Subgroups — #34203564

Order 20224.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098686	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098372	C₂	1	1	yes	maximal, characteristic
3	#23765110	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098341	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761508	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1776	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579183	C₂²	1	1	yes	characteristic
9	C₂×#14579167	—	2	1	no	—
10	C₂×#14579101	C₂²	1	1	yes	characteristic
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	#19098686	—	2	2	no	—
13	#19098686	C₄	1	4	yes	complemented
14	C₂×#14579097	C₂²	1	1	yes	characteristic
15	C₂×#14579089	—	2	1	no	—
16	#19098372	C₄	1	4	yes	complemented
17	#19098366	C₄	1	2	yes	complemented
18	#19098363	C₄	1	2	yes	complemented
19	#19098341	—	2	2	no	—
20	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
24	#340×C₇₉	—	2	1	no	—
25	C₂×#11929624	—	2	1	no	—
26	#14579183	—	2	1	no	—
27	#14579183	—	2	1	no	—
28	#14579183	C₂×C₄	1	2	yes	—
29	#14579183	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×#11929613	—	2	1	no	—
32	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
33	#14579167	—	2	4	no	—
34	#14579167	—	4	2	no	—
35	C₂×#11929608	—	2	1	no	—
36	C₂×#11929607	—	4	1	no	—
37	C₂×C₈×D₇₉	—	4	1	no	—
38	#14579101	—	2	1	no	—
39	#14579101	—	2	1	no	—
40	#14579101	—	2	1	no	—
41	#14579101	C₂×C₄	1	2	yes	—
42	C₂×C₇₉⋊C₁₆	—	2	1	no	—
43	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
44	#14579097	—	2	1	no	—
45	#14579097	C₂×C₄	1	2	yes	—
46	#14579089	—	2	4	no	—
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
51	C₂×C₄²×C₇₉	—	4	1	no	—
52	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
53	#107×C₇₉	—	2	1	no	—
54	#107×C₇₉	C₂×C₄	1	2	yes	—
55	#106×C₇₉	—	2	1	no	—
56	#106×C₇₉	C₂×C₄	1	2	yes	—
57	#97×C₇₉	C₂×C₄	1	2	yes	—
58	#97×C₇₉	D₄	1	2	yes	—
59	#95×C₇₉	—	4	1	no	—
60	#11929624	—	2	2	no	—
61	#11929624	—	2	2	no	—
62	#11929624	—	4	1	no	—
63	#11929624	—	4	1	no	—
64	C₂×Dic₁₅₈	—	2	1	no	—
65	C₂×D₃₁₆	—	2	1	no	—
66	C₂×C₂²⋊D₇₉	—	4	1	no	—
67	C₂×C₄×D₇₉	—	4	1	no	—
68	Dic₃₁₆	—	2	1	no	—
69	Dic₃₁₆	C₄²	1	2	yes	complemented
70	#11929613	—	2	2	no	—
71	#11929613	—	2	2	no	—
72	D₆₃₂	—	2	1	no	—
73	D₆₃₂	C₄²	1	2	yes	complemented
74	#11929608	—	2	2	no	—
75	#11929608	—	4	1	no	—
76	#11929607	—	4	4	no	—
77	C₂×C₇₉⋊C₈	—	4	1	no	—
78	C₈×D₇₉	—	4	4	no	—
79	C₇₉⋊C₁₆	—	2	2	no	—
80	C₇₉⋊C₁₆	—	2	2	no	—
81	C₂²×C₄×C₇₉	—	2	1	no	—
82	C₂²×C₄×C₇₉	—	4	1	no	—
83	C₂²×C₄×C₇₉	—	4	1	no	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	—	2	1	no	—
86	C₂×C₈×C₇₉	—	2	1	no	—
87	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₇₉	—	2	2	no	—
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	—	4	1	no	—
97	C₄⋊C₄×C₇₉	—	4	1	no	—
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	—	8	2	no	—
100	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×D₇₉	—	4	1	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	4	1	no	—
105	C₂×C₄×C₇₉	—	4	1	no	—
