Subgroups — #34206282

Order 20224.

290 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765160	C₂	1	2	yes	maximal
2	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
3	#23763279	C₂	1	1	yes	maximal, characteristic
4	#23761490	C₂	1	2	yes	maximal
5	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14579097	C₄	1	1	yes	characteristic
9	#19098601	C₄	1	1	yes	characteristic
10	#19098518	C₂²	1	2	yes	—
11	Dic₇₉⋊C₁₆	—	2	4	no	—
12	#19098339	—	2	2	no	—
13	#19098323	C₄	1	2	yes	—
14	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#364×C₇₉	—	2	2	no	—
17	C₂×#11929608	C₂×C₄	1	2	yes	—
18	#14579120	C₂×C₄	1	2	yes	—
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	C₂×C₇₉⋊C₁₆	—	2	1	no	—
21	C₂×C₇₉⋊C₁₆	—	2	1	no	—
22	C₂×C₇₉⋊C₁₆	—	2	1	no	—
23	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
24	C₂×C₇₉⋊C₁₆	Q₈	1	2	yes	—
25	#14579097	—	2	2	no	—
26	#14579095	C₈	1	2	yes	—
27	#14579093	C₈	1	2	yes	—
28	#14579071	—	2	2	no	—
29	C₈×Dic₇₉	—	2	2	no	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₇₉	—	2	2	no	—
34	M₃₂×C₇₉	—	2	2	no	—
35	C₂×C₁₆×C₇₉	—	2	2	no	—
36	C₄×C₈×C₇₉	D₄	1	2	yes	—
37	C₄×C₈×C₇₉	Q₈	1	2	yes	—
38	#11929608	—	2	2	no	—
39	#11929608	—	2	2	no	—
40	C₂×C₇₉⋊C₈	—	2	2	no	—
41	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
42	C₂²⋊Dic₇₉	—	2	2	no	—
43	C₄×Dic₇₉	—	2	2	no	—
44	C₇₉⋊C₁₆	—	2	2	no	—
45	C₇₉⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₉	—	2	1	no	—
47	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
52	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
58	C₄²×C₇₉	—	2	1	no	—
59	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₇₉	—	4	2	no	—
61	C₂³×C₇₉	#131	1	1	yes	characteristic
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	4	2	no	—
64	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
70	C₂×Dic₇₉	—	2	4	no	—
71	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
72	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
73	C₈×C₇₉	#121	1	1	yes	characteristic
74	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₇₉	#125	1	1	yes	characteristic
76	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₇₉⋊C₈	—	4	2	no	—
78	C₂²×C₇₉	—	2	2	no	—
79	C₂²×C₇₉	#424	1	1	yes	characteristic
80	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
81	C₂²×C₇₉	#446	1	1	yes	characteristic
82	C₄×C₇₉	—	4	2	no	—
83	C₄×C₇₉	#389	1	1	yes	characteristic
84	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₇₉	#436	1	1	yes	characteristic
86	C₄×C₇₉	C₈×Q₈	1	1	yes	characteristic
87	Dic₇₉	—	4	4	no	—
88	#13162	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	—	2	1	no	—
90	C₂×C₇₉	—	2	1	no	—
91	C₂×C₇₉	#1939	1	1	yes	characteristic
92	C₂×C₇₉	#2171	1	2	yes	—
93	C₂×#364	D₇₉	1	1	yes	characteristic
94	#2151	—	79	2	no	—
95	C₂×#346	—	79	1	no	—
96	#1772	—	79	1	no	—
97	#1379	—	79	2	no	—
98	C₇₉	#13162	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
100	C₂²×C₁₆	—	79	1	no	—
101	C₂²×C₁₆	—	79	1	no	—
102	C₂×#108	—	79	1	no	—
103	#427	—	79	1	no	—
104	#405	—	79	2	no	—
105	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
106	#364	—	2	2	no	—
107	C₄⋊C₁₆	—	158	4	no	—
108	#346	—	158	2	no	—
109	#339	—	79	2	no	—
110	C₂×M₁₆	—	79	2	no	—
111	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
113	#117	—	79	2	no	—
114	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	79	2	no	—
119	C₂×C₁₆	—	79	2	no	—
120	C₂×C₁₆	—	158	1	no	—
121	C₂×C₁₆	—	158	1	no	—
122	C₂×C₁₆	—	158	1	no	—
123	C₂×C₁₆	—	158	1	no	—
124	#108	—	158	2	no	—
125	#107	—	79	2	no	—
126	#106	—	79	2	no	—
127	#97	—	158	2	no	—
128	C₄×C₈	—	158	2	no	—
129	C₄×C₈	Dic₁₅₈	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
133	M₁₆	—	158	2	no	—
134	M₁₆	—	158	2	no	—
135	C₂×C₈	—	158	2	no	—
136	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
137	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
138	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
139	C₂×C₈	#11929607	1	2	yes	—
140	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
141	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
144	C₂×C₈	#11929623	1	1	yes	characteristic
145	C₂×C₈	#11929625	1	1	yes	characteristic
146	C₄⋊C₄	—	79	4	no	—
147	C₂²⋊C₄	—	158	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	158	2	no	—
150	C₄²	Dic₇₉⋊C₄	1	2	yes	—
151	C₁₆	—	4	2	no	—
152	C₁₆	—	158	2	no	—
153	C₁₆	—	158	2	no	—
154	C₂³	#14579169	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	158	4	no	—
158	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#14579123	1	1	yes	characteristic
160	C₂×C₄	#14579126	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
162	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
163	C₂×C₄	#14579169	1	1	yes	characteristic
164	C₈	—	316	2	no	—
165	C₈	Dic₇₉⋊C₈	1	2	yes	—
166	C₈	Dic₇₉⋊C₈	1	2	yes	—
167	C₈	#14579132	1	1	yes	characteristic
168	C₈	#14579137	1	1	yes	characteristic
169	C₈	#14579152	1	1	yes	characteristic
170	C₈	#14579160	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#19098591	1	1	yes	characteristic
173	C₂²	#19098628	1	1	yes	characteristic
174	C₂²	#19098690	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	316	4	no	—
177	C₄	#19098459	1	1	yes	characteristic
178	C₄	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
179	C₄	#19098640	1	1	yes	characteristic
180	C₄	#19098687	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#23764116	1	1	yes	characteristic
184	C₂	#23765241	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂³×C₇₉	#131
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#13162
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
98. C₇₉	#13162	semidirect

Maximal subgroups

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

subgroup	index
#23765160	2
C₂×#19098339	2
#23763279	2
#23761490	2
C₂×#364×C₇₉	2
#13162	79