Subgroups — #34253997

Order 20224.

782 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772366	C₂	1	2	yes	maximal
2	C₄×#14579167	C₂	1	1	yes	maximal, characteristic
3	#23761704	C₂	1	2	yes	maximal, complemented
4	#23761698	C₂	1	2	yes	maximal, complemented
5	#23761689	C₂	1	2	yes	maximal, complemented
6	#23761686	C₂	1	4	yes	maximal, complemented
7	#23761683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579167	C₄	1	2	yes	—
10	C₄×#11929624	C₂²	1	1	yes	characteristic
11	#19098693	C₄	1	4	yes	—
12	#19098642	C₄	1	4	yes	—
13	#19098634	C₄	1	8	yes	—
14	#19098616	C₂²	1	2	yes	—
15	#19098614	C₂²	1	4	yes	—
16	#19098518	C₄	1	2	yes	—
17	#19098517	C₄	1	4	yes	—
18	C₄×#11929608	C₂²	1	1	yes	characteristic
19	C₄×#11929607	C₂²	1	4	yes	—
20	C₄×#11929607	C₂²	1	2	yes	—
21	C₄×C₈×D₇₉	C₂²	1	2	yes	—
22	#19098285	C₂²	1	4	yes	—
23	#19098284	C₂²	1	2	yes	—
24	#19098284	C₂²	1	4	yes	—
25	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
28	#322×C₇₉	C₂²	1	2	yes	—
29	#322×C₇₉	C₂²	1	1	yes	characteristic
30	#322×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
32	#14579167	—	2	8	no	—
33	C₂×#11929608	C₂×C₄	1	2	yes	—
34	C₂×#11929607	C₂×C₄	1	4	yes	—
35	C₂×#11929607	C₂×C₄	1	8	yes	—
36	C₂×C₈×D₇₉	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
38	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
39	C₄×C₂²⋊D₇₉	C₂×C₄	1	4	yes	—
40	#14579120	C₂×C₄	1	1	yes	characteristic
41	#14579118	C₂×C₄	1	2	yes	—
42	C₄²×D₇₉	C₂³	1	2	yes	—
43	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₇₉⋊C₈	C₂×C₄	1	8	yes	—
45	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
46	(C₂×D₇₉)⋊C₈	C₂×C₄	1	8	yes	—
47	#14579072	C₂×C₄	1	2	yes	—
48	#14579071	C₂×C₄	1	8	yes	—
49	#14579071	C₂×C₄	1	4	yes	—
50	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
51	C₄×C₇₉⋊C₈	C₂³	1	4	yes	—
52	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₇₉	C₂³	1	2	yes	—
61	C₄×C₈×C₇₉	C₂³	1	2	yes	—
62	#11929624	—	2	4	no	—
63	C₂×Dic₁₅₈	C₄²	1	1	yes	characteristic
64	C₂×D₃₁₆	C₄²	1	1	yes	characteristic
65	C₂×C₂²⋊D₇₉	C₄²	1	2	yes	—
66	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×D₇₉	C₂²×C₄	1	4	yes	—
68	#11929608	—	2	4	no	—
69	#11929607	—	2	8	no	—
70	#11929607	—	4	8	no	—
71	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
73	C₈×D₇₉	—	2	8	no	—
74	C₄⋊Dic₇₉	C₄²	1	2	yes	—
75	Dic₇₉⋊C₄	C₄²	1	4	yes	—
76	C₂²⋊Dic₇₉	C₄²	1	2	yes	—
77	(C₂×D₇₉)⋊C₄	C₄²	1	4	yes	—
78	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
79	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	—	4	4	no	—
83	C₂×C₈×C₇₉	M₁₆	1	8	yes	complemented
84	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
92	C₂²×D₇₉	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₇₉	—	2	4	no	—
94	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
97	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
99	C₂²⋊D₇₉	—	2	4	no	—
100	C₂×Dic₇₉	C₂×C₄²	1	4	yes	—
101	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
102	D₃₁₆	—	2	2	no	—
103	C₄×D₇₉	—	2	4	no	—
104	C₄×D₇₉	—	2	8	no	—
105	Dic₁₅₈	—	2	2	no	—
106	C₈×C₇₉	—	2	4	no	—
107	C₈×C₇₉	—	2	4	no	—
108	C₈×C₇₉	C₂×M₁₆	1	4	yes	—
109	C₈×C₇₉	C₂×M₁₆	1	4	yes	—
110	C₇₉⋊C₈	—	2	4	no	—
111	C₇₉⋊C₈	—	2	8	no	—
112	C₂²×C₇₉	—	2	2	no	—
113	C₂²×C₇₉	C₄×M₁₆	1	2	yes	—
114	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
115	C₂×D₇₉	—	2	2	no	—
116	C₂×D₇₉	—	2	4	no	—
117	C₄×C₇₉	—	2	2	no	—
118	C₄×C₇₉	C₄×M₁₆	1	2	yes	—
119	C₄×C₇₉	C₂²×M₁₆	1	2	yes	—
120	C₄×C₇₉	#568	1	4	yes	—
121	Dic₇₉	—	2	2	no	—
122	Dic₇₉	—	2	4	no	—
123	#17336	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	—	2	2	no	—
125	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
126	C₂×C₇₉	#2875	1	2	yes	—
127	D₇₉	—	4	2	no	—
128	#2965	—	79	2	no	—
129	C₄×#131	—	79	1	no	—
130	#1455	—	79	2	no	—
131	#1453	—	79	2	no	—
132	#1451	—	79	2	no	—
133	#1451	—	79	4	no	—
134	#1450	—	79	1	no	—
135	C₂×#322	D₇₉	1	1	yes	characteristic, complemented
