Subgroups — C₄×#19151422 #36469815

Order 22784.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151422	C₂	1	2	yes	maximal
2	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
4	#26018352	C₂	1	2	yes	maximal
5	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	1	yes	characteristic
7	C₂×#15667952	C₂²	1	1	yes	characteristic
8	C₂×#15667944	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₈₉	—	2	1	no	—
10	#19151422	C₄	1	8	yes	direct product factor
11	#19151374	C₄	1	2	yes	complemented
12	C₄×#13083549	C₄	1	2	yes	complemented
13	#19151361	—	2	1	no	—
14	C₂×#108×C₈₉	C₂²	1	2	yes	—
15	C₄×M₁₆×C₈₉	—	2	1	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#341×C₈₉	—	2	1	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄×Dic₈₉	—	4	1	no	—
20	#15667952	—	2	2	no	—
21	#15667952	C₂×C₄	1	4	yes	—
22	C₂×#13083549	C₂×C₄	1	2	yes	—
23	C₂×#13083549	C₂×C₄	1	2	yes	—
24	#15667944	—	2	2	no	—
25	#15667944	C₂×C₄	1	4	yes	—
26	#15667919	—	4	1	no	—
27	C₂×M₁₆×C₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
31	#108×C₈₉	—	2	2	no	—
32	#108×C₈₉	C₂×C₄	1	4	yes	—
33	#97×C₈₉	—	4	1	no	—
34	C₄×C₈×C₈₉	—	4	1	no	—
35	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₄×Dic₈₉	—	8	2	no	—
43	#13083549	—	2	2	no	—
44	#13083549	C₄²	1	4	yes	complemented
45	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₈₉	—	2	4	no	—
49	M₁₆×C₈₉	—	4	2	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	4	2	no	—
52	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₈₉	#41	1	2	yes	—
58	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
61	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
62	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	4	1	no	—
65	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#135	1	2	yes	—
73	C₂×Dic₈₉	—	4	4	no	—
74	C₂×Dic₈₉	—	8	2	no	—
75	C₂×Dic₈₉	—	8	2	no	—
76	C₈×C₈₉	—	2	1	no	—
77	C₈×C₈₉	—	2	1	no	—
78	C₈×C₈₉	—	4	4	no	—
79	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
80	C₈×C₈₉	#117	1	1	yes	characteristic
81	C₈×C₈₉	C₄×Q₈	1	2	yes	—
82	C₂²×C₈₉	—	2	1	no	—
83	C₂²×C₈₉	—	2	1	no	—
84	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
85	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	#365	1	2	yes	complemented
90	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₈₉	#427	1	1	yes	characteristic
92	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
93	Dic₈₉	—	8	4	no	—
94	C₄×#365	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	—	2	2	no	—
96	C₂×C₈₉	C₄×#106	1	1	yes	characteristic
97	C₂×C₈₉	C₂×#365	1	1	yes	characteristic
98	C₂×C₈₉	#2155	1	1	yes	characteristic
99	C₂×#365	—	89	2	no	—
100	C₄×M₃₂	—	89	1	no	—
101	C₄×#108	D₈₉	1	1	yes	characteristic
102	C₄×#106	—	89	1	no	—
103	#1383	—	89	2	no	—
104	C₈₉	C₄×#365	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	—	89	1	no	—
106	C₂×M₃₂	—	89	1	no	—
107	C₂×#108	C₂×D₈₉	1	2	yes	—
108	C₂×#106	—	89	2	no	—
109	C₄×M₁₆	—	2	1	no	—
110	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	178	1	no	—
112	#365	—	89	8	no	—
113	#346	—	89	2	no	—
114	C₄×C₁₆	—	89	2	no	—
115	#341	—	2	1	no	—
116	#340	—	178	1	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	178	2	no	—
121	C₂×C₄²	—	356	1	no	—
122	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
123	M₃₂	—	89	4	no	—
124	M₃₂	—	178	2	no	—
125	C₂×C₁₆	—	89	2	no	—
126	C₂×C₁₆	—	89	2	no	—
127	#108	—	2	2	no	—
128	#108	C₄×D₈₉	1	4	yes	—
129	#106	—	89	4	no	—
130	#106	—	178	2	no	—
131	#97	—	4	1	no	—
132	C₄×C₈	—	4	1	no	—
133	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
134	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
135	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
137	#95	—	356	1	no	—
138	C₂²×C₄	—	356	2	no	—
139	C₂²×C₄	#13083561	1	1	yes	characteristic
140	C₂²×C₄	#13083561	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	M₁₆	—	2	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	#13083553	1	1	yes	characteristic
147	C₂×C₈	#13083553	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
150	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
151	C₂×C₈	#13083574	1	2	yes	—
152	C₄⋊C₄	—	356	2	no	—
153	C₄⋊C₄	—	178	4	no	—
154	C₄²	—	712	2	no	—
155	C₄²	#13083553	1	1	yes	characteristic
156	C₄²	#13083553	1	1	yes	characteristic
157	C₄²	#13083562	1	1	yes	characteristic
158	C₄²	#13083562	1	1	yes	characteristic
159	C₁₆	—	89	4	no	—
160	C₁₆	—	178	2	no	—
161	C₂³	C₂×#13083561	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	356	4	no	—
165	C₂×C₄	—	712	2	no	—
166	C₂×C₄	—	712	2	no	—
167	C₂×C₄	#15667943	1	1	yes	characteristic
168	C₂×C₄	#15667943	1	1	yes	characteristic
169	C₂×C₄	#15667943	1	1	yes	characteristic
170	C₂×C₄	#15667943	1	1	yes	characteristic
171	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
172	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
173	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
174	C₂×C₄	#15668032	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	4	no	—
178	C₈	C₄²×D₈₉	1	1	yes	characteristic
179	C₈	#15667969	1	1	yes	characteristic
180	C₈	C₄×Dic₁₇₈	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#15667943	1	1	yes	characteristic
184	C₂²	C₄×#13083561	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	712	4	no	—
189	C₄	#19151422	1	2	yes	direct product factor
190	C₄	C₄×#13083553	1	1	yes	characteristic
191	C₄	#19151634	1	1	yes	characteristic
192	C₄	C₄×#13083562	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#15667943	1	1	yes	characteristic
195	C₂	C₂×#19151422	1	1	yes	characteristic
196	C₂	#26022026	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667943
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	C₄×#365
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
10. #19151422	C₄	direct (N × H)
11. #19151374	C₄	semidirect
12. C₄×#13083549	C₄	semidirect
44. #13083549	C₄²	semidirect
89. C₄×C₈₉	#365	semidirect
104. C₈₉	C₄×#365	semidirect
189. C₄	#19151422	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×#19151422	2
C₄×#15667952	2
C₄×#15667944	2
#26018352	2
C₄×#108×C₈₉	2
C₄×#365	89

Subgroups — C4×#19151422 #36469815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151422 #36469815