Subgroups — #34206855

Order 20224.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765156	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579093	C₂	1	1	yes	maximal, characteristic
3	#23761505	C₂	1	4	yes	maximal
4	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₂²	1	2	yes	—
6	C₂×#14579093	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₇₉	—	2	1	no	—
8	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
10	#19098326	—	2	1	no	—
11	#533×C₇₉	—	2	1	no	—
12	#500×C₇₉	C₄	1	1	yes	characteristic, complemented
13	#500×C₇₉	C₄	1	1	yes	characteristic, complemented
14	#499×C₇₉	—	2	1	no	—
15	#499×C₇₉	C₄	1	2	yes	complemented
16	#498×C₇₉	—	2	1	no	—
17	#427×C₇₉	—	2	1	no	—
18	#427×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄×Dic₇₉	—	4	1	no	—
22	#14579101	—	2	4	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	#14579093	—	2	2	no	—
25	#14579093	Q₈	1	4	yes	complemented
26	C₇₉⋊#95	—	4	1	no	—
27	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₇₉	—	2	1	no	—
30	C₄⋊Q₈×C₇₉	—	2	1	no	—
31	#125×C₇₉	—	2	2	no	—
32	C₂²⋊Q₈×C₇₉	—	4	1	no	—
33	C₂²⋊Q₈×C₇₉	—	4	1	no	—
34	C₄×Q₈×C₇₉	—	4	1	no	—
35	C₄×Q₈×C₇₉	—	4	1	no	—
36	#117×C₇₉	—	2	2	no	—
37	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
38	#107×C₇₉	—	2	1	no	—
39	#107×C₇₉	—	2	1	no	—
40	#107×C₇₉	C₂×C₄	1	2	yes	—
41	#107×C₇₉	C₂×C₄	1	2	yes	—
42	#106×C₇₉	—	2	2	no	—
43	#106×C₇₉	—	2	2	no	—
44	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	2	4	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₄×Dic₇₉	—	8	2	no	—
52	C₇₉⋊C₁₆	—	4	2	no	—
53	C₂×Q₈×C₇₉	—	4	1	no	—
54	C₂×Q₈×C₇₉	—	4	1	no	—
55	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	—	2	2	no	—
59	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₇₉	#41	1	2	yes	—
65	C₄⋊C₄×C₇₉	—	2	2	no	—
66	C₄⋊C₄×C₇₉	—	2	2	no	—
67	C₄⋊C₄×C₇₉	—	4	2	no	—
68	C₄⋊C₄×C₇₉	—	4	2	no	—
69	C₄⋊C₄×C₇₉	—	4	1	no	—
70	C₄⋊C₄×C₇₉	—	4	1	no	—
71	C₂²⋊C₄×C₇₉	—	4	2	no	—
72	C₄²×C₇₉	—	4	2	no	—
73	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
76	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
77	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₇₉	—	8	2	no	—
79	Q₈×C₇₉	—	8	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	4	1	no	—
82	C₂×C₄×C₇₉	—	4	2	no	—
83	C₂×C₄×C₇₉	—	4	2	no	—
84	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
87	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#137	1	2	yes	—
92	C₂×Dic₇₉	—	4	4	no	—
93	C₂×Dic₇₉	—	8	2	no	—
94	C₂×Dic₇₉	—	8	2	no	—
95	C₈×C₇₉	—	2	1	no	—
96	C₈×C₇₉	—	2	1	no	—
97	C₈×C₇₉	C₄×Q₈	1	2	yes	—
98	C₈×C₇₉	#124	1	1	yes	characteristic
99	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×C₇₉	—	2	1	no	—
101	C₂²×C₇₉	—	2	1	no	—
102	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
103	C₂²×C₇₉	#474	1	2	yes	—
104	C₄×C₇₉	—	2	1	no	—
105	C₄×C₇₉	—	2	1	no	—
106	C₄×C₇₉	—	2	1	no	—
107	C₄×C₇₉	—	8	2	no	—
108	C₄×C₇₉	—	8	2	no	—
109	C₄×C₇₉	#360	1	2	yes	—
110	C₄×C₇₉	#391	1	1	yes	characteristic
111	C₄×C₇₉	#419	1	1	yes	characteristic
112	C₄×C₇₉	#475	1	2	yes	—
113	Dic₇₉	—	8	4	no	—
114	#13278	—	79	1	no	maximal, Sylow
115	C₂×C₇₉	—	2	2	no	—
116	C₂×C₇₉	#1988	1	1	yes	characteristic
117	C₂×C₇₉	C₂×#360	1	1	yes	characteristic
118	C₂×C₇₉	#2145	1	1	yes	characteristic
119	#3160	D₇₉	1	1	yes	characteristic
120	#2150	—	79	1	no	—
121	C₄×#106	—	79	1	no	—
122	#1386	—	79	4	no	—
123	C₇₉	#13278	1	1	yes	characteristic, complemented, Sylow
124	#533	—	2	1	no	—
