Subgroups — #34186968

Order 20224.

298 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	#23764175	C₂	1	2	yes	maximal
2	#23762156	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
4	#23761296	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579096	—	2	2	no	—
7	#19098587	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	#19098327	—	2	2	no	—
10	#19098314	—	2	1	no	—
11	#19098314	—	2	1	no	—
12	#19098314	C₄	1	2	yes	—
13	#19098314	C₄	1	2	yes	—
14	#431×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#322×C₇₉	—	2	1	no	—
18	C₈²×C₇₉	—	2	1	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	#14579096	—	4	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
27	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₉	—	2	1	no	—
31	C₂²⋊C₈×C₇₉	Q₈	1	2	yes	—
32	#97×C₇₉	—	2	1	no	—
33	C₄×C₈×C₇₉	—	2	1	no	—
34	C₄×C₈×C₇₉	—	2	1	no	—
35	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
39	#11929608	—	4	2	no	—
40	#11929608	—	4	2	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₂×C₇₉⋊C₈	—	4	2	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	M₁₆×C₇₉	—	2	2	no	—
49	M₁₆×C₇₉	SD₁₆	1	4	yes	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	—	2	2	no	—
53	C₂×C₈×C₇₉	—	4	1	no	—
54	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₇₉	#41	1	1	yes	characteristic
61	C₄²×C₇₉	#41	1	1	yes	characteristic
62	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	4	1	no	—
66	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₇₉	#135	1	2	yes	—
74	C₈×C₇₉	—	2	1	no	—
75	C₈×C₇₉	—	2	1	no	—
76	C₈×C₇₉	—	2	1	no	—
77	C₈×C₇₉	—	2	4	no	—
78	C₈×C₇₉	—	4	2	no	—
79	C₈×C₇₉	#108	1	2	yes	complemented
80	C₇₉⋊C₈	—	4	4	no	—
81	C₇₉⋊C₈	—	8	2	no	—
82	C₂²×C₇₉	—	2	1	no	—
83	C₂²×C₇₉	—	2	1	no	—
84	C₂²×C₇₉	#389	1	1	yes	characteristic
85	C₂²×C₇₉	#498	1	1	yes	characteristic
86	C₂²×C₇₉	#499	1	1	yes	characteristic
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	—	2	1	no	—
90	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
91	C₄×C₇₉	#430	1	1	yes	characteristic
92	C₄×C₇₉	C₄×SD₁₆	1	2	yes	—
93	C₄×C₇₉	#465	1	2	yes	—
94	#9623	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	2	1	no	—
96	C₂×C₇₉	—	2	1	no	—
97	C₂×C₇₉	#1941	1	1	yes	characteristic
98	C₂×C₇₉	#1950	1	1	yes	characteristic
99	C₂×C₇₉	#1953	1	1	yes	characteristic
100	#1950	—	79	2	no	—
101	#1566	—	79	1	no	—
102	C₂×#334	—	79	1	no	—
103	C₈×M₁₆	D₇₉	1	1	yes	characteristic
104	#1301	—	79	2	no	—
105	C₇₉	#9623	1	1	yes	characteristic, complemented, Sylow
106	#431	C₂×D₇₉	1	1	yes	characteristic
107	C₂×#108	—	158	2	no	—
108	#423	—	79	2	no	—
109	C₂×C₄⋊C₈	—	79	2	no	—
110	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
112	#341	—	158	2	no	—
113	#334	—	79	2	no	—
114	#334	—	79	2	no	—
115	#334	—	158	1	no	—
116	#334	—	158	1	no	—
117	#322	—	2	1	no	—
118	C₈²	—	2	1	no	—
119	C₂×M₁₆	—	158	2	no	—
120	C₂×M₁₆	D₃₁₆	1	2	yes	—
121	C₂²×C₈	—	158	2	no	—
122	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
123	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
125	#108	—	316	4	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	—	79	4	no	—
128	C₄⋊C₈	—	158	2	no	—
129	C₄⋊C₈	—	158	2	no	—
130	C₄⋊C₈	—	158	2	no	—
131	C₂²⋊C₈	Dic₁₅₈	1	2	yes	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	316	2	no	—
144	M₁₆	—	316	2	no	—
145	M₁₆	#11929613	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	316	2	no	—
151	C₂×C₈	—	316	2	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	158	4	no	—
154	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
159	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
160	C₄²	#11929624	1	1	yes	characteristic
161	C₄²	#11929624	1	1	yes	characteristic
162	C₂³	#14579161	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
169	C₂×C₄	#14579135	1	1	yes	characteristic
170	C₂×C₄	#14579140	1	1	yes	characteristic
171	C₂×C₄	#14579153	1	1	yes	characteristic
172	C₂×C₄	C₂×#11929613	1	2	yes	—
173	C₂×C₄	#14579183	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	2	no	—
179	C₈	—	316	4	no	—
180	C₈	—	632	2	no	—
181	C₈	#14579096	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#19098461	1	1	yes	characteristic
185	C₂²	#19098961	1	1	yes	characteristic
186	C₂²	#19098965	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#14579096	1	1	yes	characteristic
191	C₄	#19098611	1	1	yes	characteristic
192	C₄	C₄×#11929613	1	2	yes	—
193	C₄	#19098792	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#23764128	1	1	yes	characteristic
197	C₂	#23764175	1	1	yes	characteristic
198	C₂	#23764190	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₈×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₈×M₁₆	D₇₉
hypocenter	C₇₉	#9623
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
79. C₈×C₇₉	#108	semidirect
105. C₇₉	#9623	semidirect
181. C₈	#14579096	semidirect

Maximal subgroups

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

subgroup	index
#23764175	2
#23762156	2
C₂×#19098314	2
#23761296	2
C₈×M₁₆×C₇₉	2
#9623	79