Subgroups — #36571235

Order 22784.

1086 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021176	C₂	1	3	yes	maximal
4	#26021055	C₂	1	3	yes	maximal
5	#26020257	C₂	1	3	yes	maximal
6	C₄×#15667936	C₂	1	3	yes	maximal
7	C₂×#557×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	—	2	1	no	—
9	Q₈×#10623494	—	2	4	no	—
10	C₂×C₄⋊#10623494	C₂²	1	3	yes	—
11	C₂×#15667936	C₂²	1	3	yes	—
12	C₂×#15667936	C₂²	1	3	yes	—
13	C₂×#15667936	C₂²	1	6	yes	—
14	C₂×C₄×#10623494	C₂²	1	3	yes	—
15	#19151497	—	2	3	no	—
16	#19151468	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	3	yes	—
18	#19151360	C₂²	1	3	yes	—
19	#19151360	C₂²	1	1	yes	characteristic
20	#557×C₈₉	—	2	4	no	—
21	#557×C₈₉	C₄	1	8	yes	complemented
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	3	yes	—
23	C₂×#125×C₈₉	—	2	3	no	—
24	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	3	yes	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×#10623494	C₂³	1	3	yes	—
28	C₂²×#10623494	C₂³	1	1	yes	characteristic
29	Q₈×Dic₈₉	—	4	4	no	—
30	C₂×C₄⋊Dic₈₉	—	2	3	no	—
31	C₂×C₄⋊Dic₈₉	—	2	3	no	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	3	yes	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₈₉	—	2	3	no	—
35	C₄⋊#10623494	—	2	6	no	—
36	#15667936	—	2	6	no	—
37	#15667936	—	2	6	no	—
38	#15667936	—	4	6	no	—
39	C₄×#10623494	—	2	6	no	—
40	#15667919	—	2	3	no	—
41	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₈₉	C₂³	1	3	yes	—
43	C₄⋊Q₈×C₈₉	—	2	6	no	—
44	C₄⋊Q₈×C₈₉	C₂×C₄	1	12	yes	—
45	#125×C₈₉	—	2	24	no	—
46	C₄×Q₈×C₈₉	—	2	8	no	—
47	C₄×Q₈×C₈₉	—	2	6	no	—
48	C₄×Q₈×C₈₉	C₂×C₄	1	12	yes	—
49	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₄×Q₈×C₈₉	D₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
54	C₂×C₄⋊C₄×C₈₉	D₄	1	3	yes	—
55	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	—	2	3	no	—
57	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
58	C₂²×Dic₈₉	—	2	1	no	—
59	C₂²×Dic₈₉	—	2	3	no	—
60	C₂×#10623494	—	2	1	no	—
61	C₂×#10623494	—	2	1	no	—
62	C₂×#10623494	—	2	3	no	—
63	C₂×#10623494	—	2	3	no	—
64	C₂×#10623494	—	2	3	no	—
65	C₂×#10623494	#41	1	2	yes	—
66	C₄⋊Dic₈₉	—	2	1	no	—
67	C₄⋊Dic₈₉	—	2	3	no	—
68	C₄⋊Dic₈₉	—	2	3	no	—
69	C₄⋊Dic₈₉	—	4	6	no	—
70	C₄⋊Dic₈₉	—	8	3	no	—
71	C₄⋊Dic₈₉	#41	1	2	yes	—
72	C₄×Dic₈₉	—	4	6	no	—
73	C₂×Q₈×C₈₉	—	2	3	no	—
74	C₂×Q₈×C₈₉	—	2	3	no	—
75	C₂×Q₈×C₈₉	—	2	3	no	—
76	C₂×Q₈×C₈₉	—	2	3	no	—
77	C₂×Q₈×C₈₉	—	2	3	no	—
78	C₂×Q₈×C₈₉	C₂²⋊C₄	1	4	yes	—
79	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	—
80	C₂×Q₈×C₈₉	Q₁₆	1	4	yes	—
81	C₂×Q₈×C₈₉	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₈₉	—	2	1	no	—
83	C₂²×C₄×C₈₉	—	2	3	no	—
84	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₈₉	—	2	12	no	—
88	C₄⋊C₄×C₈₉	—	2	12	no	—
89	C₄⋊C₄×C₈₉	—	2	12	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	12	yes	—
92	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
93	C₄²×C₈₉	—	2	12	no	—
94	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
95	C₄²×C₈₉	C₂×D₄	1	6	yes	—
96	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₈₉	—	2	6	no	—
98	Q₈×C₈₉	—	2	6	no	—
99	Q₈×C₈₉	—	4	3	no	—
100	Q₈×C₈₉	—	4	3	no	—
101	Q₈×C₈₉	#103	1	16	yes	complemented
102	C₂×C₄×C₈₉	—	2	3	no	—
103	C₂×C₄×C₈₉	—	2	3	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	6	no	—