106	C₂×C₄×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	—	4	2	no	—
108	C₂×C₄×C₇₉	—	8	1	no	—
109	C₂×C₄×C₇₉	—	8	1	no	—
110	C₂×C₄×C₇₉	—	8	1	no	—
111	C₂×C₄×C₇₉	—	8	1	no	—
112	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
118	C₂²⋊D₇₉	—	4	2	no	—
119	C₂²⋊D₇₉	—	4	2	no	—
120	C₂×Dic₇₉	—	4	1	no	—
121	D₃₁₆	—	2	2	no	—
122	D₃₁₆	—	4	1	no	—
123	C₄×D₇₉	—	4	2	no	—
124	C₄×D₇₉	—	4	2	no	—
125	Dic₁₅₈	—	2	2	no	—
126	Dic₁₅₈	—	4	1	no	—
127	C₈×C₇₉	—	2	2	no	—
128	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₇₉	#117	1	1	yes	characteristic
130	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
132	C₇₉⋊C₈	—	4	2	no	—
133	C₂²×C₇₉	—	2	1	no	—
134	C₂²×C₇₉	—	2	1	no	—
135	C₂²×C₇₉	#417	1	1	yes	characteristic
136	C₂²×C₇₉	#442	1	1	yes	characteristic
137	C₂²×C₇₉	#442	1	1	yes	characteristic
138	C₂×D₇₉	—	4	2	no	—
139	C₂×D₇₉	—	8	1	no	—
140	C₂×D₇₉	—	8	1	no	—
141	C₄×C₇₉	—	4	2	no	—
142	C₄×C₇₉	—	8	2	no	—
143	C₄×C₇₉	—	8	2	no	—
144	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₇₉	#418	1	1	yes	characteristic
146	C₄×C₇₉	#440	1	1	yes	characteristic
147	C₄×C₇₉	#440	1	1	yes	characteristic
148	Dic₇₉	—	4	2	no	—
149	#12650	—	79	1	no	maximal, Sylow
150	C₂×C₇₉	—	2	1	no	—
151	C₂×C₇₉	—	2	1	no	—
152	C₂×C₇₉	#1762	1	1	yes	characteristic
153	C₂×C₇₉	#2180	1	1	yes	characteristic
154	C₂×C₇₉	AGL(1,ℤ/16)	1	1	yes	characteristic
155	D₇₉	—	8	2	no	—
156	C₂×#444	—	79	1	no	—
157	C₂×#360	—	79	1	no	—
158	#2139	—	79	1	no	—
159	C₂×#347	—	79	1	no	—
160	#1776	D₇₉	1	1	yes	characteristic, complemented
161	#1387	—	79	1	no	—
162	#1383	—	79	1	no	—
163	C₇₉	#12650	1	1	yes	characteristic, complemented, Sylow
164	C₂×#135	—	79	1	no	—
165	C₂×#131	—	158	1	no	—
166	C₂×M₃₂	—	79	1	no	—
167	C₂×M₃₂	—	79	1	no	—
168	#444	—	79	4	no	—
169	#444	—	158	2	no	—
170	C₂×#108	—	79	1	no	—
171	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
172	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
173	C₂×#104	—	158	1	no	—
174	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
175	C₄×C₄⋊C₄	—	2	1	no	—
176	#360	—	79	4	no	—
177	#358	—	79	2	no	—
178	#357	—	79	2	no	—
179	#347	—	158	2	no	—
180	#340	—	2	1	no	—
181	C₂×#41	—	158	1	no	—
182	#135	—	79	2	no	—
183	#135	—	79	2	no	—
184	#135	—	158	1	no	—
185	#135	—	158	1	no	—
186	C₂×Q₁₆	—	79	1	no	—
187	C₂×SD₁₆	—	158	1	no	—
188	C₂×D₈	—	79	1	no	—
189	#131	—	316	2	no	—
190	#131	—	158	4	no	—
191	C₂×M₁₆	—	316	1	no	—
192	C₂×M₁₆	—	158	1	no	—
193	C₂²×C₈	—	316	1	no	—
194	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄²	—	4	1	no	—
199	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
200	M₃₂	—	79	2	no	—
201	M₃₂	—	158	1	no	—
202	M₃₂	—	158	1	no	—
203	M₃₂	—	158	1	no	—
204	C₂×C₁₆	—	79	2	no	—
205	C₂×C₁₆	—	158	1	no	—
206	#108	—	79	2	no	—
207	#108	—	158	1	no	—
208	#107	—	2	1	no	—
209	#107	C₄×D₇₉	1	2	yes	—
210	#106	—	2	1	no	—
211	#106	C₄×D₇₉	1	2	yes	—
212	#104	—	158	4	no	—
213	#97	C₄×D₇₉	1	2	yes	—
214	#97	D₃₁₆	1	2	yes	—
215	#95	—	4	1	no	—
216	#41	—	316	1	no	—
217	#41	—	316	1	no	—
218	#41	—	158	2	no	—
219	#41	—	158	2	no	—
