Subgroups — C₄×#19151978 #36562865

Order 22784.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151978	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	4	yes	maximal
3	C₄×#15668012	C₂	1	2	yes	maximal
4	#26021113	C₂	1	4	yes	maximal
5	#26021054	C₂	1	2	yes	maximal
6	#26020412	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₅₆	C₂²	1	4	yes	—
9	C₂×#15668012	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₇₈	C₂²	1	4	yes	—
11	#19151978	C₄	1	8	yes	direct product factor
12	C₄×Dic₃₅₆	—	2	16	no	—
13	C₂×#15667946	C₂²	1	4	yes	—
14	C₂×#15667937	C₂²	1	8	yes	—
15	#19151504	C₂²	1	4	yes	—
16	#19151496	—	2	4	no	—
17	#19151468	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₈₉	—	2	4	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
23	C₄³×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₇₈	C₂³	1	4	yes	—
25	C₂×Dic₃₅₆	—	2	8	no	—
26	C₂×Dic₃₅₆	—	2	8	no	—
27	C₂×Dic₃₅₆	C₂×C₄	1	16	yes	—
28	#15668012	—	2	4	no	—
29	#15668012	C₂×C₄	1	8	yes	—
30	C₄×Dic₁₇₈	—	2	16	no	—
31	C₂×C₄⋊Dic₈₉	—	2	4	no	—
32	C₂×C₄⋊Dic₈₉	—	2	4	no	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
34	C₂×#13083553	—	2	4	no	—
35	C₂×C₄×Dic₈₉	—	2	4	no	—
36	#15667946	—	2	8	no	—
37	#15667937	—	2	16	no	—
38	#15667919	—	2	8	no	—
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
41	C₂×C₄²×C₈₉	D₄	1	2	yes	—
42	C₂×C₄²×C₈₉	D₄	1	2	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₉	—	2	4	no	—
47	C₄×C₈×C₈₉	—	2	2	no	—
48	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₈₉	D₄	1	8	yes	—
50	C₂×Dic₁₇₈	—	2	8	no	—
51	C₂×Dic₁₇₈	—	2	8	no	—
52	C₂×Dic₁₇₈	—	2	4	no	—
53	C₂×Dic₁₇₈	C₂²×C₄	1	8	yes	—
54	C₂²×Dic₈₉	—	2	4	no	—
55	C₂²×Dic₈₉	—	2	4	no	—
56	Dic₃₅₆	—	2	32	no	—
57	C₄⋊Dic₈₉	—	2	16	no	—
58	C₄⋊Dic₈₉	—	2	8	no	—
59	C₄⋊Dic₈₉	—	4	8	no	—
60	#13083553	—	4	8	no	—
61	C₄×Dic₈₉	—	4	8	no	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₈₉	#41	1	4	yes	—
73	C₄²×C₈₉	—	2	4	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	Q₁₆	1	8	yes	complemented
76	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
77	C₄²×C₈₉	C₂×D₄	1	4	yes	—
78	C₄²×C₈₉	#41	1	2	yes	—
79	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₈₉	#127	1	2	yes	—
85	C₂×C₄×C₈₉	#127	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
88	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
89	C₂×C₄×C₈₉	#135	1	4	yes	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
94	C₂×Dic₈₉	—	2	8	no	—
95	C₂×Dic₈₉	—	4	4	no	—
96	C₂×Dic₈₉	—	4	4	no	—
97	C₂×Dic₈₉	—	4	4	no	—
98	C₂×Dic₈₉	—	4	8	no	—
99	Dic₁₇₈	—	2	16	no	—
100	Dic₁₇₈	—	4	8	no	—
101	C₈×C₈₉	—	2	4	no	—
102	C₈×C₈₉	C₄×D₄	1	8	yes	—
103	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
105	C₂²×C₈₉	#532	1	1	yes	characteristic
106	C₂²×C₈₉	C₂²×Q₁₆	1	2	yes	—
107	C₂²×C₈₉	C₂×#135	1	2	yes	—
108	C₄×C₈₉	—	2	2	no	—
109	C₄×C₈₉	—	2	4	no	—
110	C₄×C₈₉	C₄×Q₁₆	1	8	yes	complemented
111	C₄×C₈₉	#494	1	4	yes	complemented
112	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
113	C₄×C₈₉	#532	1	2	yes	—
114	Dic₈₉	—	4	8	no	—
115	Dic₈₉	—	8	4	no	—
116	C₄×#494	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
118	C₂×C₈₉	C₂×C₄×Q₁₆	1	2	yes	—
119	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
120	C₂×C₈₉	#3147	1	1	yes	characteristic
121	C₂×C₈₉	#3170	1	2	yes	—
122	C₂×#494	—	89	1	no	—
123	C₂×C₄×Q₁₆	—	89	4	no	—
124	C₄×C₄⋊Q₈	—	89	2	no	—
125	#1969	—	89	4	no	—
126	#1956	—	89	2	no	—
127	#1845	—	89	1	no	—
128	C₄²×C₈	D₈₉	1	1	yes	characteristic
129	C₈₉	C₄×#494	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	89	4	no	—
131	C₂×C₄⋊Q₈	—	89	2	no	—
132	C₂×C₄×Q₈	—	89	4	no	—
133	#494	—	89	8	no	—
134	C₄×Q₁₆	—	178	16	no	—
135	C₂×#107	—	89	4	no	—
136	C₂×#103	—	89	8	no	—
