Subgroups — #34307221

Order 20224.

822 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₇₉	C₂	1	2	yes	maximal
2	#23764159	C₂	1	6	yes	maximal
3	#23763528	C₂	1	3	yes	maximal
4	C₄×#14579095	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₇₉	—	2	2	no	—
7	C₂×#14579095	C₂²	1	6	yes	—
8	#19098462	—	2	6	no	—
9	#19098433	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₇₉	C₂²	1	6	yes	—
11	#19098326	C₂²	1	3	yes	—
12	#19098326	C₂²	1	6	yes	—
13	#19098326	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₇₉	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
18	Q₈×Dic₇₉	—	4	8	no	—
19	C₂×C₄⋊Dic₇₉	—	2	6	no	—
20	C₂×C₄⋊Dic₇₉	—	2	6	no	—
21	C₂×C₄⋊Dic₇₉	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
23	C₂×C₄×Dic₇₉	—	2	6	no	—
24	#14579095	—	2	12	no	—
25	C₇₉⋊#95	—	2	6	no	—
26	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₄×Q₈×C₇₉	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₇₉	D₄	1	3	yes	—
32	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
33	C₄⋊C₈×C₇₉	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₇₉	C₂×C₄	1	12	yes	—
35	C₂²×Dic₇₉	—	2	2	no	—
36	C₂²×Dic₇₉	—	2	6	no	—
37	C₄⋊Dic₇₉	—	2	6	no	—
38	C₄⋊Dic₇₉	—	2	2	no	—
39	C₄⋊Dic₇₉	—	2	6	no	—
40	C₄⋊Dic₇₉	—	4	12	no	—
41	C₄⋊Dic₇₉	—	8	6	no	—
42	C₄⋊Dic₇₉	#41	1	4	yes	—
43	C₄×Dic₇₉	—	4	12	no	—
44	C₂×Q₈×C₇₉	C₄⋊C₄	1	4	yes	—
45	C₂×Q₈×C₇₉	D₈	1	4	yes	—
46	C₂×Q₈×C₇₉	Q₁₆	1	4	yes	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₇₉	—	2	6	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₉	C₂²×C₄	1	6	yes	—
54	C₄⋊C₄×C₇₉	C₄⋊C₄	1	12	yes	—
55	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
56	C₄²×C₇₉	C₂×Q₈	1	6	yes	—
57	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₇₉	#107	1	16	yes	complemented
59	C₂×C₄×C₇₉	—	2	6	no	—
60	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₇₉	C₂×D₈	1	6	yes	—
63	C₂×C₄×C₇₉	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
67	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
68	C₂×Dic₇₉	—	2	4	no	—
69	C₂×Dic₇₉	—	4	2	no	—
70	C₂×Dic₇₉	—	4	2	no	—
71	C₂×Dic₇₉	—	4	6	no	—
72	C₂×Dic₇₉	—	4	6	no	—
73	C₂×Dic₇₉	—	4	6	no	—
74	C₈×C₇₉	—	2	6	no	—
75	C₈×C₇₉	—	2	2	no	—
76	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₇₉	#548	1	2	yes	—
78	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
79	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
80	C₂²×C₇₉	#577	1	2	yes	—
81	C₄×C₇₉	—	2	6	no	—
82	C₄×C₇₉	C₂×#107	1	12	yes	—
83	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
84	C₄×C₇₉	#519	1	1	yes	characteristic
85	C₄×C₇₉	#557	1	2	yes	—
86	Dic₇₉	—	4	4	no	—
87	Dic₇₉	—	8	6	no	—
88	#22740	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	#2351	1	1	yes	characteristic
90	C₂×C₇₉	C₂²×#107	1	1	yes	characteristic
91	C₂×C₇₉	#2912	1	1	yes	characteristic
92	C₂×C₇₉	#3358	1	2	yes	—
93	C₂×C₇₉	#3363	1	2	yes	—
94	C₂×C₈×Q₈	D₇₉	1	1	yes	characteristic
95	Q₈×C₄⋊C₄	—	79	2	no	—
96	#1947	—	79	6	no	—
97	#1834	—	79	3	no	—
98	C₄×#107	—	79	3	no	—
99	C₇₉	#22740	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×Q₈	—	158	2	no	—
