Subgroups — #36469473

Order 22784.

490 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×#19151552	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151379	C₂	1	2	yes	maximal
3	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018335	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂×#15667969	C₄	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	#19151552	—	2	1	no	—
11	#19151552	—	2	1	no	—
12	#19151552	—	2	1	no	—
13	#19151552	—	2	1	no	—
14	C₂×#15667923	C₄	1	1	yes	characteristic
15	C₂×#15667922	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
17	#19151379	C₄	1	8	yes	complemented
18	#19151374	—	2	1	no	—
19	C₄×#13083549	—	2	1	no	—
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#346×C₈₉	—	2	1	no	—
23	C₄×C₁₆×C₈₉	—	2	1	no	—
24	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×#13083557	—	2	1	no	—
26	C₂×#13083557	—	2	1	no	—
27	C₂×#13083557	—	2	2	no	—
28	C₂×#13083557	C₂×C₄	1	4	yes	—
29	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₈×D₈₉	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	2	no	—
34	C₂×C₈×D₈₉	C₈	1	4	yes	—
35	#15667969	—	2	1	no	—
36	#15667969	—	2	1	no	—
37	#15667969	—	2	2	no	—
38	C₂×#13083553	C₈	1	2	yes	—
39	C₂×#13083551	C₈	1	2	yes	—
40	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	#15667952	—	2	2	no	—
42	#15667952	C₂×C₄	1	4	yes	—
43	C₂×#13083549	—	2	2	no	—
44	#15667923	—	2	1	no	—
45	#15667923	—	2	1	no	—
46	#15667922	—	2	1	no	—
47	#15667922	—	2	1	no	—
48	C₈×Dic₈₉	—	2	1	no	—
49	C₈×Dic₈₉	—	2	1	no	—
50	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₈₉	—	2	2	no	—
54	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
55	C₂×C₁₆×C₈₉	—	2	2	no	—
56	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	C₄²	1	2	yes	—
68	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
69	#13083557	—	2	4	no	—
70	#13083557	—	2	4	no	—
71	C₂×#10623494	—	2	1	no	—
72	C₂×#10623494	—	2	1	no	—
73	C₂×#10623494	—	2	1	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₂×#10623494	C₄²	1	2	yes	—
76	C₂×#10623494	C₂×C₈	1	2	yes	—
77	C₈×D₈₉	—	2	4	no	—
78	C₈×D₈₉	—	2	4	no	—
79	#13083553	—	2	2	no	—
80	#13083553	—	2	2	no	—
81	#13083551	—	2	2	no	—
82	#13083551	—	2	2	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	#13083549	—	2	4	no	—
86	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
91	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	M₁₆	1	2	yes	—
99	C₁₆×C₈₉	—	2	4	no	—
100	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
101	C₂²×D₈₉	—	2	1	no	—
102	C₂²×D₈₉	—	2	1	no	—
103	C₂²×D₈₉	—	4	2	no	—
104	C₂²×D₈₉	C₄×C₈	1	2	yes	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
114	C₂×Dic₈₉	—	2	1	no	—
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	2	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	C₄×C₈	1	2	yes	—
119	C₄×D₈₉	—	2	2	no	—
120	C₄×D₈₉	—	2	2	no	—
121	C₄×D₈₉	—	2	2	no	—
122	C₄×D₈₉	—	2	2	no	—
123	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
124	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
125	C₈×C₈₉	C₄×D₄	1	2	yes	—
126	C₈×C₈₉	C₄×D₄	1	2	yes	—
127	#10623494	—	2	2	no	—
128	#10623494	—	2	2	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₈₉	#431	1	1	yes	characteristic
133	C₂×D₈₉	—	2	2	no	—
134	C₂×D₈₉	—	4	2	no	—
135	C₂×D₈₉	—	4	2	no	—
136	C₂×D₈₉	—	4	2	no	—
137	C₄×C₈₉	—	2	2	no	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
141	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₈₉	#431	1	1	yes	characteristic
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	—	4	2	no	—
145	#12602	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
149	C₂×C₈₉	#2115	1	2	yes	—
150	D₈₉	—	4	2	no	—
151	C₂×#405	—	89	1	no	—
152	C₂×#349	—	89	2	no	—
153	C₄×M₃₂	—	89	1	no	—
154	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
155	#1377	—	89	2	no	—
156	C₈₉	#12602	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	89	1	no	—
158	C₂³×C₈	—	89	1	no	—
159	C₂×#117	—	89	1	no	—
160	C₂×M₃₂	—	89	2	no	—
161	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
162	#405	—	178	1	no	—
163	#405	—	178	1	no	—
164	#405	—	178	1	no	—
165	#405	—	178	1	no	—
166	C₂×#97	—	89	1	no	—
167	C₂×#97	—	89	1	no	—
168	C₂×C₄×C₈	—	89	1	no	—