106	C₂×C₄×C₈₉	—	4	3	no	—
107	C₂×C₄×C₈₉	—	4	3	no	—
108	C₂×C₄×C₈₉	—	4	1	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
111	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
112	C₂×C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
113	C₂×C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
114	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×Dic₈₉	—	2	2	no	—
121	C₂×Dic₈₉	—	4	1	no	—
122	C₂×Dic₈₉	—	4	1	no	—
123	C₂×Dic₈₉	—	4	3	no	—
124	C₂×Dic₈₉	—	4	3	no	—
125	C₂×Dic₈₉	—	4	3	no	—
126	#10623494	—	2	2	no	—
127	#10623494	—	4	3	no	—
128	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₈₉	#548	1	1	yes	characteristic
130	C₂²×C₈₉	#550	1	1	yes	characteristic
131	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
132	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂²×C₈₉	#577	1	1	yes	characteristic
134	C₂²×C₈₉	#578	1	1	yes	characteristic
135	C₄×C₈₉	—	2	6	no	—
136	C₄×C₈₉	—	4	6	no	—
137	C₄×C₈₉	—	4	2	no	—
138	C₄×C₈₉	C₂×#103	1	12	yes	—
139	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
140	C₄×C₈₉	#520	1	1	yes	characteristic
141	C₄×C₈₉	#548	1	1	yes	characteristic
142	C₄×C₈₉	#550	1	1	yes	characteristic
143	Dic₈₉	—	4	2	no	—
144	Dic₈₉	—	8	3	no	—
145	#22542	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	#2341	1	1	yes	characteristic
147	C₂×C₈₉	C₂²×#103	1	1	yes	characteristic
148	C₂×C₈₉	#2898	1	1	yes	characteristic
149	C₂×C₈₉	#3359	1	1	yes	characteristic
150	C₂×C₈₉	#3363	1	1	yes	characteristic
151	C₂×C₈₉	#3392	1	1	yes	characteristic
152	C₂×C₈₉	#3395	1	1	yes	characteristic
153	C₂×#557	D₈₉	1	1	yes	characteristic
154	C₂×C₈×Q₈	—	89	1	no	—
155	Q₈×C₄⋊C₄	—	89	1	no	—
156	#1981	—	89	3	no	—
157	#1956	—	89	3	no	—
158	#1803	—	89	3	no	—
159	C₄×#103	—	89	3	no	—
160	C₈₉	#22542	1	1	yes	characteristic, complemented, Sylow
161	#557	—	2	4	no	—
162	#557	Dic₈₉	1	8	yes	complemented
163	C₂×C₄⋊Q₈	C₂×D₈₉	1	3	yes	—
164	C₂×#125	—	2	3	no	—
165	C₂×C₄×Q₈	—	2	1	no	—
166	C₂×C₄×Q₈	—	178	1	no	—
167	C₂×C₄×Q₈	C₂×D₈₉	1	3	yes	—
168	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
169	C₈×Q₈	—	178	4	no	—
170	C₂×C₄⋊C₈	—	89	3	no	—
171	C₂×#103	—	89	6	no	—
172	C₂×#103	—	89	3	no	—
173	C₂×#103	—	89	3	no	—
174	C₂×C₄×C₈	—	89	3	no	—
175	#389	—	178	3	no	—
176	#384	—	89	3	no	—
177	C₄×C₄⋊C₄	—	89	3	no	—
178	#340	—	89	3	no	—
179	#340	—	89	1	no	—
180	C₂²×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²×Q₈	C₂²×D₈₉	1	3	yes	—
182	C₂²×C₈	—	89	1	no	—
183	C₂²×C₈	—	89	3	no	—
184	C₄⋊Q₈	—	2	6	no	—
185	C₄⋊Q₈	C₂×Dic₈₉	1	12	yes	—
186	#125	—	2	24	no	—
187	C₄×Q₈	—	2	8	no	—
188	C₄×Q₈	—	2	6	no	—
189	C₄×Q₈	—	356	4	no	—
190	C₄×Q₈	C₂×Dic₈₉	1	12	yes	—
191	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
192	C₄×Q₈	C₂²⋊D₈₉	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	3	no	—
194	C₂×C₄⋊C₄	—	2	3	no	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	—	89	1	no	—
197	C₂×C₄⋊C₄	—	89	3	no	—
198	C₂×C₄⋊C₄	—	178	3	no	—
199	C₂×C₄⋊C₄	—	178	3	no	—
200	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	3	yes	—
201	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
202	C₂×C₄²	—	2	3	no	—
203	C₂×C₄²	—	178	3	no	—
204	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
205	C₄⋊C₈	—	178	6	no	—
206	#103	—	356	6	no	—
207	#103	—	178	6	no	—
208	#103	—	178	6	no	—
