Subgroups — C₂²×#19151421 #36707435

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667944	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151421	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	—	2	1	no	—
6	C₂²×C₄⋊Dic₈₉	—	2	1	no	—
7	C₂×#15667948	—	2	3	no	—
8	C₂×#15667948	C₂²	1	6	yes	—
9	C₂×#15667944	—	2	3	no	—
10	C₂×#15667944	C₂²	1	6	yes	—
11	#19151421	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₈₉	C₄	1	6	yes	complemented
14	C₂×M₃₂×C₈₉	C₂²	1	6	yes	—
15	C₂²×C₁₆×C₈₉	C₄	1	2	yes	complemented
16	C₂³×Dic₈₉	—	4	1	no	—
17	C₂×#13083558	—	2	6	no	—
18	C₂×#13083558	—	4	3	no	—
19	C₂²×#10623494	—	4	1	no	—
20	C₂×C₄⋊Dic₈₉	—	2	6	no	—
21	C₂×C₄⋊Dic₈₉	—	4	3	no	—
22	#15667948	—	2	6	no	—
23	#15667948	C₂³	1	4	yes	—
24	#15667944	—	2	6	no	—
25	#15667944	C₂³	1	4	yes	—
26	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₈₉	D₄	1	3	yes	—
30	C₂²×C₈×C₈₉	Q₈	1	3	yes	—
31	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
33	M₃₂×C₈₉	C₂×C₄	1	12	yes	complemented
34	M₃₂×C₈₉	C₂³	1	4	yes	—
35	C₂×C₁₆×C₈₉	C₂×C₄	1	12	yes	complemented
36	C₂²×Dic₈₉	—	4	6	no	—
37	C₂²×Dic₈₉	—	8	4	no	—
38	#13083558	—	2	4	no	—
39	#13083558	—	4	6	no	—
40	C₂×#10623494	—	4	6	no	—
41	C₄⋊Dic₈₉	—	2	4	no	—
42	C₄⋊Dic₈₉	—	4	6	no	—
43	C₂⁴×C₈₉	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₄⋊C₄	1	3	yes	—
45	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	SD₁₆	1	3	yes	—
47	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	SD₁₆	1	3	yes	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
50	C₂×C₈×C₈₉	C₄⋊C₄	1	3	yes	—
51	C₂×C₈×C₈₉	C₄⋊C₄	1	6	yes	—
52	C₂×C₈×C₈₉	C₄⋊C₄	1	3	yes	—
53	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
56	C₂×C₈×C₈₉	C₂×Q₈	1	3	yes	—
57	C₂×C₈×C₈₉	C₂×Q₈	1	3	yes	—
58	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₁₆×C₈₉	C₂²×C₄	1	8	yes	complemented
60	C₂³×C₈₉	—	2	4	no	—
61	C₂³×C₈₉	#106	1	3	yes	—
62	C₂³×C₈₉	#106	1	1	yes	characteristic
63	C₂³×C₈₉	C₂×SD₁₆	1	3	yes	—
64	C₂×C₄×C₈₉	#106	1	6	yes	—
65	C₂×C₄×C₈₉	#106	1	3	yes	—
66	C₂×C₄×C₈₉	#106	1	3	yes	—
67	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₈₉	C₂×SD₁₆	1	3	yes	—
70	C₂×C₄×C₈₉	C₂×SD₁₆	1	3	yes	—
71	C₂×C₄×C₈₉	C₂×SD₁₆	1	3	yes	—
72	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₈₉	—	4	4	no	—
74	C₂×Dic₈₉	—	8	12	no	—
75	C₈×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
76	C₈×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
77	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
79	#10623494	—	4	4	no	—
80	C₂²×C₈₉	—	2	6	no	—
81	C₂²×C₈₉	—	2	6	no	—
82	C₂²×C₈₉	#365	1	4	yes	complemented
83	C₂²×C₈₉	C₂×#106	1	3	yes	—
84	C₂²×C₈₉	C₂×#106	1	3	yes	—
85	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
86	C₄×C₈₉	C₂×#106	1	3	yes	—
87	C₄×C₈₉	C₂×#106	1	3	yes	—
88	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
90	Dic₈₉	—	8	4	no	—
91	C₂²×#365	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	—	2	1	no	—
93	C₂×C₈₉	—	2	3	no	—
94	C₂×C₈₉	C₂×#365	1	6	yes	complemented
95	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
96	C₂²×M₃₂	D₈₉	1	1	yes	characteristic
97	C₂²×#108	—	89	1	no	—
98	C₂²×#106	—	89	1	no	—
99	C₂×#365	—	89	12	no	—
100	C₈₉	C₂²×#365	1	1	yes	characteristic, complemented, Sylow
101	C₂²×M₁₆	—	178	1	no	—
102	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
103	C₂²×C₄⋊C₄	—	178	1	no	—
104	C₂×M₃₂	Dic₈₉	1	6	yes	complemented
105	C₂×M₃₂	C₂×D₈₉	1	6	yes	—
106	C₂²×C₁₆	Dic₈₉	1	2	yes	complemented
107	C₂×#108	—	89	6	no	—
108	C₂×#108	—	178	3	no	—
109	C₂×#106	—	89	6	no	—
110	C₂×#106	—	178	3	no	—
111	#365	—	89	16	no	—
112	C₂³×C₄	—	356	1	no	—
