Subgroups — #34203571

Order 20224.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#23765147	C₂	1	1	yes	maximal, characteristic
3	#23765127	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098341	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761511	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761503	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1778	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	—
14	C₂×#14579097	C₂²	1	1	yes	characteristic
15	C₂×#14579089	—	2	1	no	—
16	#19098378	C₄	1	2	yes	complemented
17	#19098375	C₄	1	2	yes	complemented
18	#19098369	C₄	1	4	yes	complemented
19	#19098341	—	2	2	no	—
20	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₇₉	—	2	1	no	—
23	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
24	#341×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	C₂×M₁₆×C₇₉	—	2	1	no	—
48	C₂×M₁₆×C₇₉	—	2	1	no	—
49	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
52	#108×C₇₉	—	2	1	no	—
53	#108×C₇₉	—	2	1	no	—
54	#108×C₇₉	C₂×C₄	1	2	yes	—
55	#108×C₇₉	C₂×C₄	1	2	yes	—
56	#97×C₇₉	—	4	1	no	—
57	#97×C₇₉	C₂×C₄	1	2	yes	—
58	#97×C₇₉	D₄	1	2	yes	—
59	C₄×C₈×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	—
70	#11929613	—	2	2	no	—
71	#11929613	—	2	2	no	—
72	D₆₃₂	—	2	1	no	—
73	D₆₃₂	C₄²	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
83	M₁₆×C₇₉	—	2	2	no	—
84	M₁₆×C₇₉	—	2	2	no	—
85	M₁₆×C₇₉	—	4	1	no	—
86	M₁₆×C₇₉	—	4	1	no	—
87	C₂×C₈×C₇₉	—	2	1	no	—
88	C₂×C₈×C₇₉	—	2	1	no	—
89	C₂×C₈×C₇₉	—	4	1	no	—
90	C₂×C₈×C₇₉	—	4	1	no	—
91	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
100	C₂³×C₇₉	#137	1	1	yes	characteristic
101	C₂²×D₇₉	—	4	1	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	8	1	no	—
104	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂²⋊D₇₉	—	4	2	no	—
111	C₂²⋊D₇₉	—	4	2	no	—
112	C₂×Dic₇₉	—	4	1	no	—
113	D₃₁₆	—	2	2	no	—
114	D₃₁₆	—	4	1	no	—
115	C₄×D₇₉	—	4	2	no	—
116	C₄×D₇₉	—	4	2	no	—
117	Dic₁₅₈	—	2	2	no	—
118	Dic₁₅₈	—	4	1	no	—
119	C₈×C₇₉	—	2	2	no	—
120	C₈×C₇₉	—	4	2	no	—
121	C₈×C₇₉	—	4	2	no	—
122	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
123	C₈×C₇₉	#117	1	1	yes	characteristic
124	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
125	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
126	C₇₉⋊C₈	—	4	2	no	—
127	C₂²×C₇₉	—	4	1	no	—
128	C₂²×C₇₉	#417	1	1	yes	characteristic
129	C₂²×C₇₉	#440	1	1	yes	characteristic
130	C₂²×C₇₉	#440	1	1	yes	characteristic
131	C₂×D₇₉	—	4	2	no	—
132	C₂×D₇₉	—	8	2	no	—
133	C₄×C₇₉	—	4	2	no	—
134	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₇₉	#418	1	1	yes	characteristic
136	C₄×C₇₉	#442	1	1	yes	characteristic
137	C₄×C₇₉	#442	1	1	yes	characteristic
138	Dic₇₉	—	4	2	no	—
139	#12651	—	79	1	no	maximal, Sylow
140	C₂×C₇₉	—	2	1	no	—
141	C₂×C₇₉	—	2	1	no	—
142	C₂×C₇₉	—	2	1	no	—
143	C₂×C₇₉	#1762	1	1	yes	characteristic
144	D₇₉	—	8	2	no	—
145	C₂×#444	—	79	1	no	—
146	#2148	—	79	1	no	—
147	#2143	—	79	1	no	—
148	C₂×#347	—	79	1	no	—
149	#1778	D₇₉	1	1	yes	characteristic, complemented
150	#1388	—	79	1	no	—
151	#1385	—	79	1	no	—
152	C₇₉	#12651	1	1	yes	characteristic, complemented, Sylow
153	C₂×#135	—	79	1	no	—
154	C₂×#131	—	158	1	no	—
155	C₂×M₃₂	—	79	1	no	—
156	C₂×M₃₂	—	79	1	no	—
157	#444	—	79	4	no	—
158	#444	—	158	2	no	—
159	C₂×#108	—	79	1	no	—
160	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
161	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
162	C₂×#104	—	158	1	no	—
163	C₄×M₁₆	—	2	1	no	—
164	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
165	#362	—	79	2	no	—
166	#361	—	79	2	no	—
167	#359	—	79	4	no	—