136	C₇₉	#17336	1	1	yes	characteristic, complemented, Sylow
137	C₂×#131	—	79	2	no	—
138	C₄×#41	—	79	1	no	—
139	#446	—	79	4	no	—
140	#436	—	79	4	no	—
141	#435	—	79	8	no	—
142	#431	—	79	2	no	—
143	#431	—	79	4	no	—
144	#405	—	79	2	no	—
145	#405	—	79	4	no	—
146	C₄×M₁₆	—	79	1	no	—
147	C₄×M₁₆	—	79	2	no	—
148	C₄×M₁₆	—	79	4	no	—
149	C₂×C₄×C₈	—	79	2	no	—
150	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
152	#322	—	79	2	no	—
153	#322	—	79	4	no	—
154	#322	—	79	4	no	—
155	#322	C₂×D₇₉	1	2	yes	—
156	#322	C₂×D₇₉	1	1	yes	characteristic
157	#322	C₂×D₇₉	1	1	yes	characteristic
158	C₈²	—	79	2	no	—
159	C₂×#41	—	79	1	no	—
160	#131	—	158	8	no	—
161	C₂×M₁₆	—	79	2	no	—
162	C₂×M₁₆	—	79	8	no	—
163	C₂×M₁₆	—	79	4	no	—
164	C₂²×C₈	—	79	4	no	—
165	C₂²×C₈	C₄×D₇₉	1	4	yes	—
166	C₂²×C₈	C₄×D₇₉	1	2	yes	—
167	C₄×Q₈	—	79	2	no	—
168	C₄×D₄	—	79	4	no	—
169	C₄×D₄	—	79	2	no	—
170	#117	—	79	1	no	—
171	#117	—	79	2	no	—
172	C₂×C₄²	—	79	2	no	—
173	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
174	C₄⋊C₈	—	79	8	no	—
175	C₄⋊C₈	—	79	4	no	—
176	C₂²⋊C₈	—	79	4	no	—
177	C₂²⋊C₈	—	79	8	no	—
178	#97	—	79	4	no	—
179	#97	—	79	8	no	—
180	#97	—	79	2	no	—
181	C₄×C₈	—	79	4	no	—
182	C₄×C₈	—	79	4	no	—
183	C₄×C₈	—	79	2	no	—
184	C₄×C₈	C₄×D₇₉	1	2	yes	—
185	C₄×C₈	C₄×D₇₉	1	4	yes	—
186	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₇₉	1	2	yes	—
189	C₄×C₈	C₂²×D₇₉	1	2	yes	—
190	#41	—	158	4	no	—
191	C₂×Q₈	—	79	1	no	—
192	C₂×D₄	—	79	2	no	—
193	C₂×D₄	—	79	1	no	—
194	C₂²×C₄	—	79	4	no	—
195	C₂²×C₄	—	79	2	no	—
196	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
198	M₁₆	—	316	8	no	—
199	M₁₆	—	158	4	no	—
200	M₁₆	—	158	8	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	79	4	no	—
203	C₂×C₈	—	79	8	no	—
204	C₂×C₈	—	158	8	no	—
205	C₂×C₈	#11929607	1	8	yes	complemented
206	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
207	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
208	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
210	C₄⋊C₄	—	79	4	no	—
211	C₄⋊C₄	—	79	2	no	—
212	C₂²⋊C₄	—	79	2	no	—
213	C₂²⋊C₄	—	79	4	no	—
214	C₄²	—	79	2	no	—
215	C₄²	—	79	4	no	—
216	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
217	C₄²	C₂×C₄×D₇₉	1	2	yes	—
218	C₄²	C₂³×D₇₉	1	1	yes	characteristic
219	C₂³	—	79	2	no	—
220	C₂³	C₄²×D₇₉	1	1	yes	characteristic
221	Q₈	—	158	2	no	—
222	D₄	—	158	4	no	—
223	D₄	—	158	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	79	4	no	—
226	C₂×C₄	—	79	2	no	—
227	C₂×C₄	—	158	4	no	—
228	C₂×C₄	—	158	8	no	—
229	C₂×C₄	C₄²×D₇₉	1	2	yes	—
230	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
231	C₂×C₄	C₂×#11929607	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	2	4	no	—
236	C₈	—	158	4	no	—
237	C₈	—	158	8	no	—
238	C₈	C₂×#11929607	1	4	yes	—
239	C₈	C₂×#11929607	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	158	2	no	—
242	C₂²	—	158	4	no	—
243	C₂²	C₄×#11929607	1	2	yes	—
244	C₂²	C₂×C₄²×D₇₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	158	2	no	—
247	C₄	—	158	4	no	—
248	C₄	C₄×#11929607	1	2	yes	—
249	C₄	C₂²×#11929607	1	2	yes	—
250	C₄	#19099329	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	316	2	no	—
253	C₂	C₂×C₄×#11929607	1	1	yes	characteristic
254	C₂	#23771825	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×#322×C₇₉	C₂
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#17336
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #23761704	C₂	semidirect
4. #23761698	C₂	semidirect
5. #23761689	C₂	semidirect
6. #23761686	C₂	semidirect
7. #23761683	C₂	semidirect
8. C₂×#322×C₇₉	C₂	semidirect
83. C₂×C₈×C₇₉	M₁₆	semidirect
135. C₂×#322	D₇₉	semidirect
136. C₇₉	#17336	semidirect
205. C₂×C₈	#11929607	semidirect

Maximal subgroups

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

subgroup	index
#23772366	2
C₄×#14579167	2
#23761704	2
#23761698	2
#23761689	2
#23761686	2
#23761683	2
C₂×#322×C₇₉	2
#17336	79