137	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
141	#391	—	89	4	no	—
142	#389	—	178	4	no	—
143	#384	—	89	2	no	—
144	C₄×C₄⋊C₄	—	178	4	no	—
145	C₄³	C₂×D₈₉	1	1	yes	characteristic
146	C₂²×Q₈	—	89	4	no	—
147	C₂×Q₁₆	—	89	16	no	—
148	C₂×Q₁₆	—	178	8	no	—
149	C₂×Q₁₆	—	178	8	no	—
150	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
151	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
152	C₄⋊Q₈	—	89	8	no	—
153	C₄⋊Q₈	—	178	4	no	—
154	C₄×Q₈	—	178	16	no	—
155	C₂×C₄⋊C₄	—	89	4	no	—
156	C₂×C₄⋊C₄	—	178	4	no	—
157	C₂×C₄⋊C₄	—	178	4	no	—
158	C₂×C₄⋊C₄	—	178	4	no	—
159	C₂×C₄²	—	178	4	no	—
160	C₂×C₄²	D₃₅₆	1	2	yes	—
161	C₂×C₄²	D₃₅₆	1	2	yes	—
162	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
165	#107	—	178	8	no	—
166	#103	—	178	16	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₄×D₈₉	1	4	yes	—
170	C₄×C₈	D₃₅₆	1	8	yes	—
171	#95	—	178	8	no	—
172	C₂×Q₈	—	89	8	no	—
173	C₂×Q₈	—	178	8	no	—
174	C₂×Q₈	—	178	8	no	—
175	C₂×Q₈	—	178	4	no	—
176	C₂²×C₄	—	178	4	no	—
177	C₂²×C₄	—	178	4	no	—
178	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
180	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
181	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
183	Q₁₆	—	178	32	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
187	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
188	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
189	C₂×C₈	#13083574	1	4	yes	—
190	C₄⋊C₄	—	356	8	no	—
191	C₄⋊C₄	—	356	8	no	—
192	C₄⋊C₄	—	178	8	no	—
193	C₄⋊C₄	—	178	16	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	356	8	no	—
197	C₄²	Dic₃₅₆	1	8	yes	complemented
198	C₄²	C₂×C₄×D₈₉	1	2	yes	—
199	C₄²	C₂×D₃₅₆	1	4	yes	—
200	C₄²	#13083574	1	2	yes	—
201	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
202	Q₈	—	356	8	no	—
203	Q₈	—	178	16	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	356	4	no	—
208	C₂×C₄	—	356	4	no	—
209	C₂×C₄	—	356	4	no	—
210	C₂×C₄	—	356	8	no	—
211	C₂×C₄	—	178	8	no	—
212	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
213	C₂×C₄	#15668009	1	2	yes	—
214	C₂×C₄	#15668009	1	2	yes	—
215	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
216	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
217	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
218	C₂×C₄	#15668034	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
221	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
222	C₂×C₄	C₂×#13083574	1	2	yes	—
223	C₈	—	2	4	no	—
224	C₈	C₄×D₃₅₆	1	8	yes	—
225	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
226	C₂²	C₂×#15668009	1	1	yes	characteristic
227	C₂²	#19152148	1	1	yes	characteristic
228	C₂²	C₂²×Dic₃₅₆	1	2	yes	—
229	C₂²	C₂×#15668034	1	2	yes	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	356	8	no	—
233	C₄	—	712	4	no	—
234	C₄	C₄×Dic₃₅₆	1	8	yes	complemented
235	C₄	#19151978	1	4	yes	direct product factor
236	C₄	C₂×C₄×D₃₅₆	1	2	yes	—
237	C₄	#19152148	1	2	yes	—
238	C₂	C₄×#15668009	1	1	yes	characteristic
239	C₂	C₂×C₄×Dic₃₅₆	1	2	yes	—
240	C₂	C₂×#19151978	1	1	yes	characteristic
241	C₂	#26030680	1	1	yes	characteristic
242	C₂	#26030838	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	C₄×#494
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. #19151978	C₄	direct (N × H)
75. C₄²×C₈₉	Q₁₆	semidirect
110. C₄×C₈₉	C₄×Q₁₆	semidirect
111. C₄×C₈₉	#494	semidirect
129. C₈₉	C₄×#494	semidirect
197. C₄²	Dic₃₅₆	semidirect
234. C₄	C₄×Dic₃₅₆	semidirect
235. C₄	#19151978	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×#19151978	2
C₂×C₄×Dic₃₅₆	2
C₄×#15668012	2
#26021113	2
#26021054	2
#26020412	2
C₄²×C₈×C₈₉	2
C₄×#494	89

Subgroups — C4×#19151978 #36562865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151978 #36562865