125	C₂×M₃₂	—	79	2	no	—
126	#500	Dic₇₉	1	1	yes	characteristic, complemented
127	#500	Dic₇₉	1	1	yes	characteristic, complemented
128	#499	—	2	1	no	—
129	#499	Dic₇₉	1	2	yes	complemented
130	#498	—	2	1	no	—
131	#427	—	2	1	no	—
132	#427	C₂×D₇₉	1	2	yes	—
133	C₂×#106	—	79	2	no	—
134	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	158	1	no	—
136	C₄⋊C₁₆	—	158	1	no	—
137	C₄⋊C₁₆	—	158	1	no	—
138	#340	—	158	1	no	—
139	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	1	no	—
142	C₄⋊Q₈	—	2	1	no	—
143	#125	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₂²⋊Q₈	—	4	1	no	—
146	C₄×Q₈	—	4	1	no	—
147	C₄×Q₈	—	4	1	no	—
148	#117	—	2	2	no	—
149	C₂×C₄⋊C₄	—	158	2	no	—
150	C₂×C₄²	—	316	1	no	—
151	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
152	M₃₂	—	158	4	no	—
153	C₂×C₁₆	—	158	2	no	—
154	#107	—	2	1	no	—
155	#107	—	2	1	no	—
156	#107	C₂×Dic₇₉	1	2	yes	—
157	#107	C₂×Dic₇₉	1	2	yes	—
158	#106	—	2	2	no	—
159	#106	—	2	2	no	—
160	#106	—	79	4	no	—
161	#106	—	158	2	no	—
162	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
163	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
166	#95	—	316	1	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂²×C₄	—	316	2	no	—
170	C₂²×C₄	#11929609	1	1	yes	characteristic
171	C₂²×C₄	#11929609	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
176	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
178	C₂×C₈	Q₈×D₇₉	1	2	yes	—
179	C₂×C₈	#11929625	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	316	2	no	—
187	C₄⋊C₄	—	158	4	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	632	2	no	—
191	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
192	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
193	C₄²	#11929612	1	1	yes	characteristic
194	C₄²	#11929612	1	1	yes	characteristic
195	C₁₆	—	316	2	no	—
196	C₂³	C₂×#11929609	1	1	yes	characteristic
197	Q₈	—	8	2	no	—
198	Q₈	—	8	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	316	4	no	—
204	C₂×C₄	—	632	2	no	—
205	C₂×C₄	—	632	2	no	—
206	C₂×C₄	#14579082	1	1	yes	characteristic
207	C₂×C₄	#14579082	1	1	yes	characteristic
208	C₂×C₄	#14579082	1	1	yes	characteristic
209	C₂×C₄	#14579082	1	1	yes	characteristic
210	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
211	C₂×C₄	#14579137	1	2	yes	—
212	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
213	C₂×C₄	#14579192	1	2	yes	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	Q₈×Dic₇₉	1	2	yes	—
217	C₈	#14579150	1	1	yes	characteristic
218	C₈	#14579159	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#14579082	1	1	yes	characteristic
222	C₂²	#19098840	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	—	632	4	no	—
229	C₄	#19098373	1	2	yes	—
230	C₄	#19098470	1	1	yes	characteristic
231	C₄	#19098572	1	1	yes	characteristic
232	C₄	#19098845	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	#23764353	1	1	yes	characteristic
235	C₂	C₂×#19098373	1	1	yes	characteristic
236	C₂	#23765136	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#13278
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
12. #500×C₇₉	C₄	semidirect
13. #500×C₇₉	C₄	semidirect
15. #499×C₇₉	C₄	semidirect
25. #14579093	Q₈	semidirect
123. C₇₉	#13278	semidirect
126. #500	Dic₇₉	semidirect
127. #500	Dic₇₉	semidirect
129. #499	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23765156	2
C₄×#14579093	2
#23761505	2
C₇₉×#3160	2
#13278	79