Subgroups — #34298650

Order 20224.

906 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765807	C₂	1	2	yes	maximal
2	#23764159	C₂	1	8	yes	maximal, complemented
3	#23763429	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579120	—	2	2	no	—
6	C₂×#14579095	C₂²	1	8	yes	—
7	#19098433	C₂²	1	4	yes	—
8	#19098429	—	2	2	no	—
9	#19098423	—	2	2	no	—
10	#19098423	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₇₉	—	2	2	no	—
12	C₄×C₄⋊Dic₇₉	C₂²	1	4	yes	—
13	#19098326	C₂²	1	8	yes	—
14	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₇₉	C₄	1	8	yes	complemented
17	C₂×C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
18	#14579120	—	4	8	no	—
19	C₂×C₄⋊Dic₇₉	—	2	4	no	—
20	C₂×C₄⋊Dic₇₉	—	2	4	no	—
21	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
22	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
23	C₂×C₄×Dic₇₉	—	2	4	no	—
24	#14579095	—	4	8	no	—
25	C₇₉⋊#95	—	2	4	no	—
26	C₇₉⋊#95	—	2	4	no	—
27	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₇₉	—	4	4	no	—
31	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₇₉	D₄	1	2	yes	—
35	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
36	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
37	C₄⋊C₈×C₇₉	C₂×C₄	1	16	yes	—
38	C₂²×Dic₇₉	—	2	8	no	—
39	C₄⋊Dic₇₉	—	2	8	no	—
40	C₄⋊Dic₇₉	—	2	8	no	—
41	C₄⋊Dic₇₉	—	4	8	no	—
42	C₄⋊Dic₇₉	—	4	8	no	—
43	C₂²⋊Dic₇₉	—	4	8	no	—
44	C₄×Dic₇₉	—	4	8	no	—
45	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	—	2	2	no	—
47	C₂²×C₄×C₇₉	—	2	2	no	—
48	C₂²×C₄×C₇₉	—	4	2	no	—
49	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₇₉	D₈	1	4	yes	—
52	C₂²×C₄×C₇₉	Q₁₆	1	4	yes	—
53	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₇₉	—	2	8	no	—
60	C₂×C₈×C₇₉	—	4	8	no	—
61	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
62	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
63	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
64	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
65	C₂³×C₇₉	—	2	2	no	—
66	C₂³×C₇₉	—	2	2	no	—
67	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₇₉	C₂×D₈	1	2	yes	—
69	C₂³×C₇₉	C₂×Q₁₆	1	2	yes	—
70	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	2	4	no	—
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	—	2	4	no	—
76	C₂×C₄×C₇₉	—	4	4	no	—
77	C₂×C₄×C₇₉	#107	1	16	yes	complemented
78	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
82	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
83	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	H(5,𝔽₂)	1	2	yes	—
86	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
87	C₂×Dic₇₉	—	4	8	no	—
88	C₂×Dic₇₉	—	4	8	no	—
89	C₂×Dic₇₉	—	4	8	no	—
90	C₈×C₇₉	—	2	8	no	—
91	C₂²×C₇₉	—	2	2	no	—
92	C₂²×C₇₉	—	2	2	no	—
93	C₂²×C₇₉	—	2	4	no	—
94	C₂²×C₇₉	—	2	4	no	—
95	C₂²×C₇₉	C₂×#107	1	4	yes	—
96	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₇₉	#537	1	2	yes	—
98	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
99	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
100	C₂²×C₇₉	#575	1	2	yes	—
101	C₄×C₇₉	—	2	2	no	—
102	C₄×C₇₉	—	2	4	no	—
103	C₄×C₇₉	C₂×#107	1	8	yes	—
104	C₄×C₇₉	#518	1	1	yes	characteristic
105	C₄×C₇₉	#519	1	1	yes	characteristic
106	C₄×C₇₉	#544	1	2	yes	—
107	Dic₇₉	—	8	8	no	—
108	#21961	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	—	2	4	no	—
110	C₂×C₇₉	#2316	1	1	yes	characteristic
111	C₂×C₇₉	C₂²×#107	1	1	yes	characteristic
112	C₂×C₇₉	#2911	1	1	yes	characteristic
113	C₂×C₇₉	#3245	1	2	yes	—
114	C₂×C₇₉	#3249	1	2	yes	—
115	C₂²×C₄⋊C₈	D₇₉	1	1	yes	characteristic
116	#2285	—	79	2	no	—
117	#1947	—	79	8	no	—
118	#1808	—	79	4	no	—
119	C₇₉	#21961	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₈	C₂×D₇₉	1	2	yes	—
121	C₂×#117	—	158	2	no	—
122	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
