—
125	Q₈×C₇₉	—	2	1	no	—
126	D₄×C₇₉	—	2	1	no	—
127	D₄×C₇₉	—	2	1	no	—
128	D₄×C₇₉	—	4	1	no	—
129	D₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	2	1	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
139	C₂²⋊D₇₉	—	2	2	no	—
140	C₂²⋊D₇₉	—	2	2	no	—
141	C₂×Dic₇₉	—	2	1	no	—
142	C₂×Dic₇₉	—	4	1	no	—
143	D₃₁₆	—	2	1	no	—
144	D₃₁₆	C₂²⋊C₈	1	2	yes	—
145	C₄×D₇₉	—	2	2	no	—
146	C₄×D₇₉	—	2	2	no	—
147	Dic₁₅₈	—	2	1	no	—
148	Dic₁₅₈	C₂²⋊C₈	1	2	yes	—
149	C₈×C₇₉	—	4	1	no	—
150	C₈×C₇₉	#120	1	1	yes	characteristic
151	C₈×C₇₉	#121	1	1	yes	characteristic
152	C₈×C₇₉	#121	1	1	yes	characteristic
153	C₈×C₇₉	#123	1	1	yes	characteristic
154	C₇₉⋊C₈	—	2	2	no	—
155	C₇₉⋊C₈	—	2	2	no	—
156	C₂²×C₇₉	—	4	1	no	—
157	C₂²×C₇₉	—	4	2	no	—
158	C₂²×C₇₉	—	4	1	no	—
159	C₂²×C₇₉	#408	1	1	yes	characteristic
160	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
161	C₂²×C₇₉	#435	1	1	yes	characteristic
162	C₂×D₇₉	—	2	2	no	—
163	C₂×D₇₉	—	8	1	no	—
164	C₄×C₇₉	—	4	1	no	—
165	C₄×C₇₉	#386	1	1	yes	characteristic
166	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
168	C₄×C₇₉	#435	1	1	yes	characteristic
169	Dic₇₉	—	2	2	no	—
170	Dic₇₉	—	8	1	no	—
171	#12979	—	79	1	no	maximal, Sylow
172	C₂×C₇₉	—	2	1	no	—
173	C₂×C₇₉	—	2	1	no	—
174	C₂×C₇₉	—	2	1	no	—
175	C₂×C₇₉	—	8	1	no	—
176	C₂×C₇₉	#1869	1	1	yes	characteristic
177	D₇₉	—	4	2	no	—
178	#3407	—	79	1	no	—
179	#2955	—	79	1	no	—
180	#2117	D₇₉	1	1	yes	characteristic, complemented
181	C₂×#350	—	79	1	no	—
182	#2113	—	79	1	no	—
183	#1379	—	79	1	no	—
184	#1378	—	79	1	no	—
185	C₇₉	#12979	1	1	yes	characteristic, complemented, Sylow
186	#575	—	79	1	no	—
187	C₂×#131	—	79	1	no	—
188	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
189	#504	—	158	2	no	—
190	#504	—	158	2	no	—
191	C₂×M₃₂	—	79	1	no	—
192	C₂×M₃₂	—	158	2	no	—
193	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
194	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
195	C₂²×C₁₆	—	79	1	no	—
196	#444	—	158	2	no	—
197	#443	—	158	2	no	—
198	#430	—	79	1	no	—
199	#421	—	79	1	no	—
200	#421	—	79	1	no	—
201	#405	—	79	1	no	—
202	#350	—	2	2	no	—
203	#350	—	158	2	no	—
204	C₂²⋊C₁₆	—	158	2	no	—
205	C₂×#41	—	79	1	no	—
206	C₂×#41	C₄×D₇₉	1	1	yes	characteristic
207	#137	—	79	2	no	—
208	#137	—	158	1	no	—
209	AGL(1,ℤ/8)	—	79	2	no	—
210	AGL(1,ℤ/8)	—	158	1	no	—
211	#135	—	158	2	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	2	1	no	—
216	#131	—	79	2	no	—
217	#131	—	79	2	no	—
218	#131	—	158	1	no	—
219	#131	—	158	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	79	1	no	—
222	C₂×M₁₆	—	79	1	no	—
223	C₂×M₁₆	—	158	1	no	—
224	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	158	1	no	—
227	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
228	#117	—	79	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	158	4	no	—
233	M₃₂	—	158	1	no	—
234	M₃₂	—	158	1	no	—
235	M₃₂	C₂²⋊D₇₉	1	2	yes	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	79	2	no	—
239	C₂×C₁₆	—	158	4	no	—
240	C₂×C₁₆	—	158	1	no	—
241	C₂×C₁₆	—	158	1	no	—
242	C₂×C₁₆	—	158	1	no	—
243	#108	—	158	2	no	—
244	#104	—	158	2	no	—
245	#104	—	158	2	no	—
246	#97	—	158	1	no	—
247	C₄×C₈	—	158	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	79	2	no	—
253	#41	—	79	2	no	—
254	#41	—	158	1	no	—
255	#41	—	158	1	no	—
256	C₂×Q₈	—	79	1	no	—
257	C₂×Q₈	C₈×D₇₉	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	79	1	no	—
