Subgroups — C₄×#19152000 #36563172

Order 22784.

774 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152000	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668004	C₂	1	2	yes	maximal
3	#26021217	C₂	1	2	yes	maximal
4	#26020985	C₂	1	2	yes	maximal
5	#26020980	C₂	1	2	yes	maximal
6	#26020416	C₂	1	1	yes	maximal, characteristic
7	C₄×#15667946	C₂	1	2	yes	maximal
8	C₄×#15667944	C₂	1	2	yes	maximal
9	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668004	C₂²	1	2	yes	—
11	#19152000	C₄	1	8	yes	direct product factor
12	C₂×#15667946	C₂²	1	2	yes	—
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667944	C₂²	1	2	yes	—
15	C₂×#15667944	C₂²	1	2	yes	—
16	#19151496	C₂²	1	4	yes	—
17	#19151482	—	2	4	no	—
18	#19151464	C₂²	1	2	yes	—
19	C₄×#13083553	—	2	4	no	—
20	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
21	#19151361	C₂²	1	4	yes	—
22	#19151361	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₄³×C₈₉	C₂²	1	1	yes	characteristic
30	#15668004	—	2	4	no	—
31	#15668004	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
34	C₂×#13083553	—	2	4	no	—
35	C₂×#13083553	—	2	4	no	—
36	C₂×C₄×Dic₈₉	—	2	4	no	—
37	#15667946	—	2	4	no	—
38	#15667946	C₂×C₄	1	8	yes	—
39	#15667944	—	2	4	no	—
40	#15667944	C₂×C₄	1	8	yes	—
41	#15667919	—	2	4	no	—
42	#15667919	—	2	4	no	—
43	#15667919	—	2	4	no	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	2	yes	—
49	C₂×C₄²×C₈₉	D₄	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₈₉	—	2	2	no	—
54	C₄×C₈×C₈₉	—	2	2	no	—
55	C₄×C₈×C₈₉	—	2	2	no	—
56	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₈₉	Q₈	1	4	yes	—
58	C₄×C₈×C₈₉	Q₈	1	4	yes	—
59	C₂²×Dic₈₉	—	2	4	no	—
60	C₂²×Dic₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	—	2	4	no	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	—	2	4	no	—
64	C₄⋊Dic₈₉	C₂²×C₄	1	8	yes	—
65	#13083553	—	2	16	no	—
66	#13083553	—	4	8	no	—
67	C₄×Dic₈₉	—	4	8	no	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	—	2	1	no	—
75	C₂×C₈×C₈₉	—	2	1	no	—
76	C₂×C₈×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₈₉	#41	1	2	yes	—
84	C₂×C₈×C₈₉	#41	1	2	yes	—
85	C₄²×C₈₉	—	2	4	no	—
86	C₄²×C₈₉	—	2	4	no	—
87	C₄²×C₈₉	—	2	1	no	—
88	C₄²×C₈₉	—	2	1	no	—
89	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
92	C₄²×C₈₉	#41	1	2	yes	—
93	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
101	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
102	C₂×C₄×C₈₉	#135	1	4	yes	—
103	C₂×C₄×C₈₉	#135	1	4	yes	—
104	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₈₉	—	2	8	no	—
111	C₂×Dic₈₉	—	4	4	no	—
112	C₂×Dic₈₉	—	4	4	no	—
113	C₂×Dic₈₉	—	4	4	no	—
114	C₂×Dic₈₉	—	4	4	no	—
115	C₂×Dic₈₉	—	4	4	no	—
116	C₈×C₈₉	—	2	2	no	—
117	C₈×C₈₉	—	2	2	no	—
118	C₈×C₈₉	C₄×Q₈	1	4	yes	—
119	C₈×C₈₉	C₄×Q₈	1	4	yes	—
120	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₈₉	#532	1	1	yes	characteristic
123	C₂²×C₈₉	C₂×#135	1	2	yes	—
124	C₂²×C₈₉	C₂×#135	1	2	yes	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	4	no	—
127	C₄×C₈₉	—	2	4	no	—
128	C₄×C₈₉	#499	1	4	yes	complemented
129	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₈₉	#533	1	1	yes	characteristic
132	C₄×C₈₉	#533	1	1	yes	characteristic
133	Dic₈₉	—	4	8	no	—
134	Dic₈₉	—	8	4	no	—
135	C₄×#499	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₈₉	C₄×#135	1	2	yes	—
138	C₂×C₈₉	C₂×#499	1	1	yes	characteristic
139	C₂×C₈₉	#3160	1	1	yes	characteristic
140	C₂×C₈₉	#3168	1	2	yes	—
141	C₂×#499	—	89	1	no	—
142	C₄×#125	—	89	2	no	—
143	#1991	—	89	2	no	—
144	#1942	—	89	2	no	—
145	#1941	—	89	2	no	—
146	#1846	—	89	1	no	—
147	C₄×#107	—	89	2	no	—
148	C₄×#106	—	89	2	no	—
149	C₄²×C₈	D₈₉	1	1	yes	characteristic
150	C₈₉	C₄×#499	1	1	yes	characteristic, complemented, Sylow
151	C₂×#125	—	89	2	no	—
152	#499	—	89	8	no	—
