Subgroups — #34206886

Order 20224.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773914	C₂	1	1	yes	maximal, characteristic
2	#23763572	C₂	1	1	yes	maximal, characteristic
3	#23761496	C₂	1	4	yes	maximal
4	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19099120	—	2	1	no	—
6	#19098968	C₄	1	2	yes	—
7	#19098965	—	2	1	no	—
8	#19098965	C₄	1	2	yes	—
9	#19098961	—	2	1	no	—
10	C₂×#14579097	—	2	1	no	—
11	C₂×#14579097	C₂²	1	2	yes	—
12	#19098601	—	2	1	no	—
13	#19098601	C₂²	1	2	yes	—
14	#19098587	—	2	1	no	—
15	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#346×C₇₉	—	2	1	no	—
18	#346×C₇₉	—	2	1	no	—
19	C₂×#11929608	—	2	2	no	—
20	#14579161	—	2	2	no	—
21	#14579153	—	2	2	no	—
22	#14579140	—	4	2	no	—
23	C₄×Dic₁₅₈	—	4	2	no	—
24	#14579120	—	2	2	no	—
25	#14579097	—	2	4	no	—
26	#14579097	—	4	2	no	—
27	#14579095	—	2	2	no	—
28	#14579095	C₂×C₄	1	4	yes	—
29	#14579093	—	2	2	no	—
30	#14579093	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
32	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₉	—	2	2	no	—
36	C₂×C₁₆×C₇₉	—	2	2	no	—
37	C₂×C₁₆×C₇₉	Q₈	1	4	yes	complemented
38	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂×Dic₁₅₈	—	4	2	no	—
43	#11929608	—	4	2	no	—
44	#11929608	—	4	2	no	—
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	C₄⋊Dic₇₉	—	2	4	no	—
47	C₄⋊Dic₇₉	—	4	2	no	—
48	Dic₇₉⋊C₄	—	4	4	no	—
49	C₂²⋊Dic₇₉	—	4	2	no	—
50	C₄×Dic₇₉	—	4	2	no	—
51	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
56	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
57	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₇₉	#41	1	2	yes	—
61	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇₉	D₈	1	2	yes	—
64	C₁₆×C₇₉	—	2	4	no	—
65	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	#102	1	2	yes	—
68	C₂×C₄×C₇₉	#102	1	2	yes	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₇₉	—	4	4	no	—
76	Dic₁₅₈	—	8	4	no	—
77	C₈×C₇₉	#103	1	2	yes	—
78	C₈×C₇₉	#103	1	2	yes	—
79	C₈×C₇₉	C₄×Q₈	1	2	yes	—
80	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
81	C₇₉⋊C₈	—	8	2	no	—
82	C₂²×C₇₉	—	2	2	no	—
83	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
84	C₂²×C₇₉	#487	1	1	yes	characteristic
85	C₂²×C₇₉	#498	1	1	yes	characteristic
86	C₄×C₇₉	—	2	2	no	—
87	C₄×C₇₉	—	2	2	no	—
88	C₄×C₇₉	#391	1	1	yes	characteristic
89	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
90	C₄×C₇₉	#486	1	1	yes	characteristic
91	C₄×C₇₉	#500	1	1	yes	characteristic
92	Dic₇₉	—	8	4	no	—
93	#13286	—	79	1	no	maximal, Sylow
94	C₂×C₇₉	—	2	1	no	—
95	C₂×C₇₉	—	2	1	no	—
96	C₂×C₇₉	#1990	1	1	yes	characteristic
97	C₂×C₇₉	#2143	1	2	yes	—
98	#3160	—	79	1	no	—
99	C₂×#346	D₇₉	1	1	yes	characteristic
100	#1848	—	79	1	no	—
101	#1382	—	79	4	no	—
102	C₇₉	#13286	1	1	yes	characteristic, complemented, Sylow
103	#533	—	158	1	no	—
104	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
105	#500	—	79	2	no	—
106	#499	—	79	2	no	—
107	#499	—	158	1	no	—
108	#498	—	158	1	no	—
109	C₂×#108	—	79	2	no	—
110	C₂×#108	—	158	1	no	—
111	#427	—	79	2	no	—
112	#427	—	158	1	no	—
113	#423	—	158	1	no	—
114	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
115	#346	—	2	1	no	—
116	#346	—	2	1	no	—
117	C₂×M₁₆	—	158	2	no	—
118	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	158	2	no	—
121	#125	—	158	2	no	—
122	C₂²⋊Q₈	—	316	2	no	—
123	C₄×Q₈	—	316	2	no	—
124	#117	—	158	2	no	—
125	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Dic₁₅₈	1	4	yes	complemented
129	#108	—	316	2	no	—
130	#108	—	158	4	no	—
131	#107	—	79	4	no	—
132	#107	—	158	2	no	—
133	#106	—	158	2	no	—
134	#106	—	158	2	no	—
135	C₄⋊C₈	—	316	2	no	—
136	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
137	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂×Q₈	—	316	2	no	—
141	C₂²×C₄	#11929613	1	2	yes	—
142	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
143	M₁₆	—	316	2	no	—
144	M₁₆	—	316	2	no	—
145	C₂×C₈	—	316	2	no	—
146	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#11929613	1	2	yes	—
149	C₂×C₈	Dic₃₁₆	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
153	C₂×C₈	#11929624	1	2	yes	—
154	C₄⋊C₄	—	316	4	no	—
155	C₄⋊C₄	—	316	2	no	—
156	C₄⋊C₄	—	158	4	no	—
157	C₂²⋊C₄	—	316	2	no	—
158	C₄²	—	316	2	no	—
159	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
160	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
161	C₄²	D₆₃₂	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#11929613	1	1	yes	characteristic
164	Q₈	—	632	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	316	4	no	—
167	C₂×C₄	#14579083	1	2	yes	—
168	C₂×C₄	#14579083	1	2	yes	—
169	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#14579149	1	1	yes	characteristic
171	C₂×C₄	#14579161	1	1	yes	characteristic
172	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
173	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
175	C₈	—	632	2	no	—
176	C₈	#14579086	1	2	yes	—
177	C₈	#14579086	1	2	yes	—
178	C₈	C₄×Dic₁₅₈	1	2	yes	—
179	C₈	#14579140	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14579083	1	1	yes	characteristic
182	C₂²	#19098914	1	1	yes	characteristic
183	C₂²	#19098961	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	632	4	no	—
187	C₄	#19098469	1	1	yes	characteristic
188	C₄	C₂×#14579086	1	1	yes	characteristic
189	C₄	#19098910	1	1	yes	characteristic
190	C₄	#19098968	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#23764361	1	1	yes	characteristic
194	C₂	#23765129	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#13286
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
37. C₂×C₁₆×C₇₉	Q₈	semidirect
102. C₇₉	#13286	semidirect
128. C₂×C₁₆	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
#23763572	2
#23761496	2
C₂×#346×C₇₉	2
#13286	79