169	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
170	#349	—	89	8	no	—
171	#346	—	2	1	no	—
172	#346	—	178	1	no	—
173	C₄×C₁₆	—	2	1	no	—
174	C₄×C₁₆	—	178	1	no	—
175	C₂³×C₄	—	89	1	no	—
176	C₂×M₁₆	—	89	4	no	—
177	C₂×M₁₆	—	178	1	no	—
178	C₂×M₁₆	—	178	1	no	—
179	C₂×M₁₆	—	178	2	no	—
180	C₂²×C₈	—	89	1	no	—
181	C₂²×C₈	—	89	1	no	—
182	C₂²×C₈	—	89	4	no	—
183	C₂²×C₈	—	178	1	no	—
184	C₂²×C₈	—	178	1	no	—
185	C₂²×C₈	—	178	2	no	—
186	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
188	#117	—	178	1	no	—
189	#117	—	178	1	no	—
190	#117	—	178	2	no	—
191	C₂×C₄⋊C₄	—	89	2	no	—
192	C₂×C₂²⋊C₄	—	89	2	no	—
193	C₂×C₄²	—	89	1	no	—
194	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
195	M₃₂	—	2	2	no	—
196	M₃₂	—	89	4	no	—
197	M₃₂	—	178	2	no	—
198	M₃₂	C₄×D₈₉	1	4	yes	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	178	2	no	—
201	#97	—	178	1	no	—
202	#97	—	178	1	no	—
203	#97	—	178	1	no	—
204	#97	—	178	1	no	—
205	C₄×C₈	—	178	1	no	—
206	C₄×C₈	—	178	1	no	—
207	C₄×C₈	D₃₅₆	1	1	yes	characteristic
208	C₄×C₈	D₃₅₆	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
211	C₂⁴	—	89	1	no	—
212	C₂²×C₄	—	89	1	no	—
213	C₂²×C₄	—	89	2	no	—
214	C₂²×C₄	—	89	2	no	—
215	C₂²×C₄	—	89	2	no	—
216	C₂²×C₄	—	178	1	no	—
217	C₂²×C₄	—	178	1	no	—
218	C₂²×C₄	—	178	1	no	—
219	C₂²×C₄	—	178	1	no	—
220	C₂²×C₄	C₈×D₈₉	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
222	M₁₆	—	178	4	no	—
223	M₁₆	—	178	4	no	—
224	C₂×C₈	—	89	2	no	—
225	C₂×C₈	—	89	2	no	—
226	C₂×C₈	—	178	1	no	—
227	C₂×C₈	—	178	1	no	—
228	C₂×C₈	—	178	1	no	—
229	C₂×C₈	—	178	1	no	—
230	C₂×C₈	—	178	4	no	—
231	C₂×C₈	—	178	4	no	—
232	C₂×C₈	#13083551	1	1	yes	characteristic
233	C₂×C₈	#13083551	1	1	yes	characteristic
234	C₂×C₈	C₈×D₈₉	1	2	yes	—
235	C₂×C₈	#13083557	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
239	C₂×C₈	#13083574	1	2	yes	—
240	C₄⋊C₄	—	178	2	no	—
241	C₄⋊C₄	—	178	2	no	—
242	C₂²⋊C₄	—	178	2	no	—
243	C₂²⋊C₄	—	178	2	no	—
244	C₄²	—	178	1	no	—
245	C₄²	—	178	1	no	—
246	C₄²	#13083551	1	1	yes	characteristic
247	C₄²	#13083551	1	1	yes	characteristic
248	C₄²	#13083557	1	2	yes	—
249	C₁₆	—	2	4	no	—
250	C₁₆	—	178	4	no	—
251	C₂³	—	89	2	no	—
252	C₂³	—	356	2	no	—
253	C₂³	—	178	1	no	—
254	C₂³	—	178	1	no	—
255	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	89	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	178	2	no	—
260	C₂×C₄	—	178	1	no	—
261	C₂×C₄	—	178	1	no	—
262	C₂×C₄	—	178	2	no	—
263	C₂×C₄	—	178	2	no	—
264	C₂×C₄	—	178	2	no	—
265	C₂×C₄	—	178	2	no	—
266	C₂×C₄	#15667924	1	2	yes	—
267	C₂×C₄	#15667924	1	2	yes	—
268	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
269	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
270	C₂×C₄	#15667969	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
272	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
273	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
274	C₈	—	178	2	no	—
275	C₈	—	178	2	no	—
276	C₈	#15667924	1	2	yes	—
277	C₈	#15667924	1	2	yes	—
278	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
279	C₈	C₄×D₃₅₆	1	2	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	356	2	no	—
282	C₂²	—	356	2	no	—
283	C₂²	—	356	2	no	—
284	C₂²	—	178	2	no	—
285	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
286	C₂²	C₂×#15667924	1	1	yes	characteristic
287	C₂²	#19151649	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	356	2	no	—
291	C₄	—	178	2	no	—
292	C₄	C₄×#13083551	1	1	yes	characteristic
293	C₄	C₄×#13083557	1	1	yes	characteristic
294	C₄	C₂×#15667924	1	1	yes	characteristic
295	C₄	#19151649	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	356	2	no	—
299	C₂	C₄×#15667924	1	1	yes	characteristic
300	C₂	#26021872	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#13083551
commutator	C₂²×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#12602
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×#15667952	C₂	semidirect
4. #26018335	C₂	semidirect
5. C₄×M₃₂×C₈₉	C₂	semidirect
17. #19151379	C₄	semidirect
154. C₄×M₃₂	D₈₉	semidirect
156. C₈₉	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151552	2
C₂×#19151379	2
C₄×#15667952	2
#26018335	2
C₄×M₃₂×C₈₉	2
#12602	89