260	C₂×D₄	—	158	1	no	—
261	C₂×D₄	C₈×D₇₉	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	158	1	no	—
264	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
265	Q₁₆	—	158	2	no	—
266	SD₁₆	—	158	2	no	—
267	SD₁₆	—	158	2	no	—
268	D₈	—	158	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	79	2	no	—
274	M₁₆	—	79	2	no	—
275	M₁₆	—	158	1	no	—
276	M₁₆	—	158	1	no	—
277	M₁₆	—	158	2	no	—
278	M₁₆	—	158	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	158	1	no	—
283	C₂×C₈	—	158	1	no	—
284	C₂×C₈	—	158	2	no	—
285	C₂×C₈	—	158	2	no	—
286	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
287	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
288	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
290	C₂×C₈	#11929623	1	1	yes	characteristic
291	C₂×C₈	#11929624	1	1	yes	characteristic
292	C₄⋊C₄	—	158	1	no	—
293	C₂²⋊C₄	—	158	1	no	—
294	C₄²	—	158	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	4	1	no	—
297	C₁₆	—	316	1	no	—
298	C₁₆	—	158	2	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	158	1	no	—
301	C₂³	#14579169	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	79	2	no	—
305	Q₈	—	158	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	1	no	—
310	D₄	—	79	2	no	—
311	D₄	—	158	2	no	—
312	D₄	—	158	2	no	—
313	D₄	—	158	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	316	1	no	—
318	C₂×C₄	—	158	1	no	—
319	C₂×C₄	—	158	2	no	—
320	C₂×C₄	—	158	2	no	—
321	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
322	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
324	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
325	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
326	C₂×C₄	#14579167	1	1	yes	characteristic
327	C₈	—	4	1	no	—
328	C₈	—	158	2	no	—
329	C₈	—	158	2	no	—
330	C₈	#14579129	1	1	yes	characteristic
331	C₈	#14579132	1	1	yes	characteristic
332	C₈	#14579134	1	1	yes	characteristic
333	C₈	#14579143	1	1	yes	characteristic
334	C₂²	—	4	1	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	632	1	no	—
338	C₂²	—	158	2	no	—
339	C₂²	#19098528	1	1	yes	characteristic
340	C₂²	(C₂²⋊D₇₉)⋊C₈	1	1	yes	characteristic
341	C₂²	#19098632	1	1	yes	characteristic
342	C₄	—	4	1	no	—
343	C₄	—	632	1	no	—
344	C₄	—	158	2	no	—
345	C₄	#19098439	1	1	yes	characteristic
346	C₄	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
347	C₄	C₈×C₂²⋊D₇₉	1	1	yes	characteristic
348	C₄	#19098637	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	8	1	no	—
353	C₂	—	316	2	no	—
354	C₂	#23763689	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579129
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2117	C₂
socle	C₂×C₇₉	#1869
prime radical	#19099361	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2117	D₇₉
hypocenter	C₇₉	#12979
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2117

Complements

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

N	H ≅ G/N	type
1. #23776502	C₂	semidirect
3. C₂×#19098348	C₂	semidirect
4. #23765014	C₂	semidirect
5. #23761490	C₂	semidirect
6. #23761487	C₂	semidirect
7. C₇₉×#2117	C₂	semidirect
180. #2117	D₇₉	semidirect
185. C₇₉	#12979	semidirect

Maximal subgroups

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

subgroup	index
#23776502	2
#23772301	2
C₂×#19098348	2
#23765014	2
#23761490	2
#23761487	2
C₇₉×#2117	2
#12979	79