220	C₂×Q₈	—	158	1	no	—
221	C₂×D₄	—	316	1	no	—
222	C₂×D₄	—	158	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	316	1	no	—
227	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
228	Q₁₆	—	79	2	no	—
229	Q₁₆	—	158	1	no	—
230	SD₁₆	—	158	2	no	—
231	SD₁₆	—	158	2	no	—
232	D₈	—	79	2	no	—
233	D₈	—	158	1	no	—
234	M₁₆	—	316	4	no	—
235	M₁₆	—	316	1	no	—
236	M₁₆	—	158	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	316	4	no	—
240	C₂×C₈	—	316	1	no	—
241	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
245	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
246	C₂×C₈	#11929624	1	1	yes	characteristic
247	C₂×C₈	#11929624	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	8	2	no	—
254	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
255	C₁₆	—	158	2	no	—
256	C₁₆	—	158	2	no	—
257	C₂³	—	316	1	no	—
258	C₂³	#14579187	1	1	yes	characteristic
259	Q₈	—	316	1	no	—
260	Q₈	—	158	2	no	—
261	D₄	—	316	2	no	—
262	D₄	—	316	2	no	—
263	D₄	—	316	1	no	—
264	D₄	—	158	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	316	1	no	—
275	C₂×C₄	—	316	2	no	—
276	C₂×C₄	—	316	2	no	—
277	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
279	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
280	C₂×C₄	#14579186	1	1	yes	characteristic
281	C₂×C₄	#14579186	1	1	yes	characteristic
282	C₂×C₄	#14579192	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	316	2	no	—
285	C₈	C₄²×D₇₉	1	1	yes	characteristic
286	C₈	#14579118	1	1	yes	characteristic
287	C₈	C₄×D₃₁₆	1	1	yes	characteristic
288	C₈	C₄×D₃₁₆	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	316	2	no	—
292	C₂²	—	632	1	no	—
293	C₂²	—	632	1	no	—
294	C₂²	#19098562	1	1	yes	characteristic
295	C₂²	#19098673	1	1	yes	characteristic
296	C₂²	#19098673	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	316	2	no	—
301	C₄	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
302	C₄	#19098567	1	1	yes	characteristic
303	C₄	#19098661	1	1	yes	characteristic
304	C₄	#19098661	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	632	2	no	—
308	C₂	#23763228	1	1	yes	characteristic
309	C₂	#23765289	1	1	yes	characteristic
310	C₂	#23765296	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098673
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1776	C₂
socle	C₂²×C₇₉	#442
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1776	D₇₉
hypocenter	C₇₉	#12650
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1776

Complements

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

N	H ≅ G/N	type
4. C₂×#19098341	C₂	semidirect
5. #23761508	C₂	semidirect
6. #23761498	C₂	semidirect
7. C₇₉×#1776	C₂	semidirect
13. #19098686	C₄	semidirect
16. #19098372	C₄	semidirect
17. #19098366	C₄	semidirect
18. #19098363	C₄	semidirect
69. Dic₃₁₆	C₄²	semidirect
73. D₆₃₂	C₄²	semidirect
160. #1776	D₇₉	semidirect
163. C₇₉	#12650	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098686	2
C₂×#19098372	2
#23765110	2
C₂×#19098341	2
#23761508	2
#23761498	2
C₇₉×#1776	2
#12650	79