Subgroups — #36584411

Order 22784.

648 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151647	C₂	1	1	yes	maximal, characteristic
2	#26021832	C₂	1	8	yes	maximal, complemented
3	C₂×#19151368	C₂	1	2	yes	maximal
4	#26019979	C₂	1	2	yes	maximal
5	C₂×#19151356	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2905	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₂²	1	2	yes	—
8	C₂²×#13083558	C₂²	1	2	yes	—
9	#19151647	—	2	2	no	—
10	#19151647	C₂²	1	4	yes	—
11	C₂×#15667948	—	2	2	no	—
12	#19151368	C₂²	1	8	yes	—
13	#19151363	C₂²	1	8	yes	—
14	#19151356	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
18	#424×C₈₉	—	2	2	no	—
19	#424×C₈₉	C₂²	1	4	yes	—
20	#423×C₈₉	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	—	2	2	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	—	2	2	no	—
29	C₂×#13083558	—	2	2	no	—
30	C₂×#13083558	—	2	2	no	—
31	C₂×#13083558	C₂³	1	4	yes	—
32	C₂×#13083558	C₂³	1	4	yes	—
33	C₂²×#10623494	—	2	2	no	—
34	C₂²×#10623494	—	2	2	no	—
35	#15667948	—	2	8	no	—
36	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₈₉	—	2	2	no	—
38	C₂×M₁₆×C₈₉	—	2	2	no	—
39	C₂×M₁₆×C₈₉	—	2	1	no	—
40	C₂×M₁₆×C₈₉	—	2	1	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
44	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₈₉	D₄	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
47	#117×C₈₉	—	2	1	no	—
48	#117×C₈₉	—	2	1	no	—
49	#117×C₈₉	Q₈	1	2	yes	—
50	#117×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₈₉	Q₈	1	2	yes	—
53	C₂×C₄²×C₈₉	—	2	1	no	—
54	C₄⋊C₈×C₈₉	—	2	8	no	—
55	#13083558	—	2	4	no	—
56	#13083558	—	2	4	no	—
57	#13083558	—	2	4	no	—
58	#13083558	—	2	4	no	—
59	#13083558	—	4	2	no	—
60	#13083558	—	4	2	no	—
61	#13083558	—	4	2	no	—
62	#13083558	—	4	2	no	—
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	—	2	4	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₂×#10623494	—	4	2	no	—
67	C₂×#10623494	—	4	2	no	—
68	C₂×#10623494	—	4	2	no	—
69	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	#41	1	2	yes	—
77	C₂²×C₄×C₈₉	#41	1	2	yes	—
78	C₂²×C₄×C₈₉	#41	1	2	yes	—
79	C₂²×C₄×C₈₉	#41	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₈₉	—	2	4	no	—
82	M₁₆×C₈₉	—	2	4	no	—
83	M₁₆×C₈₉	—	4	1	no	—
84	M₁₆×C₈₉	—	4	1	no	—
85	M₁₆×C₈₉	—	4	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	4	no	—
90	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	—	2	2	no	—
93	C₂²⋊C₄×C₈₉	—	2	2	no	—
94	C₂²⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	—	8	1	no	—
98	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₈₉	#125	1	2	yes	—
101	C₂³×C₈₉	#125	1	2	yes	—
102	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	—	8	2	no	—
105	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#123	1	2	yes	—
110	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#124	1	2	yes	—
112	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#125	1	2	yes	—
114	C₂×C₄×C₈₉	#125	1	2	yes	—
115	C₂×C₄×C₈₉	#126	1	2	yes	—
116	C₂×C₄×C₈₉	#126	1	2	yes	—
117	C₂×C₄×C₈₉	#126	1	2	yes	—
118	C₂×C₄×C₈₉	#126	1	2	yes	—
119	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
123	C₈×C₈₉	—	2	4	no	—
124	C₈×C₈₉	—	4	2	no	—
125	#10623494	—	4	4	no	—
126	#10623494	—	4	4	no	—
127	C₂²×C₈₉	—	4	1	no	—
128	C₂²×C₈₉	—	4	1	no	—
129	C₂²×C₈₉	—	4	2	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	—	4	1	no	—
132	C₂²×C₈₉	#400	1	1	yes	characteristic
133	C₂²×C₈₉	#400	1	1	yes	characteristic
134	C₂²×C₈₉	#400	1	2	yes	—
135	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₈₉	C₂×#125	1	2	yes	—
137	C₄×C₈₉	—	4	4	no	—
138	C₄×C₈₉	#400	1	2	yes	—
139	C₄×C₈₉	#400	1	1	yes	characteristic
140	C₄×C₈₉	#400	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
142	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
143	C₄×C₈₉	C₂×#126	1	2	yes	—
144	#23753	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	C₂×#400	1	1	yes	characteristic
151	C₂×#430	—	89	1	no	—
152	#2905	D₈₉	1	1	yes	characteristic
153	#2103	—	89	8	no	—
154	C₂×#344	—	89	2	no	—
155	#1738	—	89	2	no	—
156	C₂×#338	—	89	1	no	—
157	C₈₉	#23753	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	89	2	no	—
