Subgroups — #36453003

Order 22784.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151350	C₂	1	2	yes	maximal
4	#26018086	C₂	1	2	yes	maximal
5	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10623494	—	2	1	no	—
11	C₂×C₄×#10623494	C₂²	1	2	yes	—
12	#19151353	—	2	1	no	—
13	#19151353	C₄	1	2	yes	—
14	#19151350	—	2	4	no	—
15	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₄³×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	—	2	1	no	—
20	C₂²×#10623494	—	2	1	no	—
21	C₂²×#10623494	C₂×C₄	1	2	yes	—
22	C₂²×#10623494	C₂×C₄	1	2	yes	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	1	no	—
25	C₄⋊#10623494	—	2	1	no	—
26	C₄⋊#10623494	C₈	1	4	yes	complemented
27	C₄⋊#10623494	D₄	1	2	yes	—
28	C₄⋊#10623494	Q₈	1	2	yes	—
29	C₄×#10623494	—	2	2	no	—
30	C₄×#10623494	—	2	2	no	—
31	C₄×#10623494	—	4	2	no	—
32	C₂²×C₈×C₈₉	—	2	2	no	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	C₄⋊C₈×C₈₉	—	4	2	no	—
43	C₄⋊C₈×C₈₉	—	4	2	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	2	2	no	—
49	C₂×#10623494	—	2	2	no	—
50	C₂×#10623494	—	2	2	no	—
51	C₂×#10623494	—	4	1	no	—
52	C₂×#10623494	—	4	1	no	—
53	C₂×#10623494	—	4	1	no	—
54	C₂×#10623494	—	4	1	no	—
55	C₂×#10623494	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₈₉	—	4	2	no	—
64	C₂×C₈×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₄²×C₈₉	—	2	2	no	—
67	C₄²×C₈₉	—	2	1	no	—
68	C₄²×C₈₉	—	2	1	no	—
69	C₄²×C₈₉	—	2	1	no	—
70	C₄²×C₈₉	—	2	1	no	—
71	C₄²×C₈₉	—	4	1	no	—
72	C₄²×C₈₉	C₂×C₈	1	2	yes	—
73	C₄²×C₈₉	M₁₆	1	2	yes	—
74	C₄²×C₈₉	D₈	1	2	yes	—
75	C₄²×C₈₉	Q₁₆	1	2	yes	—
76	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₈₉	#41	1	1	yes	characteristic
79	C₄²×C₈₉	#41	1	1	yes	characteristic
80	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₈₉	#107	1	2	yes	—
93	C₂×C₄×C₈₉	#107	1	2	yes	—
94	C₂×C₄×C₈₉	#108	1	2	yes	—
95	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
109	C₈×C₈₉	—	8	2	no	—
110	#10623494	—	2	4	no	—
111	#10623494	—	4	2	no	—
112	#10623494	—	4	2	no	—
113	#10623494	—	4	2	no	—
114	C₂²×C₈₉	#389	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₈₉	#424	1	1	yes	characteristic
117	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
118	C₂²×C₈₉	#428	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
120	C₂²×C₈₉	#430	1	1	yes	characteristic
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	4	1	no	—
126	C₄×C₈₉	#335	1	4	yes	complemented
127	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
128	C₄×C₈₉	#436	1	1	yes	characteristic
129	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
130	C₄×C₈₉	#446	1	1	yes	characteristic
131	C₄×C₈₉	#459	1	1	yes	characteristic
132	C₄×C₈₉	#462	1	1	yes	characteristic
133	C₄×C₈₉	#474	1	1	yes	characteristic
134	C₄×C₈₉	#477	1	1	yes	characteristic
135	#9601	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
137	C₂×C₈₉	#1565	1	1	yes	characteristic
138	C₂×C₈₉	#1571	1	1	yes	characteristic
139	C₂×C₈₉	#1574	1	1	yes	characteristic
140	C₂×C₈₉	#1939	1	1	yes	characteristic
141	C₂×C₈₉	#1947	1	1	yes	characteristic
142	C₂×C₈₉	#1950	1	1	yes	characteristic
143	#1842	D₈₉	1	1	yes	characteristic
144	C₄×C₄⋊C₈	—	89	1	no	—
145	C₄×C₄⋊C₈	—	89	1	no	—
146	C₂×#335	—	89	2	no	—
147	#1277	—	89	2	no	—
148	C₈₉	#9601	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	—	89	1	no	—
152	C₂×C₄⋊C₈	—	89	1	no	—
153	C₂×C₄⋊C₈	—	89	1	no	—
154	C₂×C₄⋊C₈	—	89	1	no	—
155	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
156	C₂×C₄×C₈	—	89	2	no	—
157	C₂×C₄×C₈	—	178	1	no	—
158	C₄³	C₂×D₈₉	1	1	yes	characteristic
159	#336	—	89	2	no	—
160	#336	—	178	1	no	—
