Subgroups — #34203390

Order 20224.

382 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×#19098981	C₂	1	1	yes	maximal, characteristic
2	#23765020	C₂	1	1	yes	maximal, characteristic
3	#23765004	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763201	C₂	1	1	yes	maximal, characteristic
5	#23761490	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579167	C₂²	1	1	yes	characteristic
9	#19098981	—	2	2	no	—
10	#19098981	—	2	2	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂×#14579100	C₄	1	2	yes	complemented
13	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₉	C₄	1	2	yes	complemented
15	#19098545	C₄	1	1	yes	characteristic
16	#19098536	C₄	1	1	yes	characteristic
17	#19098518	C₂²	1	1	yes	characteristic
18	#19098344	C₄	1	4	yes	complemented
19	#19098340	—	2	2	no	—
20	#19098335	C₄	1	1	yes	characteristic
21	#19098333	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
24	#405×C₇₉	C₂²	1	1	yes	characteristic
25	#347×C₇₉	—	2	2	no	—
26	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
27	#14579167	—	2	1	no	—
28	#14579167	—	2	1	no	—
29	#14579167	—	2	1	no	—
30	#14579167	—	2	1	no	—
31	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
32	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
33	C₂×#11929607	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
35	#14579120	C₂×C₄	1	1	yes	characteristic
36	#14579101	—	2	1	no	—
37	#14579101	C₂×C₄	1	2	yes	—
38	#14579100	—	2	4	no	—
39	C₂×C₇₉⋊C₁₆	—	2	1	no	—
40	C₂×C₇₉⋊C₁₆	—	2	1	no	—
41	C₂×C₇₉⋊C₁₆	—	2	1	no	—
42	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₇₉	—	2	4	no	—
45	#14579079	—	2	1	no	—
46	#14579077	—	2	1	no	—
47	#14579075	—	2	2	no	—
48	#14579071	—	2	1	no	—
49	C₈×Dic₇₉	—	2	1	no	—
50	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₇₉	—	2	1	no	—
54	M₃₂×C₇₉	C₂×C₄	1	2	yes	—
55	M₃₂×C₇₉	D₄	1	2	yes	—
56	M₃₂×C₇₉	D₄	1	2	yes	—
57	C₂×C₁₆×C₇₉	—	2	1	no	—
58	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
59	#97×C₇₉	—	2	1	no	—
60	C₄×C₈×C₇₉	—	2	1	no	—
61	#11929624	—	2	1	no	—
62	#11929624	—	2	1	no	—
63	#11929624	—	2	1	no	—
64	#11929624	—	2	1	no	—
65	C₂×Dic₁₅₈	M₁₆	1	1	yes	characteristic
66	C₂×D₃₁₆	M₁₆	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₉	M₁₆	1	2	yes	—
68	C₂×C₄×D₇₉	C₄²	1	2	yes	—
69	#11929608	—	2	1	no	—
70	#11929608	—	2	1	no	—
71	#11929608	—	2	1	no	—
72	#11929608	—	2	1	no	—
73	#11929607	—	2	2	no	—
74	#11929607	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	2	1	no	—
76	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
77	C₈×D₇₉	—	2	2	no	—
78	C₈×D₇₉	—	2	2	no	—
79	C₄⋊Dic₇₉	—	2	1	no	—
80	C₂²⋊Dic₇₉	—	2	1	no	—
81	C₄×Dic₇₉	—	2	1	no	—
82	C₇₉⋊C₁₆	—	2	1	no	—
83	C₇₉⋊C₁₆	—	2	1	no	—
84	C₇₉⋊C₁₆	—	2	2	no	—
85	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₇₉	—	2	1	no	—
87	M₁₆×C₇₉	—	2	1	no	—
88	C₂×C₈×C₇₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₇₉	—	2	1	no	—
96	C₂²⋊C₄×C₇₉	—	2	1	no	—
97	C₄²×C₇₉	—	2	1	no	—
98	C₁₆×C₇₉	—	2	2	no	—
99	C₁₆×C₇₉	C₂²⋊C₄	1	4	yes	—
100	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₇₉	#97	1	2	yes	—
102	C₂×C₄×C₇₉	—	4	1	no	—
103	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
108	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
109	C₂²⋊D₇₉	—	2	2	no	—
110	C₂²⋊D₇₉	—	2	2	no	—
111	C₂×Dic₇₉	—	4	1	no	—
112	C₂×Dic₇₉	#97	1	2	yes	—
113	D₃₁₆	—	2	1	no	—
114	D₃₁₆	—	2	1	no	—
115	C₄×D₇₉	—	2	2	no	—
116	C₄×D₇₉	—	2	2	no	—
117	Dic₁₅₈	—	2	1	no	—
118	Dic₁₅₈	—	2	1	no	—
119	C₈×C₇₉	—	4	1	no	—
120	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₇₉	#117	1	1	yes	characteristic
122	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
124	C₇₉⋊C₈	—	2	2	no	—
125	C₇₉⋊C₈	—	4	1	no	—
126	C₂²×C₇₉	—	4	1	no	—
127	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
128	C₂²×C₇₉	#435	1	1	yes	characteristic
129	C₂²×C₇₉	#435	1	1	yes	characteristic
130	C₂×D₇₉	—	2	2	no	—
131	C₂×D₇₉	—	4	2	no	—
132	C₄×C₇₉	—	8	1	no	—
133	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₇₉	#405	1	1	yes	characteristic
135	C₄×C₇₉	#435	1	1	yes	characteristic
136	C₄×C₇₉	#435	1	1	yes	characteristic
137	Dic₇₉	—	2	2	no	—
138	Dic₇₉	—	8	1	no	—
139	#12617	—	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	#1752	1	1	yes	characteristic
144	D₇₉	—	4	2	no	—
145	C₂×#504	—	79	1	no	—
146	#2115	—	79	1	no	—
147	#2110	—	79	1	no	—
148	#2110	D₇₉	1	1	yes	characteristic, complemented
149	#1757	—	79	1	no	—
150	#1379	—	79	1	no	—
151	#1379	—	79	1	no	—
