Subgroups — #34187493

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098968	C₂	1	1	yes	maximal, characteristic
2	#23764348	C₂	1	2	yes	maximal
3	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098311	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	2	yes	—
7	#19098968	—	2	2	no	—
8	#19098968	C₄	1	4	yes	complemented
9	C₂×#14579095	—	2	1	no	—
10	C₂×#14579095	—	2	1	no	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
12	#19098325	—	2	4	no	—
13	#19098316	—	2	2	no	—
14	#19098311	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579161	—	2	2	no	—
23	#14579161	—	2	2	no	—
24	#14579161	—	2	2	no	—
25	#14579161	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	#14579095	—	2	4	no	—
29	#14579095	—	2	4	no	—
30	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₂²×C₈×C₇₉	—	2	1	no	—
33	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
36	C₂×C₄²×C₇₉	—	2	1	no	—
37	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₇₉	—	2	1	no	—
39	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
40	C₄×C₈×C₇₉	—	4	2	no	—
41	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₉	D₄	1	2	yes	—
43	#95×C₇₉	—	2	1	no	—
44	C₂×Dic₁₅₈	—	2	4	no	—
45	C₂×Dic₁₅₈	—	4	2	no	—
46	C₂×Dic₁₅₈	—	8	4	no	—
47	C₂²×Dic₇₉	—	4	2	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂×C₇₉⋊C₈	—	4	4	no	—
50	C₄⋊Dic₇₉	—	2	4	no	—
51	C₄⋊Dic₇₉	—	2	4	no	—
52	C₄⋊Dic₇₉	—	4	2	no	—
53	C₄⋊Dic₇₉	—	4	2	no	—
54	C₂²×C₄×C₇₉	—	2	2	no	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₉	—	2	2	no	—
59	C₂×C₈×C₇₉	—	2	1	no	—
60	C₂×C₈×C₇₉	—	2	1	no	—
61	C₂×C₈×C₇₉	—	4	1	no	—
62	C₂×C₈×C₇₉	—	4	1	no	—
63	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
66	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
67	C₄⋊C₄×C₇₉	—	2	2	no	—
68	C₄⋊C₄×C₇₉	Q₁₆	1	4	yes	—
69	C₄²×C₇₉	—	4	1	no	—
70	C₄²×C₇₉	—	4	1	no	—
71	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₉	#41	1	1	yes	characteristic
75	C₂³×C₇₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	—	2	4	no	—
78	C₂×C₄×C₇₉	—	4	2	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₇₉	#135	1	2	yes	—
90	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
92	C₂×Dic₇₉	—	4	4	no	—
93	C₂×Dic₇₉	—	8	4	no	—
94	Dic₁₅₈	—	8	4	no	—
95	Dic₁₅₈	—	8	4	no	—
96	C₈×C₇₉	—	2	2	no	—
97	C₈×C₇₉	—	4	2	no	—
98	C₈×C₇₉	#103	1	4	yes	—
99	C₇₉⋊C₈	—	8	2	no	—
100	C₂²×C₇₉	#390	1	1	yes	characteristic
101	C₂²×C₇₉	#461	1	1	yes	characteristic
102	C₂²×C₇₉	#462	1	1	yes	characteristic
103	C₂²×C₇₉	#467	1	2	yes	—
104	C₂²×C₇₉	#472	1	2	yes	—
105	C₄×C₇₉	—	2	1	no	—
106	C₄×C₇₉	—	2	1	no	—
107	C₄×C₇₉	—	4	4	no	—
108	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
109	C₄×C₇₉	#419	1	1	yes	characteristic
110	C₄×C₇₉	C₄×Q₁₆	1	2	yes	—
111	C₄×C₇₉	#461	1	1	yes	characteristic
112	C₄×C₇₉	#462	1	1	yes	characteristic
113	C₄×C₇₉	#494	1	1	yes	characteristic
114	C₄×C₇₉	#495	1	1	yes	characteristic
115	Dic₇₉	—	8	4	no	—
116	#9730	—	79	1	no	maximal, Sylow
117	C₂×C₇₉	#1675	1	2	yes	—
118	C₂×C₇₉	#1676	1	2	yes	—
119	C₂×C₇₉	#1969	1	1	yes	characteristic
120	C₂×C₇₉	#1978	1	1	yes	characteristic
121	C₂×C₇₉	#1981	1	1	yes	characteristic
122	C₂×#500	—	79	1	no	—
123	#1987	—	79	2	no	—
124	C₈×C₄⋊C₄	D₇₉	1	1	yes	characteristic
125	C₂×#335	—	79	1	no	—
126	C₂×#333	—	79	2	no	—
127	C₇₉	#9730	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊Q₈	—	79	2	no	—
129	#500	—	79	4	no	—
130	#500	—	158	2	no	—
131	C₂×#107	—	158	1	no	—
132	C₂×#107	—	158	1	no	—
133	C₂×C₄⋊C₈	—	79	2	no	—
134	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
