Subgroups — C₂²×#19151403 #36707305

Order 22784.

1214 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667946	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151403	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₇₈	—	2	1	no	—
6	C₂²×Dic₃₅₆	—	2	3	no	—
7	C₂²×Dic₃₅₆	—	2	4	no	—
8	C₂²×Dic₃₅₆	C₄	1	8	yes	complemented
9	C₂²×Dic₃₅₆	C₂²	1	6	yes	—
10	C₂²×C₄⋊Dic₈₉	—	2	1	no	—
11	C₂×#15667946	—	2	3	no	—
12	C₂×#15667946	C₂²	1	6	yes	—
13	#19151403	C₂²	1	16	yes	direct product factor
14	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₈₉	—	2	4	no	—
16	C₂²×C₁₆×C₈₉	C₂²	1	6	yes	—
17	C₂²×Dic₁₇₈	—	2	8	no	—
18	C₂²×Dic₁₇₈	—	2	6	no	—
19	C₂²×Dic₁₇₈	—	4	3	no	—
20	C₂²×Dic₁₇₈	—	4	4	no	—
21	C₂³×Dic₈₉	—	4	1	no	—
22	C₂³×Dic₈₉	—	4	1	no	—
23	C₂×Dic₃₅₆	—	2	24	no	—
24	C₂×Dic₃₅₆	—	2	12	no	—
25	C₂×Dic₃₅₆	—	2	6	no	—
26	C₂×Dic₃₅₆	C₂×C₄	1	24	yes	complemented
27	C₂×Dic₃₅₆	C₂³	1	4	yes	—
28	C₂×C₄⋊Dic₈₉	—	2	6	no	—
29	C₂×C₄⋊Dic₈₉	—	4	3	no	—
30	#15667946	—	2	6	no	—
31	#15667946	C₂³	1	4	yes	—
32	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₈₉	D₄	1	3	yes	—
35	C₂²×C₈×C₈₉	D₄	1	4	yes	—
36	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
37	C₂×C₁₆×C₈₉	—	2	12	no	—
38	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
39	C₂×Dic₁₇₈	—	2	24	no	—
40	C₂×Dic₁₇₈	—	2	4	no	—
41	C₂×Dic₁₇₈	—	4	24	no	—
42	C₂×Dic₁₇₈	—	4	6	no	—
43	C₂×Dic₁₇₈	—	4	12	no	—
44	C₂²×Dic₈₉	—	4	8	no	—
45	C₂²×Dic₈₉	—	4	6	no	—
46	C₂²×Dic₈₉	—	4	6	no	—
47	C₂²×Dic₈₉	—	8	4	no	—
48	Dic₃₅₆	—	2	24	no	—
49	Dic₃₅₆	—	2	4	no	—
50	Dic₃₅₆	C₂²×C₄	1	8	yes	complemented
51	C₄⋊Dic₈₉	—	2	4	no	—
52	C₄⋊Dic₈₉	—	4	6	no	—
53	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₈₉	D₈	1	3	yes	—
56	C₂²×C₄×C₈₉	SD₁₆	1	4	yes	—
57	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	12	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
60	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
63	C₁₆×C₈₉	—	2	4	no	—
64	C₂³×C₈₉	#102	1	4	yes	—
65	C₂³×C₈₉	SD₃₂	1	4	yes	—
66	C₂³×C₈₉	Q₃₂	1	4	yes	—
67	C₂³×C₈₉	C₂×D₈	1	3	yes	—
68	C₂×C₄×C₈₉	#102	1	12	yes	—
69	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
70	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
71	C₂×C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
72	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₈₉	—	4	4	no	—
74	C₂×Dic₈₉	—	4	24	no	—
75	C₂×Dic₈₉	—	4	4	no	—
76	C₂×Dic₈₉	—	8	12	no	—
77	Dic₁₇₈	—	2	8	no	—
78	Dic₁₇₈	—	4	4	no	—
79	Dic₁₇₈	—	4	24	no	—
80	C₈×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
81	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₈₉	#358	1	16	yes	complemented
84	C₂²×C₈₉	C₂×#102	1	6	yes	—
85	C₂²×C₈₉	C₂×SD₃₂	1	6	yes	—
86	C₂²×C₈₉	C₂×Q₃₂	1	6	yes	—
87	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₈₉	C₂×#102	1	6	yes	—
89	C₄×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₈₉	—	4	8	no	—
92	Dic₈₉	—	8	4	no	—
93	C₂²×#358	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	C₂×#358	1	12	yes	complemented
95	C₂×C₈₉	C₂²×#102	1	1	yes	characteristic
96	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
97	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
98	C₂³×Q₁₆	—	89	1	no	—
99	C₂³×C₁₆	D₈₉	1	1	yes	characteristic
100	C₂²×#107	—	89	1	no	—
101	C₂×#358	—	89	12	no	—
102	C₈₉	C₂²×#358	1	1	yes	characteristic, complemented, Sylow
103	C₂³×Q₈	—	178	1	no	—
104	C₂²×Q₁₆	—	89	8	no	—
105	C₂²×Q₁₆	—	89	6	no	—
106	C₂²×Q₁₆	—	178	3	no	—
107	C₂²×Q₁₆	—	178	4	no	—
