Subgroups — #34298625

Order 20224.

770 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	C₂×#19098620	C₂	1	2	yes	maximal
2	#23764124	C₂	1	8	yes	maximal, complemented
3	#23763384	C₂	1	4	yes	maximal
4	C₇₉×#2285	C₂	1	1	yes	maximal, characteristic
5	#19098620	—	2	8	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
8	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
9	C₂×#14579072	C₂²	1	4	yes	—
10	#19098318	C₂²	1	8	yes	—
11	C₂×#117×C₇₉	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	#384×C₇₉	C₄	1	2	yes	—
14	#382×C₇₉	C₄	1	2	yes	—
15	#380×C₇₉	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
17	C₂²×C₇₉⋊C₈	C₂³	1	8	yes	—
18	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
19	C₄⋊(C₇₉⋊C₈)	—	4	8	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	2	8	no	—
21	#14579072	—	2	8	no	—
22	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
24	#117×C₇₉	—	4	4	no	—
25	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	8	yes	—
27	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	4	yes	—
31	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
32	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
34	#95×C₇₉	C₂×C₄	1	8	yes	—
35	C₂×C₇₉⋊C₈	—	2	8	no	—
36	C₂×C₇₉⋊C₈	—	2	8	no	—
37	C₂×C₇₉⋊C₈	—	2	8	no	—
38	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	—	2	2	no	—
40	C₂²×C₄×C₇₉	—	2	2	no	—
41	C₂²×C₄×C₇₉	—	4	4	no	—
42	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
43	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₇₉	—	4	8	no	—
51	C₄⋊C₄×C₇₉	—	4	4	no	—
52	C₂²⋊C₄×C₇₉	—	4	4	no	—
53	C₄²×C₇₉	—	2	8	no	—
54	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
55	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
56	C₄²×C₇₉	C₂×D₄	1	4	yes	—
57	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
58	C₂³×C₇₉	—	2	1	no	—
59	C₂³×C₇₉	—	2	1	no	—
60	C₂³×C₇₉	—	2	4	no	—
61	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	4	no	—
66	C₂×C₄×C₇₉	—	2	4	no	—
67	C₂×C₄×C₇₉	—	2	4	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	—	2	8	no	—
70	C₂×C₄×C₇₉	—	2	16	no	—
71	C₂×C₄×C₇₉	—	4	8	no	—
72	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
79	C₇₉⋊C₈	—	4	8	no	—
80	C₂²×C₇₉	—	2	2	no	—
81	C₂²×C₇₉	—	2	4	no	—
82	C₂²×C₇₉	—	4	2	no	—
83	C₂²×C₇₉	—	4	2	no	—
84	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₇₉	#518	1	1	yes	characteristic
86	C₂²×C₇₉	#519	1	1	yes	characteristic
87	C₂²×C₇₉	#568	1	4	yes	—
88	C₄×C₇₉	—	2	2	no	—
89	C₄×C₇₉	—	2	4	no	—
90	C₄×C₇₉	—	2	4	no	—
91	C₄×C₇₉	—	4	8	no	—
92	C₄×C₇₉	#537	1	2	yes	—
93	C₄×C₇₉	#544	1	2	yes	—
94	#21956	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	4	2	no	—
96	C₂×C₇₉	#2316	1	1	yes	characteristic
97	C₂×C₇₉	#2905	1	2	yes	—
98	C₂×C₇₉	#2980	1	2	yes	—
99	C₂×C₇₉	#2987	1	2	yes	—
100	C₂×#432	—	79	2	no	—
101	#2285	D₇₉	1	1	yes	characteristic
102	#1940	—	79	8	no	—
103	#1798	—	79	4	no	—
104	C₇₉	#21956	1	1	yes	characteristic, complemented, Sylow
105	C₂×#117	C₂×D₇₉	1	2	yes	—
106	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
107	#432	—	158	8	no	—
108	C₂×C₄⋊C₈	—	79	8	no	—
109	C₂×C₄⋊C₈	—	79	4	no	—
110	C₂×C₂²⋊C₈	—	79	4	no	—
111	C₂×#97	—	79	4	no	—
112	#384	Dic₇₉	1	2	yes	—
113	#382	Dic₇₉	1	2	yes	—
114	#380	Dic₇₉	1	4	yes	—
115	C₄×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
116	#336	—	79	8	no	—
117	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
118	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
119	C₂²×C₈	—	79	8	no	—
120	#117	—	4	4	no	—
121	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₇₉	1	8	yes	—
123	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	4	yes	—
124	C₂×C₄²	Dic₁₅₈	1	4	yes	—
125	C₂×C₄²	D₃₁₆	1	4	yes	—
126	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
129	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
130	C₄⋊C₈	—	316	8	no	—
131	C₄⋊C₈	—	158	8	no	—
132	C₂²⋊C₈	—	158	8	no	—
133	#97	—	158	8	no	—
134	#95	C₂×Dic₇₉	1	8	yes	—
135	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	C₄⋊Dic₇₉	1	8	yes	—
140	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
146	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
147	C₂×C₈	—	158	8	no	—
148	C₂×C₈	—	158	8	no	—
149	C₂×C₈	—	158	8	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₄⋊Dic₇₉	1	4	yes	—
155	C₄²	C₄⋊Dic₇₉	1	4	yes	—
156	C₄²	C₂×D₃₁₆	1	4	yes	—
157	C₄²	C₂×Dic₁₅₈	1	4	yes	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	4	no	—
161	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
162	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
175	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
177	C₂×C₄	#14579206	1	2	yes	—
178	C₂×C₄	#14579209	1	2	yes	—
179	C₈	—	316	8	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
185	C₂²	#19099038	1	1	yes	characteristic
186	C₂²	#19099041	1	1	yes	characteristic
187	C₂²	#19099330	1	4	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	4	8	no	—
192	C₄	#19099144	1	2	yes	—
193	C₄	#19099194	1	2	yes	—
194	C₂	—	4	2	no	—
195	C₂	#23766018	1	1	yes	characteristic
196	C₂	#23771964	1	2	yes	—
197	C₂	#23772471	1	2	yes	—
198	C₂	#23772506	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator, socle	C₂³×C₇₉	C₂³×C₄
Fitting	C₇₉×#2285	C₂
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2285	D₇₉
hypocenter	C₇₉	#21956
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
2. #23764124	C₂	semidirect
104. C₇₉	#21956	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098620	2
#23764124	2
#23763384	2
C₇₉×#2285	2
#21956	79