113	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
114	C₂×M₁₆	—	356	3	no	—
115	C₂×M₁₆	—	178	6	no	—
116	C₂²×C₈	—	356	1	no	—
117	C₂²×C₈	Dic₁₇₈	1	3	yes	—
118	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
119	C₂²×C₈	D₃₅₆	1	3	yes	—
120	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
122	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
123	C₂×C₄⋊C₄	—	356	3	no	—
124	C₂×C₄⋊C₄	—	178	6	no	—
125	M₃₂	C₂×Dic₈₉	1	12	yes	complemented
126	M₃₂	C₂²×D₈₉	1	4	yes	—
127	C₂×C₁₆	C₂×Dic₈₉	1	12	yes	complemented
128	#108	—	89	4	no	—
129	#108	—	178	6	no	—
130	#106	—	89	4	no	—
131	#106	—	178	6	no	—
132	C₂⁴	#13083563	1	1	yes	characteristic
133	C₂²×C₄	—	356	6	no	—
134	C₂²×C₄	—	712	4	no	—
135	C₂²×C₄	C₄⋊Dic₈₉	1	3	yes	—
136	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
137	C₂²×C₄	#13083563	1	3	yes	—
138	C₂²×C₄	#13083563	1	1	yes	characteristic
139	C₂²×C₄	#13083563	1	3	yes	—
140	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
141	M₁₆	—	356	6	no	—
142	M₁₆	—	178	4	no	—
143	C₂×C₈	—	356	6	no	—
144	C₂×C₈	C₄⋊Dic₈₉	1	3	yes	—
145	C₂×C₈	C₄⋊Dic₈₉	1	6	yes	—
146	C₂×C₈	C₄⋊Dic₈₉	1	3	yes	—
147	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
148	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
149	C₂×C₈	C₂×D₃₅₆	1	3	yes	—
150	C₂×C₈	C₂×Dic₁₇₈	1	3	yes	—
151	C₂×C₈	C₂×Dic₁₇₈	1	3	yes	—
152	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
153	C₄⋊C₄	—	356	6	no	—
154	C₄⋊C₄	—	178	4	no	—
155	C₁₆	C₂²×Dic₈₉	1	8	yes	complemented
156	C₂³	—	2	4	no	—
157	C₂³	#15667944	1	3	yes	—
158	C₂³	#15667944	1	1	yes	characteristic
159	C₂³	C₂×#13083563	1	3	yes	—
160	C₂×C₄	—	356	4	no	—
161	C₂×C₄	—	712	12	no	—
162	C₂×C₄	#15667944	1	6	yes	—
163	C₂×C₄	#15667944	1	3	yes	—
164	C₂×C₄	#15667944	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
167	C₂×C₄	C₂×#13083563	1	3	yes	—
168	C₂×C₄	C₂×#13083563	1	3	yes	—
169	C₂×C₄	C₂×#13083563	1	3	yes	—
170	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
171	C₈	—	356	4	no	—
172	C₈	C₂×C₄⋊Dic₈₉	1	3	yes	—
173	C₈	C₂×C₄⋊Dic₈₉	1	3	yes	—
174	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	#19151421	1	4	yes	direct product factor
179	C₂²	C₂×#15667944	1	3	yes	—
180	C₂²	C₂×#15667944	1	3	yes	—
181	C₂²	C₂²×#13083563	1	1	yes	characteristic
182	C₄	—	712	4	no	—
183	C₄	C₂×#15667944	1	3	yes	—
184	C₄	C₂×#15667944	1	3	yes	—
185	C₄	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
186	C₄	C₂²×#13083563	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#19151421	1	6	yes	direct product factor
190	C₂	C₂²×#15667944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667944
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂³×C₈₉	#106
prime radical	C₂⁴×C₈₉	SD₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	C₂²×#365
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#19151421	C₂	direct (N × H)
11. #19151421	C₂²	direct (N × H)
13. C₂×M₃₂×C₈₉	C₄	semidirect
15. C₂²×C₁₆×C₈₉	C₄	semidirect
33. M₃₂×C₈₉	C₂×C₄	semidirect
35. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
59. C₁₆×C₈₉	C₂²×C₄	semidirect
82. C₂²×C₈₉	#365	semidirect
94. C₂×C₈₉	C₂×#365	semidirect
100. C₈₉	C₂²×#365	semidirect
104. C₂×M₃₂	Dic₈₉	semidirect
106. C₂²×C₁₆	Dic₈₉	semidirect
125. M₃₂	C₂×Dic₈₉	semidirect
127. C₂×C₁₆	C₂×Dic₈₉	semidirect
155. C₁₆	C₂²×Dic₈₉	semidirect
178. C₂²	#19151421	direct (N × H)
189. C₂	C₂×#19151421	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667948	2
C₂²×#15667944	2
C₂×#19151421	2
C₂²×M₃₂×C₈₉	2
C₂²×#365	89

Subgroups — C22×#19151421 #36707435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151421 #36707435