Subgroups — #34275970

Order 20224.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773979	C₂	1	4	yes	maximal, complemented
2	#23773828	C₂	1	2	yes	maximal, complemented
3	#23763018	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763015	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763006	C₂	1	2	yes	maximal, complemented
6	#23762992	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579183	—	2	4	no	—
9	#19099113	C₂²	1	4	yes	—
10	#19098968	C₂²	1	4	yes	—
11	#19098947	—	2	4	no	—
12	#19098943	C₂²	1	2	yes	—
13	#19098940	C₂²	1	4	yes	—
14	#19098940	C₂²	1	2	yes	—
15	#19098936	C₂²	1	4	yes	—
16	#19098910	C₂²	1	8	yes	—
17	#19098799	—	2	8	no	—
18	C₄×D₆₃₂	—	2	8	no	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₈²×C₇₉	C₂²	1	2	yes	—
22	C₈²×C₇₉	C₂²	1	1	yes	characteristic
23	C₈²×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×#11929624	—	2	4	no	—
25	#14579183	—	4	16	no	—
26	C₂×Dic₃₁₆	—	2	4	no	—
27	C₂×#11929613	—	2	8	no	—
28	C₂×D₆₃₂	—	2	8	no	—
29	C₂×D₆₃₂	—	2	4	no	—
30	#14579161	C₂³	1	4	yes	—
31	#14579158	C₂³	1	4	yes	—
32	#14579149	—	2	4	no	—
33	#14579135	—	2	8	no	—
34	C₄×D₃₁₆	—	2	8	no	—
35	#14579095	—	2	8	no	—
36	#14579083	—	2	16	no	—
37	C₂²×C₈×C₇₉	D₄	1	4	yes	—
38	C₂²×C₈×C₇₉	D₄	1	2	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₇₉	D₄	1	4	yes	—
41	C₄×C₈×C₇₉	D₄	1	2	yes	—
42	C₄×C₈×C₇₉	C₂³	1	2	yes	—
43	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₇₉	C₂³	1	2	yes	—
46	#11929624	—	4	16	no	—
47	C₂×Dic₁₅₈	—	2	4	no	—
48	C₂×D₃₁₆	—	2	4	no	—
49	C₂×D₃₁₆	—	2	8	no	—
50	C₂×C₂²⋊D₇₉	—	4	4	no	—
51	C₂×C₄×D₇₉	—	4	4	no	—
52	Dic₃₁₆	—	4	8	no	—
53	#11929613	—	4	16	no	—
54	D₆₃₂	—	4	8	no	—
55	D₆₃₂	—	4	16	no	—
56	C₄⋊Dic₇₉	—	2	8	no	—
57	(C₂×D₇₉)⋊C₄	—	4	8	no	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	—	2	8	no	—
61	C₂×C₈×C₇₉	D₈	1	8	yes	complemented
62	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
66	C₄²×C₇₉	C₂×D₄	1	2	yes	—
67	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
69	C₂³×C₇₉	#127	1	1	yes	characteristic
70	C₂²×D₇₉	—	4	4	no	—
71	C₂×C₄×C₇₉	—	2	4	no	—
72	C₂×C₄×C₇₉	#127	1	2	yes	—
73	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
77	C₂²⋊D₇₉	—	8	8	no	—
78	C₂×Dic₇₉	—	4	4	no	—
79	D₃₁₆	—	4	8	no	—
80	D₃₁₆	—	4	16	no	—
81	C₄×D₇₉	—	8	8	no	—
82	Dic₁₅₈	—	4	8	no	—
83	C₈×C₇₉	—	2	4	no	—
84	C₈×C₇₉	C₂×D₈	1	4	yes	—
85	C₈×C₇₉	C₂×D₈	1	4	yes	—
86	C₈×C₇₉	#135	1	8	yes	complemented
87	C₂²×C₇₉	—	2	2	no	—
88	C₂²×C₇₉	#493	1	2	yes	—
89	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
90	C₂×D₇₉	—	8	4	no	—
91	C₂×D₇₉	—	8	8	no	—
92	C₄×C₇₉	—	2	2	no	—
93	C₄×C₇₉	#493	1	2	yes	—
94	C₄×C₇₉	C₂²×D₈	1	2	yes	—
95	C₄×C₇₉	C₂×#135	1	4	yes	—
96	Dic₇₉	—	8	4	no	—
97	#19603	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	2	no	—
99	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
100	C₂×C₇₉	#3147	1	2	yes	—
101	D₇₉	—	16	4	no	—
102	#3169	—	79	4	no	—
103	#3147	—	79	2	no	—
104	#1713	—	79	1	no	—
105	#1712	—	79	1	no	—
106	#1709	—	79	2	no	—
107	#1706	—	79	4	no	—
108	C₂×C₈²	D₇₉	1	1	yes	characteristic, complemented
109	C₇₉	#19603	1	1	yes	characteristic, complemented, Sylow
110	C₂×#135	—	158	4	no	—
111	#532	—	79	4	no	—
112	#500	—	79	4	no	—
113	#495	—	158	4	no	—
114	#494	—	79	2	no	—
115	#493	—	79	2	no	—
116	#493	—	79	4	no	—