153	C₂×#107	—	89	2	no	—
154	C₂×#107	—	89	2	no	—
155	C₂×#106	—	89	2	no	—
156	C₂×#106	—	89	2	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
163	#389	—	89	4	no	—
164	#387	—	178	4	no	—
165	#382	—	89	2	no	—
166	C₄×C₄⋊C₄	—	89	4	no	—
167	C₄×C₄⋊C₄	—	178	4	no	—
168	C₄³	C₂×D₈₉	1	1	yes	characteristic
169	#340	—	89	4	no	—
170	#340	—	89	4	no	—
171	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
175	#125	—	89	8	no	—
176	#125	—	178	4	no	—
177	C₂×C₄⋊C₄	—	89	4	no	—
178	C₂×C₄⋊C₄	—	89	4	no	—
179	C₂×C₄⋊C₄	—	178	4	no	—
180	C₂×C₄⋊C₄	—	178	4	no	—
181	C₂×C₄²	—	178	4	no	—
182	C₂×C₄²	D₃₅₆	1	2	yes	—
183	C₂×C₄²	D₃₅₆	1	2	yes	—
184	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	#107	—	89	8	no	—
188	#107	—	178	4	no	—
189	#106	—	89	8	no	—
190	#106	—	178	4	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	Dic₁₇₈	1	4	yes	—
195	C₄×C₈	Dic₁₇₈	1	4	yes	—
196	C₄×C₈	C₄×D₈₉	1	4	yes	—
197	#95	—	178	4	no	—
198	#95	—	178	4	no	—
199	#95	—	178	4	no	—
200	C₂²×C₄	—	178	4	no	—
201	C₂²×C₄	—	178	4	no	—
202	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
203	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
205	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
206	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
213	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
214	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
215	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
216	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
217	C₂×C₈	#13083574	1	2	yes	—
218	C₂×C₈	#13083574	1	2	yes	—
219	C₄⋊C₄	—	89	8	no	—
220	C₄⋊C₄	—	356	8	no	—
221	C₄⋊C₄	—	178	4	no	—
222	C₄⋊C₄	—	178	4	no	—
223	C₄⋊C₄	—	178	16	no	—
224	C₄⋊C₄	—	178	4	no	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	356	8	no	—
230	C₄²	C₂×C₄×D₈₉	1	2	yes	—
231	C₄²	C₂×Dic₁₇₈	1	2	yes	—
232	C₄²	C₂×Dic₁₇₈	1	2	yes	—
233	C₄²	#13083574	1	2	yes	—
234	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	356	4	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	—	356	4	no	—
243	C₂×C₄	—	356	4	no	—
244	C₂×C₄	—	356	4	no	—
245	C₂×C₄	—	178	8	no	—
246	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
247	C₂×C₄	#15668012	1	2	yes	—
248	C₂×C₄	#15668012	1	2	yes	—
249	C₂×C₄	#15668034	1	4	yes	—
250	C₂×C₄	#15668034	1	4	yes	—
251	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
254	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
255	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	C₄×Dic₁₇₈	1	4	yes	—
260	C₈	C₄×Dic₁₇₈	1	4	yes	—
261	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
262	C₂²	C₂×#15668012	1	1	yes	characteristic
263	C₂²	#19152148	1	1	yes	characteristic
264	C₂²	C₂×#15668034	1	2	yes	—
265	C₂²	C₂×#15668034	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	356	8	no	—
270	C₄	—	712	4	no	—
271	C₄	#19152000	1	4	yes	direct product factor
272	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
273	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
274	C₄	#19152155	1	1	yes	characteristic
275	C₄	#19152155	1	1	yes	characteristic
276	C₂	C₄×#15668012	1	1	yes	characteristic
277	C₂	C₄×#15668034	1	2	yes	—
278	C₂	C₂×#19152000	1	1	yes	characteristic
279	C₂	#26030766	1	1	yes	characteristic
280	C₂	#26030822	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×#499
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. #19152000	C₄	direct (N × H)
128. C₄×C₈₉	#499	semidirect
150. C₈₉	C₄×#499	semidirect
271. C₄	#19152000	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×#19152000	2
C₄×#15668004	2
#26021217	2
#26020985	2
#26020980	2
#26020416	2
C₄×#15667946	2
C₄×#15667944	2
C₄²×C₈×C₈₉	2
C₄×#499	89

Subgroups — C4×#19152000 #36563172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19152000 #36563172