161	#335	—	178	4	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	89	2	no	—
164	C₂²×C₈	—	89	2	no	—
165	C₂²×C₈	—	178	1	no	—
166	C₂²×C₈	—	178	1	no	—
167	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
168	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	4	2	no	—
177	C₄⋊C₈	—	4	2	no	—
178	C₄⋊C₈	—	89	4	no	—
179	C₄⋊C₈	—	89	2	no	—
180	C₄⋊C₈	—	89	2	no	—
181	C₄⋊C₈	—	178	2	no	—
182	C₄⋊C₈	—	178	1	no	—
183	C₄⋊C₈	—	178	1	no	—
184	C₄×C₈	—	356	2	no	—
185	C₄×C₈	—	178	2	no	—
186	C₄×C₈	—	178	2	no	—
187	C₂²×C₄	#13083553	1	1	yes	characteristic
188	C₂²×C₄	#13083553	1	1	yes	characteristic
189	C₂²×C₄	#13083553	1	1	yes	characteristic
190	C₂²×C₄	#13083553	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	89	4	no	—
198	C₂×C₈	—	356	1	no	—
199	C₂×C₈	—	356	1	no	—
200	C₂×C₈	—	356	1	no	—
201	C₂×C₈	—	356	1	no	—
202	C₂×C₈	—	178	2	no	—
203	C₂×C₈	—	178	2	no	—
204	C₂×C₈	—	178	2	no	—
205	C₂×C₈	—	178	2	no	—
206	C₂×C₈	—	178	2	no	—
207	C₂×C₈	—	178	2	no	—
208	C₂×C₈	—	178	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₈×D₈₉	1	2	yes	—
216	C₄²	#13083557	1	2	yes	—
217	C₄²	#13083559	1	2	yes	—
218	C₄²	#13083564	1	2	yes	—
219	C₄²	D₄×D₈₉	1	1	yes	characteristic
220	C₄²	Q₈×D₈₉	1	1	yes	characteristic
221	C₄²	#13083573	1	1	yes	characteristic
222	C₄²	#13083575	1	1	yes	characteristic
223	C₂³	C₂×#13083553	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	#15667941	1	2	yes	—
234	C₂×C₄	#15667941	1	2	yes	—
235	C₂×C₄	#15667945	1	2	yes	—
236	C₂×C₄	#15667945	1	2	yes	—
237	C₂×C₄	#15667947	1	2	yes	—
238	C₂×C₄	#15667974	1	1	yes	characteristic
239	C₂×C₄	#15667977	1	1	yes	characteristic
240	C₂×C₄	#15667983	1	1	yes	characteristic
241	C₂×C₄	#15667988	1	1	yes	characteristic
242	C₂×C₄	#15668003	1	1	yes	characteristic
243	C₂×C₄	#15668011	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
245	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
246	C₂×C₄	#15668020	1	1	yes	characteristic
247	C₂×C₄	#15668020	1	1	yes	characteristic
248	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
249	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
250	C₂×C₄	#15668038	1	1	yes	characteristic
251	C₂×C₄	#15668043	1	1	yes	characteristic
252	C₈	—	8	2	no	—
253	C₈	—	356	2	no	—
254	C₈	—	356	2	no	—
255	C₈	—	356	2	no	—
256	C₈	—	178	4	no	—
257	C₂²	#19151494	1	1	yes	characteristic
258	C₂²	C₂×#15667941	1	1	yes	characteristic
259	C₂²	#19151626	1	1	yes	characteristic
260	C₂²	C₂×#15667945	1	1	yes	characteristic
261	C₂²	#19151638	1	1	yes	characteristic
262	C₂²	C₂×#15667947	1	1	yes	characteristic
263	C₂²	#19151646	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#19151350	1	4	yes	complemented
270	C₄	#19151663	1	1	yes	characteristic
271	C₄	#19151675	1	1	yes	characteristic
272	C₄	#19151722	1	1	yes	characteristic
273	C₄	#19151725	1	1	yes	characteristic
274	C₄	#19151790	1	1	yes	characteristic
275	C₄	#19151805	1	1	yes	characteristic
276	C₄	#19151875	1	1	yes	characteristic
277	C₄	#19151890	1	1	yes	characteristic
278	C₂	C₂×#19151350	1	1	yes	characteristic
279	C₂	#26019002	1	1	yes	characteristic
280	C₂	#26019028	1	1	yes	characteristic
281	C₂	#26019043	1	1	yes	characteristic
282	C₂	#26020968	1	1	yes	characteristic
283	C₂	#26021008	1	1	yes	characteristic
284	C₂	#26021025	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#9601
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
26. C₄⋊#10623494	C₈	semidirect
126. C₄×C₈₉	#335	semidirect
148. C₈₉	#9601	semidirect
269. C₄	#19151350	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10623494	2
C₂×#19151350	2
#26018086	2
C₈₉×#1842	2
#9601	89