Subgroups — C₂×#23761473 #34199849

Order 20224.

262 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×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
3	#23761473	C₂	1	4	yes	maximal, direct product factor
4	C₂×#335×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
8	C₄×C₇₉⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#19098316	—	2	1	no	—
11	#19098316	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#335×C₇₉	—	2	1	no	—
15	#335×C₇₉	C₂²	1	2	yes	—
16	C₂²×C₇₉⋊C₈	—	4	1	no	—
17	C₂×C₇₉⋊C₁₆	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	C₈	1	4	yes	complemented
19	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₇₉	—	4	1	no	—
24	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	2	no	—
28	C₄⋊C₈×C₇₉	—	4	1	no	—
29	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	8	2	no	—
35	C₇₉⋊C₁₆	—	2	2	no	—
36	C₇₉⋊C₁₆	C₂×C₈	1	4	yes	complemented
37	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	—	4	2	no	—
42	C₂×C₈×C₇₉	—	8	2	no	—
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₇₉	D₈	1	1	yes	characteristic
51	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
52	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₇₉	#107	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#108	1	2	yes	—
60	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
61	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	—	2	1	no	—
67	C₈×C₇₉	—	8	2	no	—
68	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
69	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
70	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
71	C₇₉⋊C₈	—	8	2	no	—
72	C₂²×C₇₉	#335	1	2	yes	—
73	C₂²×C₇₉	#367	1	2	yes	—
74	C₂²×C₇₉	#368	1	2	yes	—
75	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
76	C₄×C₇₉	—	2	1	no	—
77	C₄×C₇₉	—	2	1	no	—
78	C₄×C₇₉	#335	1	2	yes	—
79	C₄×C₇₉	#367	1	2	yes	—
80	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
82	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
83	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
84	C₂×#1372	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	#1372	1	4	yes	complemented
86	C₂×C₇₉	C₂×#335	1	1	yes	characteristic
87	C₂×C₇₉	C₂×#367	1	1	yes	characteristic
88	C₂×C₇₉	C₂×#368	1	1	yes	characteristic
89	C₂×C₄×C₁₆	—	79	1	no	—
90	C₂×#335	—	79	1	no	—
91	C₂×#335	D₇₉	1	1	yes	characteristic
92	#1372	—	79	4	no	—
93	C₇₉	C₂×#1372	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₁₆	—	79	1	no	—
95	C₂²×C₁₆	—	79	1	no	—
96	C₂×C₄⋊C₈	—	2	1	no	—
97	C₂×C₄⋊C₈	—	158	1	no	—
98	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
99	C₄×C₁₆	—	79	2	no	—
100	C₄×C₁₆	—	79	2	no	—
101	#335	—	2	1	no	—
102	#335	—	79	2	no	—
103	#335	—	158	1	no	—
104	#335	C₂×D₇₉	1	2	yes	—
105	C₂²×C₈	—	4	1	no	—
106	C₂²×C₈	—	316	1	no	—
107	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
108	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
109	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×C₁₆	—	79	4	no	—
111	C₂×C₁₆	—	79	2	no	—
112	C₂×C₁₆	—	79	2	no	—
113	C₂×C₁₆	—	158	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	316	1	no	—
117	C₄⋊C₈	—	158	2	no	—
118	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
122	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	#11929609	1	1	yes	characteristic
124	C₂²×C₄	#11929614	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	316	2	no	—
129	C₂×C₈	—	632	2	no	—
130	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
131	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
132	C₂×C₈	C₈×D₇₉	1	2	yes	—
133	C₂×C₈	#11929607	1	2	yes	—
134	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
136	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
137	C₄²	#11929609	1	1	yes	characteristic
138	C₄²	#11929614	1	1	yes	characteristic
139	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
140	C₁₆	—	79	4	no	—
141	C₁₆	—	158	2	no	—
142	C₂³	#14579094	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
145	C₂×C₄	#14579094	1	1	yes	characteristic
146	C₂×C₄	#14579094	1	1	yes	characteristic
147	C₂×C₄	#14579094	1	1	yes	characteristic
148	C₂×C₄	#14579096	1	2	yes	—
149	C₂×C₄	#14579104	1	2	yes	—
150	C₂×C₄	#14579105	1	2	yes	—
151	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
153	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	8	2	no	—
157	C₈	—	632	2	no	—
158	C₈	Dic₇₉⋊C₈	1	2	yes	—
159	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
160	C₈	C₂×#11929607	1	1	yes	characteristic
161	C₂²	#19098315	1	2	yes	—
162	C₂²	#19098390	1	2	yes	—
163	C₂²	#19098392	1	2	yes	—
164	C₂²	C₂×#14579094	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	#19098315	1	2	yes	—
168	C₄	#19098390	1	2	yes	—
169	C₄	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
170	C₄	C₂×#14579096	1	1	yes	characteristic
171	C₄	C₂×#14579104	1	1	yes	characteristic
172	C₄	C₂×#14579105	1	1	yes	characteristic
173	C₂	#23761473	1	4	yes	direct product factor
174	C₂	C₂×#19098315	1	1	yes	characteristic
175	C₂	C₂×#19098390	1	1	yes	characteristic
176	C₂	C₂×#19098392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11929609
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#335×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#107
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#335	D₇₉
hypocenter	C₇₉	C₂×#1372
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #23761473	C₂	direct (N × H)
18. C₂×C₇₉⋊C₁₆	C₈	semidirect
36. C₇₉⋊C₁₆	C₂×C₈	semidirect
85. C₂×C₇₉	#1372	semidirect
93. C₇₉	C₂×#1372	semidirect
173. C₂	#23761473	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×#19098316	2
#23761473	2
C₂×#335×C₇₉	2
C₂×#1372	79

Subgroups — C2×#23761473 #34199849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761473 #34199849