209	C₄×C₈	—	178	6	no	—
210	#95	—	178	3	no	—
211	C₂×Q₈	—	2	3	no	—
212	C₂×Q₈	—	2	3	no	—
213	C₂×Q₈	—	2	3	no	—
214	C₂×Q₈	—	2	3	no	—
215	C₂×Q₈	—	2	3	no	—
216	C₂×Q₈	C₂²⋊Dic₈₉	1	4	yes	—
217	C₂×Q₈	#13083561	1	4	yes	—
218	C₂×Q₈	#13083564	1	4	yes	—
219	C₂×Q₈	C₂²×Dic₈₉	1	6	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	178	3	no	—
223	C₂²×C₄	—	178	1	no	—
224	C₂²×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
226	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
227	C₂×C₈	—	89	2	no	—
228	C₂×C₈	—	178	3	no	—
229	C₂×C₈	—	178	3	no	—
230	C₂×C₈	—	178	3	no	—
231	C₂×C₈	—	178	1	no	—
232	C₂×C₈	—	178	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	12	no	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	89	2	no	—
238	C₄⋊C₄	—	356	6	no	—
239	C₄⋊C₄	—	178	3	no	—
240	C₄⋊C₄	—	178	3	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	712	3	no	—
243	C₄⋊C₄	C₂²⋊Dic₈₉	1	12	yes	—
244	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
245	C₄²	—	2	12	no	—
246	C₄²	—	356	6	no	—
247	C₄²	C₂²×Dic₈₉	1	6	yes	—
248	C₄²	C₂×C₂²⋊D₈₉	1	6	yes	—
249	C₂³	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
250	Q₈	—	2	6	no	—
251	Q₈	—	2	6	no	—
252	Q₈	—	4	3	no	—
253	Q₈	—	4	3	no	—
254	Q₈	#15667936	1	16	yes	complemented
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	—	356	1	no	—
264	C₂×C₄	—	356	1	no	—
265	C₂×C₄	—	356	3	no	—
266	C₂×C₄	—	356	3	no	—
267	C₂×C₄	—	356	3	no	—
268	C₂×C₄	—	178	2	no	—
269	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	6	yes	—
270	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	6	yes	—
271	C₂×C₄	C₂×#13083561	1	6	yes	—
272	C₂×C₄	C₂×#13083564	1	6	yes	—
273	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
276	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
277	C₂×C₄	#15668058	1	1	yes	characteristic
278	C₂×C₄	#15668059	1	1	yes	characteristic
279	C₈	—	356	3	no	—
280	C₈	—	178	2	no	—
281	C₂²	C₂²×C₂²⋊Dic₈₉	1	1	yes	characteristic
282	C₂²	#19152255	1	1	yes	characteristic
283	C₂²	#19152267	1	1	yes	characteristic
284	C₂²	C₂²×#13083561	1	1	yes	characteristic
285	C₂²	C₂²×#13083564	1	1	yes	characteristic
286	C₂²	#19152406	1	1	yes	characteristic
287	C₂²	#19152413	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	4	6	no	—
290	C₄	—	4	2	no	—
291	C₄	—	356	2	no	—
292	C₄	—	712	3	no	—
293	C₄	C₂×#15667936	1	12	yes	—
294	C₄	#41×Dic₈₉	1	1	yes	characteristic
295	C₄	#19152081	1	1	yes	characteristic
296	C₄	#19152255	1	1	yes	characteristic
297	C₄	#19152267	1	1	yes	characteristic
298	C₂	#26023093	1	1	yes	characteristic
299	C₂	C₂²×#15667936	1	1	yes	characteristic
300	C₂	#26028785	1	1	yes	characteristic
301	C₂	#26032890	1	1	yes	characteristic
302	C₂	#26032926	1	1	yes	characteristic
303	C₂	#26033229	1	1	yes	characteristic
304	C₂	#26033255	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#557×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#557	D₈₉
hypocenter	C₈₉	#22542
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
21. #557×C₈₉	C₄	semidirect
101. Q₈×C₈₉	#103	semidirect
160. C₈₉	#22542	semidirect
162. #557	Dic₈₉	semidirect
254. Q₈	#15667936	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10623494	2
Q₈×C₄⋊Dic₈₉	2
#26021176	2
#26021055	2
#26020257	2
C₄×#15667936	2
C₂×#557×C₈₉	2
#22542	89