101	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
102	C₈×Q₈	Dic₇₉	1	8	yes	complemented
103	C₂×#107	—	79	6	no	—
104	C₂×C₄⋊C₈	C₂×D₇₉	1	3	yes	—
105	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
106	#389	—	158	6	no	—
107	#384	—	79	6	no	—
108	C₄×C₄⋊C₄	—	79	6	no	—
109	#340	—	79	3	no	—
110	#340	—	79	6	no	—
111	#340	—	79	1	no	—
112	C₂²×Q₈	D₃₁₆	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₉	1	3	yes	—
115	C₄×Q₈	—	316	8	no	—
116	C₄×Q₈	Dic₁₅₈	1	4	yes	—
117	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
118	C₂×C₄⋊C₄	—	79	2	no	—
119	C₂×C₄⋊C₄	—	79	6	no	—
120	C₂×C₄⋊C₄	—	158	6	no	—
121	C₂×C₄⋊C₄	—	158	6	no	—
122	C₂×C₄⋊C₄	D₃₁₆	1	3	yes	—
123	C₂×C₄²	—	158	6	no	—
124	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
125	#107	—	158	12	no	—
126	C₄⋊C₈	C₂×Dic₇₉	1	12	yes	—
127	C₄×C₈	C₂×Dic₇₉	1	12	yes	—
128	#95	—	158	6	no	—
129	C₂×Q₈	C₄⋊Dic₇₉	1	4	yes	—
130	C₂×Q₈	D₆₃₂	1	4	yes	—
131	C₂×Q₈	Dic₃₁₆	1	4	yes	—
132	C₂²×C₄	—	158	6	no	—
133	C₂²×C₄	—	158	2	no	—
134	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
135	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
136	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
140	C₂×C₈	C₂²×Dic₇₉	1	6	yes	—
141	C₄⋊C₄	—	316	12	no	—
142	C₄⋊C₄	—	79	4	no	—
143	C₄⋊C₄	—	632	6	no	—
144	C₄⋊C₄	—	158	6	no	—
145	C₄⋊C₄	—	158	6	no	—
146	C₄⋊C₄	—	158	2	no	—
147	C₄⋊C₄	C₄⋊Dic₇₉	1	12	yes	—
148	C₄²	—	316	12	no	—
149	C₄²	C₂²×Dic₇₉	1	6	yes	—
150	C₄²	C₂×Dic₁₅₈	1	6	yes	—
151	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
152	Q₈	#14579095	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	316	2	no	—
155	C₂×C₄	—	316	2	no	—
156	C₂×C₄	—	316	6	no	—
157	C₂×C₄	—	316	6	no	—
158	C₂×C₄	—	316	6	no	—
159	C₂×C₄	—	158	4	no	—
160	C₂×C₄	C₂×C₄⋊Dic₇₉	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₇₉	1	6	yes	—
162	C₂×C₄	C₂×D₆₃₂	1	6	yes	—
163	C₂×C₄	C₂×Dic₃₁₆	1	6	yes	—
164	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
166	C₂×C₄	#41×D₇₉	1	2	yes	—
167	C₂×C₄	#14579209	1	2	yes	—
168	C₈	—	2	6	no	—
169	C₈	—	2	2	no	—
170	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
171	C₂²	#19099225	1	2	yes	—
172	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
173	C₂²	C₂²×Dic₃₁₆	1	1	yes	characteristic
174	C₂²	#19099375	1	2	yes	—
175	C₄	—	2	6	no	—
176	C₄	—	316	4	no	—
177	C₄	—	632	6	no	—
178	C₄	C₂×#14579095	1	12	yes	—
179	C₄	#41×Dic₇₉	1	1	yes	characteristic
180	C₄	#19099041	1	1	yes	characteristic
181	C₄	#19099289	1	2	yes	—
182	C₂	#23766321	1	1	yes	characteristic
183	C₂	C₂²×#14579095	1	1	yes	characteristic
184	C₂	#23771995	1	1	yes	characteristic
185	C₂	#23776036	1	2	yes	—
186	C₂	#23776079	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₈×Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₉
hypocenter	C₇₉	#22740
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₇₉	C₄	semidirect
58. Q₈×C₇₉	#107	semidirect
99. C₇₉	#22740	semidirect
102. C₈×Q₈	Dic₇₉	semidirect
152. Q₈	#14579095	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₇₉	2
#23764159	2
#23763528	2
C₄×#14579095	2
C₂×C₈×Q₈×C₇₉	2
#22740	79