135	C₂×C₄×C₈	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
138	#340	—	158	4	no	—
139	#336	—	2	1	no	—
140	#335	—	158	2	no	—
141	#333	—	158	4	no	—
142	C₂²×Q₈	—	158	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	158	2	no	—
145	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
147	C₄⋊Q₈	—	79	4	no	—
148	C₄⋊Q₈	—	158	2	no	—
149	C₄⋊Q₈	—	158	2	no	—
150	C₄⋊Q₈	—	158	2	no	—
151	C₂×C₄⋊C₄	—	158	2	no	—
152	C₂×C₄⋊C₄	—	158	2	no	—
153	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
156	#107	—	158	4	no	—
157	#107	—	158	4	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	158	2	no	—
160	C₄⋊C₈	—	158	2	no	—
161	C₄⋊C₈	D₃₁₆	1	2	yes	—
162	C₄×C₈	—	4	2	no	—
163	C₄×C₈	C₄×D₇₉	1	2	yes	—
164	C₄×C₈	D₃₁₆	1	2	yes	—
165	#95	—	2	1	no	—
166	C₂×Q₈	—	316	2	no	—
167	C₂×Q₈	—	632	4	no	—
168	C₂×Q₈	—	158	4	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	316	2	no	—
171	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
173	C₂²×C₄	#11929624	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	316	2	no	—
180	C₂×C₈	—	316	4	no	—
181	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
182	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
183	C₂×C₈	#11929612	1	2	yes	—
184	C₂×C₈	#11929614	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	316	2	no	—
187	C₄⋊C₄	—	316	2	no	—
188	C₄⋊C₄	—	158	4	no	—
189	C₄⋊C₄	—	158	4	no	—
190	C₄⋊C₄	Dic₃₁₆	1	4	yes	—
191	C₄²	—	4	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
194	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
195	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
196	C₄²	#11929624	1	1	yes	characteristic
197	C₂³	#14579135	1	1	yes	characteristic
198	Q₈	—	632	4	no	—
199	Q₈	—	632	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	316	4	no	—
205	C₂×C₄	—	632	4	no	—
206	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
208	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
209	C₂×C₄	#14579135	1	1	yes	characteristic
210	C₂×C₄	#14579149	1	1	yes	characteristic
211	C₂×C₄	#14579158	1	1	yes	characteristic
212	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
213	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
214	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
215	C₂×C₄	#14579183	1	2	yes	—
216	C₂×C₄	#14579191	1	1	yes	characteristic
217	C₂×C₄	#14579191	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₈	—	632	2	no	—
221	C₈	#14579084	1	4	yes	—
222	C₂²	#19098465	1	1	yes	characteristic
223	C₂²	#19098768	1	1	yes	characteristic
224	C₂²	#19098773	1	1	yes	characteristic
225	C₂²	#19098804	1	2	yes	—
226	C₂²	#19098833	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	4	no	—
230	C₄	—	632	4	no	—
231	C₄	C₂×#14579084	1	1	yes	characteristic
232	C₄	#19098571	1	1	yes	characteristic
233	C₄	C₄×Dic₃₁₆	1	2	yes	—
234	C₄	#19098768	1	1	yes	characteristic
235	C₄	#19098773	1	1	yes	characteristic
236	C₄	#19098943	1	1	yes	characteristic
237	C₄	#19098947	1	1	yes	characteristic
238	C₂	#23762799	1	2	yes	—
239	C₂	#23762804	1	2	yes	—
240	C₂	#23764261	1	1	yes	characteristic
241	C₂	#23764310	1	1	yes	characteristic
242	C₂	#23764323	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₈×C₄⋊C₄×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#9730
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. #19098968	C₄	semidirect
127. C₇₉	#9730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098968	2
#23764348	2
C₂×#19098316	2
C₂×#19098311	2
C₈×C₄⋊C₄×C₇₉	2
#9730	79