Subgroups — #34288505

Order 20224.

778 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771964	C₂	1	2	yes	maximal
2	#23763541	C₂	1	4	yes	maximal, complemented
3	#23763538	C₂	1	4	yes	maximal, complemented
4	#23763127	C₂	1	2	yes	maximal
5	C₂×#19098328	C₂	1	2	yes	maximal
6	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929608	C₂²	1	4	yes	—
8	#19098590	—	2	8	no	—
9	#19098587	C₂²	1	8	yes	—
10	#19098328	C₂²	1	4	yes	—
11	#19098328	C₂²	1	8	yes	—
12	#19098328	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
16	#405×C₇₉	C₄	1	8	yes	—
17	C₂×#97×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
19	C₂×#11929608	—	2	4	no	—
20	C₂×#11929608	—	2	8	no	—
21	C₂×#11929608	—	2	8	no	—
22	C₂×#11929608	C₂³	1	8	yes	—
23	C₂²×C₇₉⋊C₈	—	2	4	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	16	no	—
25	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₇₉	—	2	4	no	—
27	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₇₉	—	2	4	no	—
29	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
33	#117×C₇₉	C₂×C₄	1	4	yes	—
34	#117×C₇₉	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₇₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
38	#97×C₇₉	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	—
40	#11929608	—	2	8	no	—
41	#11929608	—	2	8	no	—
42	#11929608	—	4	8	no	—
43	C₂×C₇₉⋊C₈	—	2	8	no	—
44	C₂×C₇₉⋊C₈	—	4	8	no	—
45	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	#41	1	4	yes	—
51	C₂²×C₄×C₇₉	#41	1	4	yes	—
52	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₇₉	—	2	8	no	—
54	C₂×C₈×C₇₉	—	2	8	no	—
55	C₂×C₈×C₇₉	—	2	2	no	—
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	—	2	4	no	—
58	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
61	C₄⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
62	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
63	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
64	C₄²×C₇₉	C₂×D₄	1	4	yes	—
65	C₂³×C₇₉	—	4	2	no	—
66	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₇₉	#125	1	4	yes	—
68	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	—	4	4	no	—
70	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₇₉	#117	1	4	yes	—
73	C₂×C₄×C₇₉	#117	1	4	yes	—
74	C₂×C₄×C₇₉	#124	1	8	yes	—
75	C₂×C₄×C₇₉	#125	1	4	yes	—
76	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
80	C₈×C₇₉	—	2	2	no	—
81	C₈×C₇₉	—	2	2	no	—
82	C₈×C₇₉	—	2	4	no	—
83	C₇₉⋊C₈	—	4	8	no	—
84	C₂²×C₇₉	—	2	2	no	—
85	C₂²×C₇₉	—	2	4	no	—
86	C₂²×C₇₉	—	4	4	no	—
87	C₂²×C₇₉	#382	1	4	yes	—
88	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₇₉	C₂×#125	1	2	yes	—
90	C₄×C₇₉	—	2	8	no	—
91	C₄×C₇₉	#382	1	4	yes	—
92	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
93	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
94	C₄×C₇₉	C₂×#124	1	2	yes	—
95	#20768	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	2	2	no	—
97	C₂×C₇₉	—	2	4	no	—
98	C₂×C₇₉	C₂×#382	1	1	yes	characteristic
99	C₂×C₇₉	#2916	1	2	yes	—
100	#2905	—	79	2	no	—
101	C₂×#405	D₇₉	1	1	yes	characteristic
102	#1838	—	79	4	no	—
103	#1837	—	79	4	no	—
104	#1738	—	79	2	no	—
105	C₂×#341	—	79	2	no	—
106	C₇₉	#20768	1	1	yes	characteristic, complemented, Sylow
107	C₂²×M₁₆	—	79	4	no	—
108	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
109	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
110	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
111	#424	—	158	8	no	—
112	#423	—	79	8	no	—
113	#405	Dic₇₉	1	8	yes	—
114	C₂×#97	C₂×D₇₉	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
116	#341	—	79	4	no	—
117	#341	—	79	8	no	—
118	#341	—	79	4	no	—
119	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	79	8	no	—
122	C₂×M₁₆	—	158	8	no	—
123	C₂×M₁₆	—	158	8	no	—
124	C₂×M₁₆	—	158	4	no	—
125	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	158	4	no	—
128	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
129	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
132	#117	D₃₁₆	1	4	yes	—
133	#117	C₂×Dic₇₉	1	4	yes	—
134	C₂×C₄⋊C₄	Dic₁₅₈	1	2	yes	—
135	C₂×C₂²⋊C₄	Dic₁₅₈	1	2	yes	—
136	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
137	C₄⋊C₈	—	158	16	no	—
138	#97	C₂×Dic₇₉	1	8	yes	—
139	C₄×C₈	C₂×Dic₇₉	1	8	yes	—
140	C₂⁴	C₂×Dic₁₅₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
142	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
143	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
144	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
145	C₂²×C₄	#11929624	1	4	yes	—
146	C₂²×C₄	#11929624	1	4	yes	—
147	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
148	M₁₆	—	2	8	no	—
149	M₁₆	—	316	8	no	—
150	M₁₆	—	158	8	no	—
151	M₁₆	—	158	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	316	8	no	—
157	C₂×C₈	—	158	8	no	—
158	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
159	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
160	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
161	C₄⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
162	C₂²⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
163	C₄²	C₂²×Dic₇₉	1	4	yes	—
164	C₄²	C₂×D₃₁₆	1	4	yes	—
165	C₂³	—	4	2	no	—
166	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
167	C₂³	#14579153	1	4	yes	—
168	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
172	C₂×C₄	#14579120	1	4	yes	—
173	C₂×C₄	#14579120	1	4	yes	—
174	C₂×C₄	#14579149	1	8	yes	—
175	C₂×C₄	#14579153	1	4	yes	—
176	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂×#11929624	1	2	yes	—
179	C₂×C₄	C₂×#11929624	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	316	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	#19098429	1	4	yes	—
188	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
189	C₂²	C₂×#14579153	1	2	yes	—
190	C₄	—	2	8	no	—
191	C₄	#19098429	1	4	yes	—
192	C₄	C₂×#14579120	1	1	yes	characteristic
193	C₄	C₂×#14579120	1	1	yes	characteristic
194	C₄	C₂×#14579149	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	—	2	2	no	—
197	C₂	C₂×#19098429	1	1	yes	characteristic
198	C₂	#23772011	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	#20768
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #23763541	C₂	semidirect
3. #23763538	C₂	semidirect
106. C₇₉	#20768	semidirect

Maximal subgroups

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

subgroup	index
#23771964	2
#23763541	2
#23763538	2
#23763127	2
C₂×#19098328	2
C₂×#405×C₇₉	2
#20768	79