108	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
109	C₂²×C₄⋊C₄	—	178	1	no	—
110	C₂²×C₁₆	—	2	4	no	—
111	C₂²×C₁₆	C₂×D₈₉	1	6	yes	—
112	C₂×#107	—	89	6	no	—
113	C₂×#107	—	178	3	no	—
114	#358	—	89	16	no	—
115	C₂²×Q₈	—	356	3	no	—
116	C₂²×Q₈	—	356	4	no	—
117	C₂²×Q₈	—	178	8	no	—
118	C₂²×Q₈	—	178	6	no	—
119	C₂³×C₄	—	356	1	no	—
120	C₂³×C₄	—	356	1	no	—
121	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
122	C₂×Q₁₆	—	89	24	no	—
123	C₂×Q₁₆	—	89	4	no	—
124	C₂×Q₁₆	—	178	24	no	—
125	C₂×Q₁₆	—	178	12	no	—
126	C₂×Q₁₆	—	178	6	no	—
127	C₂²×C₈	C₄×D₈₉	1	4	yes	—
128	C₂²×C₈	D₃₅₆	1	3	yes	—
129	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
130	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
131	C₂×C₄⋊C₄	—	356	3	no	—
132	C₂×C₄⋊C₄	—	178	6	no	—
133	C₂×C₁₆	—	2	12	no	—
134	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
135	#107	—	89	4	no	—
136	#107	—	178	6	no	—
137	C₂⁴	D₇₁₂	1	1	yes	characteristic
138	C₂×Q₈	—	356	6	no	—
139	C₂×Q₈	—	356	12	no	—
140	C₂×Q₈	—	356	24	no	—
141	C₂×Q₈	—	178	4	no	—
142	C₂×Q₈	—	178	24	no	—
143	C₂²×C₄	—	356	6	no	—
144	C₂²×C₄	—	356	8	no	—
145	C₂²×C₄	—	356	6	no	—
146	C₂²×C₄	—	712	4	no	—
147	C₂²×C₄	#13083551	1	4	yes	—
148	C₂²×C₄	D₇₁₂	1	3	yes	—
149	C₂²×C₄	#13083563	1	4	yes	—
150	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
151	Q₁₆	—	89	8	no	—
152	Q₁₆	—	178	4	no	—
153	Q₁₆	—	178	24	no	—
154	C₂×C₈	#13083551	1	12	yes	—
155	C₂×C₈	C₂×C₄×D₈₉	1	6	yes	—
156	C₂×C₈	C₂×C₂²⋊D₈₉	1	6	yes	—
157	C₂×C₈	C₂×D₃₅₆	1	3	yes	—
158	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
159	C₄⋊C₄	—	356	6	no	—
160	C₄⋊C₄	—	178	4	no	—
161	C₁₆	—	2	4	no	—
162	C₂³	#15667934	1	4	yes	—
163	C₂³	#15667957	1	4	yes	—
164	C₂³	Dic₇₁₂	1	4	yes	—
165	C₂³	C₂×D₇₁₂	1	3	yes	—
166	Q₈	—	356	24	no	—
167	Q₈	—	356	4	no	—
168	Q₈	—	178	8	no	—
169	C₂×C₄	—	356	24	no	—
170	C₂×C₄	—	356	4	no	—
171	C₂×C₄	—	356	4	no	—
172	C₂×C₄	—	712	12	no	—
173	C₂×C₄	#15667934	1	12	yes	—
174	C₂×C₄	C₂×#13083551	1	6	yes	—
175	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
176	C₂×C₄	C₂×#13083563	1	6	yes	—
177	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
178	C₈	C₂×#13083551	1	6	yes	—
179	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
180	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²	#19151403	1	16	yes	direct product factor
182	C₂²	C₂×#15667934	1	6	yes	—
183	C₂²	C₂×#15667957	1	6	yes	—
184	C₂²	C₂×Dic₇₁₂	1	6	yes	—
185	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
186	C₄	—	356	8	no	—
187	C₄	—	712	4	no	—
188	C₄	C₂×#15667934	1	6	yes	—
189	C₄	C₂²×#13083551	1	1	yes	characteristic
190	C₄	C₂²×#13083563	1	1	yes	characteristic
191	C₂	C₂×#19151403	1	12	yes	direct product factor
192	C₂	C₂²×#15667934	1	1	yes	characteristic
193	C₂	C₂²×#15667957	1	1	yes	characteristic
194	C₂	C₂²×Dic₇₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	C₂²×#358
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#19151403	C₂	direct (N × H)
8. C₂²×Dic₃₅₆	C₄	semidirect
13. #19151403	C₂²	direct (N × H)
26. C₂×Dic₃₅₆	C₂×C₄	semidirect
50. Dic₃₅₆	C₂²×C₄	semidirect
83. C₂²×C₈₉	#358	semidirect
94. C₂×C₈₉	C₂×#358	semidirect
102. C₈₉	C₂²×#358	semidirect
181. C₂²	#19151403	direct (N × H)
191. C₂	C₂×#19151403	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₅₆	2
C₂²×#15667946	2
C₂×#19151403	2
C₂³×C₁₆×C₈₉	2
C₂²×#358	89

Subgroups — C22×#19151403 #36707305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151403 #36707305