168	#347	—	158	2	no	—
169	#341	—	2	1	no	—
170	C₂×#41	—	158	1	no	—
171	#135	—	79	2	no	—
172	#135	—	79	2	no	—
173	#135	—	158	1	no	—
174	#135	—	158	1	no	—
175	C₂×Q₁₆	—	79	1	no	—
176	C₂×SD₁₆	—	158	1	no	—
177	C₂×D₈	—	79	1	no	—
178	#131	—	316	2	no	—
179	#131	—	158	4	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	316	1	no	—
183	C₂×M₁₆	—	158	1	no	—
184	C₂²×C₈	—	316	1	no	—
185	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
188	M₃₂	—	79	2	no	—
189	M₃₂	—	158	1	no	—
190	M₃₂	—	158	1	no	—
191	M₃₂	—	158	1	no	—
192	C₂×C₁₆	—	79	2	no	—
193	C₂×C₁₆	—	158	1	no	—
194	#108	—	2	1	no	—
195	#108	—	2	1	no	—
196	#108	—	79	2	no	—
197	#108	—	158	1	no	—
198	#108	C₄×D₇₉	1	2	yes	—
199	#108	C₄×D₇₉	1	2	yes	—
200	#104	—	158	4	no	—
201	#97	—	4	1	no	—
202	#97	C₄×D₇₉	1	2	yes	—
203	#97	D₃₁₆	1	2	yes	—
204	C₄×C₈	—	4	1	no	—
205	#41	—	316	1	no	—
206	#41	—	316	1	no	—
207	#41	—	158	2	no	—
208	#41	—	158	2	no	—
209	C₂×Q₈	—	158	1	no	—
210	C₂×D₄	—	316	1	no	—
211	C₂×D₄	—	158	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	316	1	no	—
214	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
215	Q₁₆	—	79	2	no	—
216	Q₁₆	—	158	1	no	—
217	SD₁₆	—	158	2	no	—
218	SD₁₆	—	158	2	no	—
219	D₈	—	79	2	no	—
220	D₈	—	158	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	316	4	no	—
226	M₁₆	—	316	1	no	—
227	M₁₆	—	158	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	316	4	no	—
233	C₂×C₈	—	316	1	no	—
234	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
239	C₂×C₈	#11929624	1	1	yes	characteristic
240	C₂×C₈	#11929624	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
243	C₁₆	—	158	2	no	—
244	C₁₆	—	158	2	no	—
245	C₂³	—	316	1	no	—
246	C₂³	#14579192	1	1	yes	characteristic
247	Q₈	—	316	1	no	—
248	Q₈	—	158	2	no	—
249	D₄	—	316	2	no	—
250	D₄	—	316	2	no	—
251	D₄	—	316	1	no	—
252	D₄	—	158	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	316	1	no	—
256	C₂×C₄	—	316	2	no	—
257	C₂×C₄	—	316	2	no	—
258	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
261	C₂×C₄	#14579186	1	1	yes	characteristic
262	C₂×C₄	#14579186	1	1	yes	characteristic
263	C₂×C₄	#14579187	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	316	2	no	—
268	C₈	C₄²×D₇₉	1	1	yes	characteristic
269	C₈	#14579118	1	1	yes	characteristic
270	C₈	C₄×D₃₁₆	1	1	yes	characteristic
271	C₈	C₄×D₃₁₆	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	316	2	no	—
274	C₂²	—	632	2	no	—
275	C₂²	#19098562	1	1	yes	characteristic
276	C₂²	#19098661	1	1	yes	characteristic
277	C₂²	#19098661	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	316	2	no	—
280	C₄	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
281	C₄	#19098567	1	1	yes	characteristic
282	C₄	#19098673	1	1	yes	characteristic
283	C₄	#19098673	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	632	2	no	—
288	C₂	#23763228	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19098673
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1778	C₂
socle	C₂×C₇₉	#1762
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1778	D₇₉
hypocenter	C₇₉	#12651
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1778

Complements

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

N	H ≅ G/N	type
4. C₂×#19098341	C₂	semidirect
5. #23761511	C₂	semidirect
6. #23761503	C₂	semidirect
7. C₇₉×#1778	C₂	semidirect
16. #19098378	C₄	semidirect
17. #19098375	C₄	semidirect
18. #19098369	C₄	semidirect
149. #1778	D₇₉	semidirect
152. C₇₉	#12651	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
#23765147	2
#23765127	2
C₂×#19098341	2
#23761511	2
#23761503	2
C₇₉×#1778	2
#12651	79