Subgroups — #36707649

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152015	C₂	1	2	yes	maximal, complemented
2	#26022084	C₂	1	2	yes	maximal, complemented
3	#26022078	C₂	1	4	yes	maximal, complemented
4	D₄×#13083549	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021866	C₂	1	2	yes	maximal, complemented
7	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
9	#19152015	—	2	8	no	—
10	C₂×#15667952	C₂²	1	4	yes	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂²×#13083549	C₂²	1	2	yes	—
14	C₂²×#13083549	C₂²	1	4	yes	—
15	C₄⋊#13083549	C₂²	1	1	yes	characteristic
16	C₄⋊#13083549	C₂²	1	2	yes	—
17	C₄⋊#13083549	C₂²	1	1	yes	characteristic
18	C₂²⋊#13083549	C₂²	1	4	yes	—
19	C₂²⋊#13083549	C₂²	1	4	yes	—
20	#19151374	C₂²	1	2	yes	—
21	C₄×#13083549	C₂²	1	2	yes	—
22	C₂×#131×C₈₉	C₂²	1	2	yes	—
23	#532×C₈₉	C₄	1	1	yes	characteristic
24	#436×C₈₉	C₄	1	2	yes	—
25	#436×C₈₉	C₄	1	2	yes	—
26	C₈×D₄×C₈₉	C₂²	1	2	yes	—
27	C₈×D₄×C₈₉	C₂²	1	2	yes	—
28	#423×C₈₉	C₄	1	1	yes	characteristic
29	#408×C₈₉	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#15667952	—	2	8	no	—
32	#15667952	—	2	2	no	—
33	#15667952	D₄	1	4	yes	complemented
34	C₂×#13083549	—	2	8	no	—
35	C₂×#13083549	—	2	4	no	—
36	C₂×#13083549	C₂³	1	2	yes	—
37	C₂×#13083549	C₂³	1	2	yes	—
38	C₂×#13083549	C₂³	1	2	yes	—
39	C₂×#13083549	C₂³	1	2	yes	—
40	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
41	#131×C₈₉	—	2	4	no	—
42	#131×C₈₉	—	2	4	no	—
43	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
49	#127×C₈₉	C₈	1	1	yes	characteristic
50	#124×C₈₉	C₈	1	2	yes	—
51	#121×C₈₉	C₈	1	2	yes	—
52	#121×C₈₉	C₈	1	2	yes	—
53	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
60	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	#13083549	—	2	2	no	—
65	#13083549	—	2	2	no	—
66	#13083549	—	2	2	no	—
67	#13083549	C₂×D₄	1	4	yes	—
68	#41×C₈₉	—	2	4	no	—
69	#41×C₈₉	—	2	4	no	—
70	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
71	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
73	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
74	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	M₁₆×C₈₉	—	2	4	no	—
80	M₁₆×C₈₉	—	2	8	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	2	4	no	—
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₈₉	#41	1	2	yes	—
90	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
92	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
94	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
95	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
100	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
101	Q₈×C₈₉	—	2	4	no	—
102	D₄×C₈₉	—	2	8	no	—
103	D₄×C₈₉	—	2	4	no	—
104	D₄×C₈₉	—	2	4	no	—
105	D₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#131	1	2	yes	—
118	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₈₉	—	2	4	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	C₄×D₄	1	2	yes	—
123	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
125	C₂²×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	—	2	4	no	—
129	C₂²×C₈₉	—	2	4	no	—
130	C₂²×C₈₉	C₈×D₄	1	2	yes	—
131	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	4	no	—
136	C₄×C₈₉	C₈×D₄	1	2	yes	—
137	C₄×C₈₉	#504	1	2	yes	complemented
138	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
140	#33438	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	2	2	no	—
142	C₂×C₈₉	—	4	4	no	—
143	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
145	C₂×C₈₉	#3407	1	1	yes	characteristic
146	C₂×#504	—	89	2	no	—
147	#2931	D₈₉	1	1	yes	characteristic
148	#2169	—	89	2	no	—
149	#2168	—	89	4	no	—
150	D₄×C₁₆	—	89	2	no	—
151	C₂×C₄⋊C₁₆	—	89	1	no	—
152	#2113	—	89	2	no	—
153	C₄×M₃₂	—	89	1	no	—
154	C₈₉	#33438	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	C₂×D₈₉	1	2	yes	—
156	#532	Dic₈₉	1	1	yes	characteristic
157	#504	—	178	8	no	—
158	C₂×M₃₂	—	89	1	no	—
159	C₂×M₃₂	—	89	1	no	—
160	C₂×M₃₂	—	89	4	no	—
161	C₂²×C₁₆	—	89	2	no	—
162	C₂²×C₁₆	—	89	4	no	—
163	#436	Dic₈₉	1	2	yes	—
164	#436	Dic₈₉	1	2	yes	—
165	C₈×D₄	C₂×D₈₉	1	2	yes	—
166	C₈×D₄	C₂×D₈₉	1	2	yes	—