123	C₂×#107	—	79	8	no	—
124	C₂×C₄⋊C₈	Dic₇₉	1	8	yes	complemented
125	C₂×C₄⋊C₈	C₂×D₇₉	1	4	yes	—
126	#384	—	79	4	no	—
127	#382	—	158	2	no	—
128	#380	—	79	4	no	—
129	#380	—	158	2	no	—
130	C₄×C₄⋊C₄	—	79	4	no	—
131	C₄×C₄⋊C₄	—	158	2	no	—
132	#340	—	79	8	no	—
133	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
134	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
136	C₂²×C₈	—	4	4	no	—
137	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
138	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
139	#117	—	316	8	no	—
140	C₂×C₄⋊C₄	—	79	8	no	—
141	C₂×C₄⋊C₄	—	158	4	no	—
142	C₂×C₄⋊C₄	—	158	4	no	—
143	C₂×C₂²⋊C₄	—	158	4	no	—
144	C₂×C₄²	—	158	4	no	—
145	C₂×C₄²	Dic₁₅₈	1	4	yes	—
146	C₂×C₄²	D₃₁₆	1	2	yes	—
147	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
148	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
149	#107	—	316	8	no	—
150	C₄⋊C₈	C₂×Dic₇₉	1	16	yes	—
151	#95	—	158	4	no	—
152	#95	—	158	4	no	—
153	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	158	8	no	—
158	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
159	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
160	C₂²×C₄	D₆₃₂	1	4	yes	—
161	C₂²×C₄	Dic₃₁₆	1	4	yes	—
162	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
163	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
165	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
166	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
167	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	4	8	no	—
170	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
171	C₄⋊C₄	—	316	8	no	—
172	C₄⋊C₄	—	316	8	no	—
173	C₄⋊C₄	—	158	8	no	—
174	C₄⋊C₄	—	158	8	no	—
175	C₂²⋊C₄	—	316	8	no	—
176	C₄²	—	316	8	no	—
177	C₄²	C₄⋊Dic₇₉	1	8	yes	—
178	C₄²	C₂²×Dic₇₉	1	4	yes	—
179	C₄²	C₂×Dic₁₅₈	1	4	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
183	C₂³	C₂×D₆₃₂	1	2	yes	—
184	C₂³	C₂×Dic₃₁₆	1	2	yes	—
185	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	316	8	no	—
193	C₂×C₄	—	316	8	no	—
194	C₂×C₄	—	316	8	no	—
195	C₂×C₄	#14579095	1	16	yes	complemented
196	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
197	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
198	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
199	C₂×C₄	C₂×D₆₃₂	1	4	yes	—
200	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
201	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
202	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
203	C₂×C₄	#14579206	1	2	yes	—
204	C₂×C₄	#14579209	1	2	yes	—
205	C₈	—	2	8	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	C₂×#14579095	1	4	yes	—
211	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
212	C₂²	#19099144	1	2	yes	—
213	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
214	C₂²	C₂²×Dic₃₁₆	1	1	yes	characteristic
215	C₂²	#19099363	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	632	8	no	—
219	C₄	C₂×#14579095	1	8	yes	—
220	C₄	#19099038	1	1	yes	characteristic
221	C₄	#19099041	1	1	yes	characteristic
222	C₄	#19099194	1	2	yes	—
223	C₂	—	2	4	no	—
224	C₂	#23766018	1	1	yes	characteristic
225	C₂	C₂²×#14579095	1	1	yes	characteristic
226	C₂	#23771990	1	1	yes	characteristic
227	C₂	#23774710	1	2	yes	—
228	C₂	#23774745	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×C₄⋊C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#21961
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #23764159	C₂	semidirect
16. C₂×C₄⋊C₈×C₇₉	C₄	semidirect
77. C₂×C₄×C₇₉	#107	semidirect
119. C₇₉	#21961	semidirect
124. C₂×C₄⋊C₈	Dic₇₉	semidirect
195. C₂×C₄	#14579095	semidirect

Maximal subgroups

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

subgroup	index
#23765807	2
#23764159	2
#23763429	2
C₂²×C₄⋊C₈×C₇₉	2
#21961	79