152	C₇₉	#12617	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	—	79	1	no	—
154	#504	—	158	2	no	—
155	#504	—	158	2	no	—
156	C₂×M₃₂	—	79	2	no	—
157	C₂×M₃₂	—	79	1	no	—
158	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
159	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
160	C₂²×C₁₆	—	79	1	no	—
161	C₂²×C₁₆	—	79	2	no	—
162	#413	—	79	1	no	—
163	#410	—	79	1	no	—
164	#405	—	79	1	no	—
165	#405	C₂×D₇₉	1	1	yes	characteristic
166	#349	—	79	4	no	—
167	#347	—	2	2	no	—
168	#347	—	158	2	no	—
169	#344	—	79	1	no	—
170	#344	—	79	1	no	—
171	C₂×#41	—	79	1	no	—
172	#131	—	158	1	no	—
173	#131	—	158	1	no	—
174	#131	—	158	1	no	—
175	#131	—	158	1	no	—
176	C₂×M₁₆	—	79	1	no	—
177	C₂×M₁₆	—	79	2	no	—
178	C₂×M₁₆	—	79	1	no	—
179	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
180	C₂²×C₈	—	79	2	no	—
181	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
182	#117	—	79	1	no	—
183	#117	C₄×D₇₉	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	—	79	2	no	—
186	M₃₂	—	158	1	no	—
187	M₃₂	—	158	4	no	—
188	M₃₂	C₄×D₇₉	1	2	yes	—
189	M₃₂	D₃₁₆	1	2	yes	—
190	M₃₂	C₂²⋊D₇₉	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	79	1	no	—
193	C₂×C₁₆	—	79	1	no	—
194	C₂×C₁₆	—	158	1	no	—
195	C₂×C₁₆	—	158	1	no	—
196	C₂×C₁₆	—	158	1	no	—
197	C₂×C₁₆	—	158	4	no	—
198	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
199	#101	—	158	1	no	—
200	#100	—	158	1	no	—
201	#99	—	158	2	no	—
202	#97	—	2	1	no	—
203	#97	—	158	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	158	1	no	—
206	#41	—	158	1	no	—
207	#41	—	158	1	no	—
208	#41	—	158	1	no	—
209	#41	—	158	1	no	—
210	C₂×Q₈	—	79	1	no	—
211	C₂×D₄	—	79	2	no	—
212	C₂×D₄	—	79	1	no	—
213	C₂²×C₄	—	79	2	no	—
214	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	158	1	no	—
218	M₁₆	—	158	1	no	—
219	M₁₆	—	158	1	no	—
220	M₁₆	—	158	1	no	—
221	M₁₆	—	158	2	no	—
222	M₁₆	—	158	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	79	2	no	—
225	C₂×C₈	—	158	1	no	—
226	C₂×C₈	—	158	2	no	—
227	C₂×C₈	—	158	2	no	—
228	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
231	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
232	C₂×C₈	#11929624	1	1	yes	characteristic
233	C₂×C₈	#11929624	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	158	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	158	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	158	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	158	1	no	—
242	C₁₆	—	158	1	no	—
243	C₁₆	—	158	2	no	—
244	C₁₆	(C₂×D₇₉)⋊C₄	1	4	yes	—
245	C₂³	—	79	2	no	—
246	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
247	Q₈	—	158	1	no	—
248	Q₈	—	158	1	no	—
249	D₄	—	158	2	no	—
250	D₄	—	158	2	no	—
251	D₄	—	158	1	no	—
252	D₄	—	158	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	316	1	no	—
255	C₂×C₄	—	79	2	no	—
256	C₂×C₄	—	158	2	no	—
257	C₂×C₄	—	158	2	no	—
258	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
262	C₂×C₄	#14579169	1	1	yes	characteristic
263	C₂×C₄	#14579169	1	1	yes	characteristic
264	C₈	—	4	1	no	—
265	C₈	—	316	1	no	—
266	C₈	—	158	2	no	—
267	C₈	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
268	C₈	#14579118	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
270	C₈	C₄×D₃₁₆	1	1	yes	characteristic
271	C₂²	—	4	1	no	—
272	C₂²	—	316	2	no	—
273	C₂²	—	158	2	no	—
274	C₂²	#97×D₇₉	1	1	yes	characteristic
275	C₂²	#19098632	1	1	yes	characteristic
276	C₂²	#19098636	1	1	yes	characteristic
277	C₄	—	8	1	no	—
278	C₄	—	632	1	no	—
279	C₄	—	158	2	no	—
280	C₄	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
281	C₄	#19098520	1	1	yes	characteristic
282	C₄	#19098632	1	1	yes	characteristic
283	C₄	#19098636	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	316	2	no	—
288	C₂	#23763177	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579118
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2110	C₂
socle	C₂×C₇₉	#1752
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2110	D₇₉
hypocenter	C₇₉	#12617
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #23765004	C₂	semidirect
5. #23761490	C₂	semidirect
6. #23761489	C₂	semidirect
7. C₇₉×#2110	C₂	semidirect
12. C₂×#14579100	C₄	semidirect
14. C₂×C₁₆×D₇₉	C₄	semidirect
18. #19098344	C₄	semidirect
148. #2110	D₇₉	semidirect
152. C₇₉	#12617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098981	2
#23765020	2
#23765004	2
#23763201	2
#23761490	2
#23761489	2
C₇₉×#2110	2
#12617	79