Subgroups — C₂×#23761467 #34199830

Order 20224.

222 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098384	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
3	#23761467	C₂	1	4	yes	maximal, direct product factor
4	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	—	2	1	no	—
6	C₂²×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
8	#19098384	—	2	1	no	—
9	#19098384	C₂²	1	2	yes	—
10	#19098339	C₄	1	2	yes	—
11	#19098339	C₂²	1	2	yes	—
12	C₂×M₃₂×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#364×C₇₉	—	2	1	no	—
15	#364×C₇₉	C₂²	1	2	yes	—
16	#14579101	—	2	2	no	—
17	#14579101	—	4	1	no	—
18	C₂×C₇₉⋊C₁₆	—	2	1	no	—
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	C₂×C₇₉⋊C₁₆	—	4	1	no	—
21	C₂×C₇₉⋊C₁₆	C₈	1	2	yes	—
22	C₂×C₇₉⋊C₁₆	C₈	1	2	yes	—
23	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
28	M₃₂×C₇₉	—	2	2	no	—
29	M₃₂×C₇₉	—	4	1	no	—
30	C₂×C₁₆×C₇₉	—	4	1	no	—
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₇₉⋊C₁₆	—	2	2	no	—
36	C₇₉⋊C₁₆	—	4	2	no	—
37	C₇₉⋊C₁₆	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₇₉	—	2	1	no	—
39	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇₉	—	2	1	no	—
53	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
55	C₁₆×C₇₉	—	4	2	no	—
56	C₂³×C₇₉	#108	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	4	2	no	—
60	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
67	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
68	C₈×C₇₉	#106	1	1	yes	characteristic
69	C₈×C₇₉	#106	1	1	yes	characteristic
70	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
73	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂²×C₇₉	—	2	2	no	—
75	C₂²×C₇₉	#334	1	1	yes	characteristic
76	C₂²×C₇₉	#334	1	1	yes	characteristic
77	C₂²×C₇₉	C₂×#108	1	1	yes	characteristic
78	C₄×C₇₉	—	4	1	no	—
79	C₄×C₇₉	—	4	1	no	—
80	C₄×C₇₉	#334	1	1	yes	characteristic
81	C₄×C₇₉	#334	1	1	yes	characteristic
82	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
84	C₂×#1369	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	—	2	1	no	—
86	C₂×C₇₉	—	2	1	no	—
87	C₂×C₇₉	#1369	1	2	yes	complemented
88	C₂×C₇₉	C₂×#334	1	1	yes	characteristic
89	C₂×#364	—	79	1	no	—
90	C₂×#364	D₇₉	1	1	yes	characteristic
91	C₂×#346	—	79	1	no	—
92	#1369	—	79	4	no	—
93	C₇₉	C₂×#1369	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	—	2	1	no	—
95	C₂×M₃₂	—	158	1	no	—
96	C₂²×C₁₆	—	79	1	no	—
97	C₂²×C₁₆	—	79	1	no	—
98	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
99	#364	—	2	1	no	—
100	#364	—	79	2	no	—
101	#364	—	158	1	no	—
102	#364	C₂×D₇₉	1	2	yes	—
103	#346	—	79	2	no	—
104	#346	—	79	2	no	—
105	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
107	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	4	1	no	—
110	M₃₂	—	316	1	no	—
111	M₃₂	—	158	2	no	—
112	C₂×C₁₆	—	4	1	no	—
113	C₂×C₁₆	—	316	1	no	—
114	C₂×C₁₆	—	79	2	no	—
115	C₂×C₁₆	—	79	2	no	—
116	C₂×C₁₆	—	79	2	no	—
117	C₂×C₁₆	—	79	2	no	—
118	C₂×C₁₆	—	158	1	no	—
119	C₂×C₁₆	—	158	1	no	—
120	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
121	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
128	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
129	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
130	C₂×C₈	#11929607	1	1	yes	characteristic
131	C₂×C₈	#11929607	1	1	yes	characteristic
132	C₂×C₈	#11929611	1	1	yes	characteristic
133	C₂×C₈	#11929611	1	1	yes	characteristic
134	C₂×C₈	#11929612	1	1	yes	characteristic
135	C₂×C₈	#11929612	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
138	C₄²	—	2	1	no	—
139	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
141	C₁₆	—	4	2	no	—
142	C₁₆	—	316	2	no	—
143	C₁₆	—	79	4	no	—
144	C₁₆	—	158	2	no	—
145	C₂³	#14579096	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
150	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
151	C₂×C₄	#14579092	1	1	yes	characteristic
152	C₂×C₄	#14579092	1	1	yes	characteristic
153	C₂×C₄	#14579096	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
155	C₈	Dic₇₉⋊C₈	1	1	yes	characteristic
156	C₈	Dic₇₉⋊C₈	1	1	yes	characteristic
157	C₈	#14579092	1	1	yes	characteristic
158	C₈	#14579092	1	1	yes	characteristic
159	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
160	C₈	C₂×#11929607	1	1	yes	characteristic
161	C₈	C₂×#11929611	1	1	yes	characteristic
162	C₈	C₂×#11929612	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#19098313	1	1	yes	characteristic
165	C₂²	#19098313	1	1	yes	characteristic
166	C₂²	C₂×#14579096	1	1	yes	characteristic
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	#19098313	1	1	yes	characteristic
170	C₄	#19098313	1	1	yes	characteristic
171	C₄	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
172	C₄	C₂×#14579092	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#23761467	1	2	yes	direct product factor
176	C₂	C₂×#19098313	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579092
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	#334
prime radical	C₂³×C₇₉	#108
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	C₂×#1369
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #23761467	C₂	direct (N × H)
87. C₂×C₇₉	#1369	semidirect
93. C₇₉	C₂×#1369	semidirect
175. C₂	#23761467	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×#19098384	2
C₂×#19098339	2
#23761467	2
C₂×#364×C₇₉	2
C₂×#1369	79

Subgroups — C2×#23761467 #34199830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761467 #34199830