117	#492	—	79	4	no	—
118	#486	—	79	8	no	—
119	#466	—	158	8	no	—
120	C₄×D₈	—	158	8	no	—
121	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
123	C₈²	C₂×D₇₉	1	2	yes	—
124	C₈²	C₂×D₇₉	1	1	yes	characteristic
125	C₈²	C₂×D₇₉	1	1	yes	characteristic
126	C₂×#41	—	158	4	no	—
127	#135	—	316	16	no	—
128	C₂×Q₁₆	—	158	4	no	—
129	C₂×SD₁₆	—	158	8	no	—
130	C₂×D₈	—	158	4	no	—
131	C₂×D₈	—	158	8	no	—
132	C₂²×C₈	D₃₁₆	1	4	yes	—
133	C₂²×C₈	D₃₁₆	1	2	yes	—
134	C₄⋊Q₈	—	79	4	no	—
135	#127	—	79	4	no	—
136	#124	—	158	4	no	—
137	#121	—	158	8	no	—
138	C₄×D₄	—	158	8	no	—
139	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
140	#107	—	158	8	no	—
141	#102	—	158	16	no	—
142	C₄×C₈	D₃₁₆	1	4	yes	—
143	C₄×C₈	D₃₁₆	1	2	yes	—
144	C₄×C₈	C₂²×D₇₉	1	2	yes	—
145	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
147	C₄×C₈	C₂²×D₇₉	1	2	yes	—
148	#41	—	316	16	no	—
149	C₂×Q₈	—	158	4	no	—
150	C₂×D₄	—	316	4	no	—
151	C₂×D₄	—	158	8	no	—
152	C₂×D₄	—	158	4	no	—
153	C₂²×C₄	—	316	4	no	—
154	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
155	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
156	Q₁₆	—	316	8	no	—
157	SD₁₆	—	316	16	no	—
158	D₈	—	316	8	no	—
159	D₈	—	316	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₆₃₂	1	8	yes	complemented
162	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
163	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
164	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
165	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
166	C₄⋊C₄	—	158	8	no	—
167	C₂²⋊C₄	—	316	8	no	—
168	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
169	C₄²	C₂×D₃₁₆	1	2	yes	—
170	C₄²	C₂³×D₇₉	1	1	yes	characteristic
171	C₂³	—	316	4	no	—
172	C₂³	#14579158	1	1	yes	characteristic
173	Q₈	—	316	8	no	—
174	D₄	—	316	8	no	—
175	D₄	—	316	16	no	—
176	D₄	—	632	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	316	4	no	—
179	C₂×C₄	—	632	8	no	—
180	C₂×C₄	#14579158	1	2	yes	—
181	C₂×C₄	#14579158	1	1	yes	characteristic
182	C₂×C₄	C₂×D₆₃₂	1	4	yes	—
183	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₆₃₂	1	4	yes	—
187	C₈	C₂×D₆₃₂	1	4	yes	—
188	C₈	#14579183	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	632	4	no	—
191	C₂²	—	632	8	no	—
192	C₂²	#19098940	1	2	yes	—
193	C₂²	C₂×#14579158	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	632	4	no	—
196	C₄	#19098940	1	2	yes	—
197	C₄	C₂²×D₆₃₂	1	2	yes	—
198	C₄	C₂×#14579183	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	1264	4	no	—
201	C₂	C₂×#19098940	1	1	yes	characteristic
202	C₂	#23773828	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#127
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#19603
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #23773979	C₂	semidirect
2. #23773828	C₂	semidirect
3. #23763018	C₂	semidirect
4. #23763015	C₂	semidirect
5. #23763006	C₂	semidirect
6. #23762992	C₂	semidirect
7. C₂×C₈²×C₇₉	C₂	semidirect
61. C₂×C₈×C₇₉	D₈	semidirect
86. C₈×C₇₉	#135	semidirect
108. C₂×C₈²	D₇₉	semidirect
109. C₇₉	#19603	semidirect
161. C₂×C₈	D₆₃₂	semidirect
188. C₈	#14579183	semidirect

Maximal subgroups

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

subgroup	index
#23773979	2
#23773828	2
#23763018	2
#23763015	2
#23763006	2
#23762992	2
C₂×C₈²×C₇₉	2
#19603	79