159	C₂²×M₁₆	—	89	2	no	—
160	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
161	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
162	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
163	#430	—	89	4	no	—
164	#430	—	178	2	no	—
165	C₂×#108	—	178	2	no	—
166	#424	—	2	2	no	—
167	#424	C₂×D₈₉	1	4	yes	—
168	#423	—	2	2	no	—
169	#344	—	89	8	no	—
170	#341	—	89	8	no	—
171	#338	—	89	4	no	—
172	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	89	4	no	—
180	C₂×M₁₆	—	89	4	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	2	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂×M₁₆	—	178	2	no	—
185	C₂×M₁₆	—	178	2	no	—
186	C₂×M₁₆	—	178	2	no	—
187	C₂×M₁₆	—	178	2	no	—
188	C₂×M₁₆	—	178	2	no	—
189	C₂×M₁₆	—	178	2	no	—
190	C₂×M₁₆	—	178	2	no	—
191	C₂×M₁₆	C₂²⋊D₈₉	1	2	yes	—
192	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
193	C₂²×C₈	—	178	2	no	—
194	C₂²×C₈	—	178	2	no	—
195	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
196	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	Dic₁₇₈	1	2	yes	—
200	#117	C₂²×D₈₉	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	Dic₁₇₈	1	2	yes	—
203	C₂×C₄²	—	2	1	no	—
204	#108	—	178	8	no	—
205	C₄⋊C₈	—	2	8	no	—
206	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
211	C₂²×C₄	#13083573	1	2	yes	—
212	C₂²×C₄	#13083573	1	2	yes	—
213	C₂²×C₄	#13083574	1	1	yes	characteristic
214	C₂²×C₄	#13083574	1	1	yes	characteristic
215	C₂²×C₄	#13083575	1	2	yes	—
216	C₂²×C₄	#13083575	1	2	yes	—
217	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	356	2	no	—
224	M₁₆	—	356	2	no	—
225	M₁₆	—	356	2	no	—
226	M₁₆	—	356	2	no	—
227	M₁₆	—	178	4	no	—
228	M₁₆	—	178	4	no	—
229	M₁₆	—	178	4	no	—
230	M₁₆	—	178	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	356	2	no	—
236	C₂×C₈	—	356	2	no	—
237	C₂×C₈	—	356	2	no	—
238	C₂×C₈	—	356	2	no	—
239	C₂×C₈	—	178	4	no	—
240	C₂×C₈	—	178	4	no	—
241	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	C₂×Dic₁₇₈	1	4	yes	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₂³	—	8	1	no	—
249	C₂³	#15667991	1	1	yes	characteristic
250	C₂³	#15667991	1	1	yes	characteristic
251	C₂³	#15668003	1	2	yes	—
252	C₂³	#15668003	1	2	yes	—
253	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	#15667991	1	1	yes	characteristic
257	C₂×C₄	#15667991	1	1	yes	characteristic
258	C₂×C₄	#15667993	1	1	yes	characteristic
259	C₂×C₄	#15667993	1	1	yes	characteristic
260	C₂×C₄	#15667993	1	2	yes	—
261	C₂×C₄	#15668001	1	1	yes	characteristic
262	C₂×C₄	#15668001	1	2	yes	—
263	C₂×C₄	#15668001	1	1	yes	characteristic
264	C₂×C₄	#15668003	1	2	yes	—
265	C₂×C₄	#15668003	1	2	yes	—
266	C₂×C₄	#15668007	1	2	yes	—
267	C₂×C₄	#15668007	1	2	yes	—
268	C₂×C₄	#15668007	1	2	yes	—
269	C₂×C₄	#15668007	1	2	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
271	C₂×C₄	C₂×#13083573	1	2	yes	—
272	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
273	C₂×C₄	C₂×#13083575	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	356	4	no	—
277	C₈	—	356	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#19151537	1	1	yes	characteristic
284	C₂²	#19151537	1	1	yes	characteristic
285	C₂²	#19151537	1	2	yes	—
286	C₂²	C₂×#15667991	1	1	yes	characteristic
287	C₂²	C₂×#15668003	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	#19151537	1	2	yes	—
290	C₄	#19151537	1	1	yes	characteristic
291	C₄	#19151537	1	1	yes	characteristic
292	C₄	C₂×#15667993	1	1	yes	characteristic
293	C₄	C₂×#15668001	1	1	yes	characteristic
294	C₄	C₂×#15668007	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₂×#19151537	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#15668001
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#2905	C₂
socle	C₂×C₈₉	C₂×#400
prime radical	C₂⁴×C₈₉	C₂×Q₈
hypercenter	#2905	D₈₉
hypocenter	C₈₉	#23753
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #26021832	C₂	semidirect
157. C₈₉	#23753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151647	2
#26021832	2
C₂×#19151368	2
#26019979	2
C₂×#19151356	2
C₈₉×#2905	2
#23753	89