167	#423	Dic₈₉	1	1	yes	characteristic
168	#408	Dic₈₉	1	2	yes	—
169	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
170	C₄⋊C₁₆	—	89	1	no	—
171	C₄⋊C₁₆	—	89	2	no	—
172	C₄⋊C₁₆	—	89	1	no	—
173	C₂²⋊C₁₆	—	89	4	no	—
174	C₂²⋊C₁₆	—	89	4	no	—
175	#346	—	89	2	no	—
176	C₄×C₁₆	—	89	2	no	—
177	C₂×#41	C₂×Dic₈₉	1	2	yes	—
178	#131	—	2	4	no	—
179	#131	—	2	4	no	—
180	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
181	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
182	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₄⋊Q₈	#10623494	1	1	yes	characteristic
186	#127	#10623494	1	1	yes	characteristic
187	#124	#10623494	1	2	yes	—
188	#121	#10623494	1	2	yes	—
189	#121	#10623494	1	2	yes	—
190	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
191	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
192	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
193	M₃₂	—	89	4	no	—
194	M₃₂	—	178	8	no	—
195	M₃₂	—	178	2	no	—
196	C₂×C₁₆	—	89	2	no	—
197	C₂×C₁₆	—	89	2	no	—
198	C₂×C₁₆	—	89	2	no	—
199	C₂×C₁₆	—	89	2	no	—
200	C₂×C₁₆	—	178	8	no	—
201	C₂×C₁₆	—	178	4	no	—
202	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
203	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
204	C₂²⋊C₈	C₂×Dic₈₉	1	4	yes	—
205	C₂²⋊C₈	C₂×Dic₈₉	1	4	yes	—
206	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	C₂×#10623494	1	2	yes	—
213	C₂×D₄	C₂×#10623494	1	2	yes	—
214	C₂×D₄	C₂×#10623494	1	4	yes	—
215	C₂×D₄	C₂×#10623494	1	2	yes	—
216	C₂×D₄	C₂×#10623494	1	2	yes	—
217	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
218	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
229	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
230	C₂×C₈	D₄×D₈₉	1	2	yes	—
231	C₂×C₈	#13083573	1	2	yes	—
232	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
233	C₄⋊C₄	C₂×#10623494	1	2	yes	—
234	C₄⋊C₄	C₂×#10623494	1	2	yes	—
235	C₂²⋊C₄	C₂×#10623494	1	4	yes	—
236	C₂²⋊C₄	C₂×#10623494	1	4	yes	—
237	C₄²	C₂×#10623494	1	1	yes	characteristic
238	C₄²	C₂×#10623494	1	1	yes	characteristic
239	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
240	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
241	C₁₆	—	89	4	no	—
242	C₁₆	—	178	2	no	—
243	C₁₆	—	178	2	no	—
244	C₁₆	—	178	2	no	—
245	C₂³	C₂²×#10623494	1	4	yes	—
246	C₂³	C₂²×#10623494	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	D₄	—	2	8	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
257	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
258	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
259	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
260	C₂×C₄	C₂²×#10623494	1	4	yes	—
261	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
262	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
263	C₂×C₄	#15668019	1	2	yes	—
264	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	D₄×Dic₈₉	1	2	yes	—
269	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
270	C₈	C₂×#13083573	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	D₄×#10623494	1	2	yes	—
277	C₂²	C₂³×#10623494	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	D₄×#10623494	1	2	yes	—
283	C₄	#19152015	1	2	yes	complemented
284	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
285	C₄	C₂×#15668019	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×D₄×#10623494	1	1	yes	characteristic
289	C₂	C₂×#19152015	1	1	yes	characteristic
290	C₂	#26033355	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2931	C₂
prime radical	#532×C₈₉	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#33438
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×#19152015	C₂	semidirect
2. #26022084	C₂	semidirect
3. #26022078	C₂	semidirect
4. D₄×#13083549	C₂	semidirect
5. C₂×C₄⋊#13083549	C₂	semidirect
6. #26021866	C₂	semidirect
7. C₄×#15667952	C₂	semidirect
33. #15667952	D₄	semidirect
137. C₄×C₈₉	#504	semidirect
154. C₈₉	#33438	semidirect
283. C₄	#19152015	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152015	2
#26022084	2
#26022078	2
D₄×#13083549	2
C₂×C₄⋊#13083549	2
#26021866	2
C₄×#15667952	2
C₈₉×#2931	2
#33438	89