Subgroups — C₂×#23761463 #34199810

Order 20224.

286 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	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23761463	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
6	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
10	C₈×C₇₉⋊C₈	C₄	1	2	yes	—
11	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₉	—	2	1	no	—
15	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
16	C₂²×C₇₉⋊C₈	—	2	1	no	—
17	C₂²×C₇₉⋊C₈	C₈	1	1	yes	characteristic
18	C₂²×C₇₉⋊C₈	C₈	1	1	yes	characteristic
19	C₂×C₇₉⋊C₁₆	—	2	2	no	—
20	C₂×C₇₉⋊C₁₆	—	4	2	no	—
21	C₄×C₇₉⋊C₈	—	2	1	no	—
22	C₄×C₇₉⋊C₈	C₈	1	2	yes	—
23	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	—	2	2	no	—
29	C₂×C₁₆×C₇₉	—	4	2	no	—
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₇₉⋊C₈	—	2	4	no	—
35	C₂×C₇₉⋊C₈	—	2	2	no	—
36	C₂×C₇₉⋊C₈	—	2	2	no	—
37	C₂×C₇₉⋊C₈	C₁₆	1	4	yes	complemented
38	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
39	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
40	C₇₉⋊C₁₆	—	4	2	no	—
41	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₇₉	—	2	2	no	—
45	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
48	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
54	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₇₉	—	4	2	no	—
56	C₂³×C₇₉	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#106	1	2	yes	—
62	C₂×C₄×C₇₉	#108	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₇₉	M₃₂	1	2	yes	—
65	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₇₉	—	2	1	no	—
69	C₈×C₇₉	—	2	1	no	—
70	C₈×C₇₉	#106	1	2	yes	—
71	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
73	C₇₉⋊C₈	—	2	4	no	—
74	C₇₉⋊C₈	—	2	2	no	—
75	C₇₉⋊C₈	C₂×C₁₆	1	4	yes	complemented
76	C₂²×C₇₉	#334	1	2	yes	—
77	C₂²×C₇₉	C₄⋊C₁₆	1	2	yes	—
78	C₂²×C₇₉	#364	1	2	yes	—
79	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₇₉	—	2	1	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	#334	1	2	yes	—
83	C₄×C₇₉	C₄⋊C₁₆	1	2	yes	—
84	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
85	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
86	C₄×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
87	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
88	C₂×#1367	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	#1367	1	4	yes	complemented
90	C₂×C₇₉	C₂×#334	1	1	yes	characteristic
91	C₂×C₇₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
92	C₂×C₇₉	C₂×#364	1	1	yes	characteristic
93	C₂×C₄⋊C₁₆	—	79	1	no	—
94	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
95	C₂×C₈²	—	79	1	no	—
96	#1367	—	79	4	no	—
97	C₇₉	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	2	1	no	—
99	C₂²×C₁₆	—	158	1	no	—
100	C₂×C₄×C₈	—	79	1	no	—
101	C₂×C₄×C₈	—	79	1	no	—
102	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
103	C₄⋊C₁₆	—	2	1	no	—
104	C₄⋊C₁₆	—	79	2	no	—
105	C₄⋊C₁₆	—	158	1	no	—
106	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
107	C₈²	—	79	2	no	—
108	C₈²	—	79	2	no	—
109	C₂²×C₈	—	79	1	no	—
110	C₂²×C₈	—	79	1	no	—
111	C₂²×C₈	—	158	1	no	—
112	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
114	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	4	2	no	—
117	C₂×C₁₆	—	316	2	no	—
118	C₂×C₁₆	—	158	2	no	—
119	C₄×C₈	—	79	2	no	—
120	C₄×C₈	—	79	2	no	—
121	C₄×C₈	—	79	2	no	—
122	C₄×C₈	—	158	1	no	—
123	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
124	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
127	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
128	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
129	C₂²×C₄	#11929607	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	79	4	no	—
132	C₂×C₈	—	79	2	no	—
133	C₂×C₈	—	79	2	no	—
134	C₂×C₈	—	158	4	no	—
135	C₂×C₈	—	158	2	no	—
136	C₂×C₈	—	158	2	no	—
137	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
138	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#11929611	1	2	yes	—
140	C₂×C₈	#11929612	1	2	yes	—
141	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
143	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
144	C₄²	C₈×D₇₉	1	1	yes	characteristic
145	C₄²	#11929607	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
147	C₁₆	—	4	2	no	—
148	C₁₆	—	316	2	no	—
149	C₂³	Dic₇₉⋊C₈	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
152	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
153	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
154	C₂×C₄	#14579092	1	2	yes	—
155	C₂×C₄	#14579096	1	2	yes	—
156	C₂×C₄	C₁₆×D₇₉	1	2	yes	—
157	C₂×C₄	#14579100	1	2	yes	—
158	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
160	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	79	4	no	—
164	C₈	—	158	4	no	—
165	C₈	—	158	2	no	—
166	C₈	#14579092	1	2	yes	—
167	C₈	C₂×#11929611	1	1	yes	characteristic
168	C₈	C₂×#11929612	1	1	yes	characteristic
169	C₂²	#19098313	1	2	yes	—
170	C₂²	Dic₇₉⋊C₁₆	1	2	yes	—
171	C₂²	#19098383	1	2	yes	—
172	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#19098313	1	2	yes	—
176	C₄	Dic₇₉⋊C₁₆	1	2	yes	—
177	C₄	C₂×#14579092	1	1	yes	characteristic
178	C₄	C₂×#14579096	1	1	yes	characteristic
179	C₄	C₂×C₁₆×D₇₉	1	1	yes	characteristic
180	C₄	C₂×#14579100	1	1	yes	characteristic
181	C₂	#23761463	1	4	yes	direct product factor
182	C₂	C₂×#19098313	1	1	yes	characteristic
183	C₂	C₂×Dic₇₉⋊C₁₆	1	1	yes	characteristic
184	C₂	C₂×#19098383	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₄⋊C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	C₂×#1367
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #23761463	C₂	direct (N × H)
37. C₂×C₇₉⋊C₈	C₁₆	semidirect
75. C₇₉⋊C₈	C₂×C₁₆	semidirect
89. C₂×C₇₉	#1367	semidirect
97. C₇₉	C₂×#1367	semidirect
181. C₂	#23761463	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×C₈×C₇₉⋊C₈	2
#23761463	2
C₂×C₄⋊C₁₆×C₇₉	2
C₂×#1367	79

Subgroups — C2×#23761463 #34199810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761463 #34199810