Subgroups — #36573439

Order 22784.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13083558	C₂	1	2	yes	maximal
2	#26021028	C₂	1	4	yes	maximal
3	#26021028	C₂	1	2	yes	maximal
4	#26020393	C₂	1	2	yes	maximal
5	#26020390	C₂	1	1	yes	maximal, characteristic
6	#26020158	C₂	1	2	yes	maximal
7	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
8	C₂×#446×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19151729	—	2	2	no	—
10	#19151729	—	2	4	no	—
11	Q₈×#10623494	—	2	2	no	—
12	C₂×#15667948	C₂²	1	2	yes	—
13	C₂×#15667948	C₂²	1	4	yes	—
14	#19151622	C₂²	1	2	yes	—
15	#19151622	C₂²	1	4	yes	—
16	C₄×#13083558	C₂²	1	2	yes	—
17	C₄×#13083558	C₂²	1	4	yes	—
18	#19151363	C₂²	1	2	yes	—
19	#19151363	C₂²	1	2	yes	—
20	#19151363	C₂²	1	1	yes	characteristic
21	#19151363	C₂²	1	4	yes	—
22	#19151363	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	#446×C₈₉	C₄	1	4	yes	—
25	#446×C₈₉	C₄	1	4	yes	—
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
28	C₂×#97×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#13083558	C₂³	1	4	yes	—
31	C₂×#13083558	C₂³	1	2	yes	—
32	C₂×#13083558	C₂³	1	2	yes	—
33	#15667948	—	2	4	no	—
34	#15667948	—	2	8	no	—
35	C₄⋊#10623494	—	2	2	no	—
36	C₄⋊#10623494	—	2	4	no	—
37	C₄⋊#10623494	—	2	4	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	#15667923	—	2	4	no	—
40	#15667923	—	2	2	no	—
41	C₄×#10623494	—	2	2	no	—
42	C₄×#10623494	—	2	4	no	—
43	C₂²×Q₈×C₈₉	Q₈	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
47	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₈₉	D₄	1	2	yes	—
50	C₄×Q₈×C₈₉	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
55	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
58	#97×C₈₉	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	#13083558	—	2	2	no	—
62	#13083558	—	2	4	no	—
63	#13083558	—	2	2	no	—
64	#13083558	—	2	4	no	—
65	#13083558	—	2	2	no	—
66	#13083558	#41	1	4	yes	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	4	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×Q₈×C₈₉	—	4	2	no	—
71	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
91	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	4	yes	—
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
95	Q₈×C₈₉	—	2	4	no	—
96	Q₈×C₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
110	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
111	C₈×C₈₉	—	2	2	no	—
112	C₈×C₈₉	—	2	4	no	—
113	C₈×C₈₉	—	2	2	no	—
114	#10623494	—	2	4	no	—
115	#10623494	—	4	4	no	—
116	#10623494	—	4	2	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₈₉	#557	1	2	yes	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	4	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	2	4	no	—
126	C₄×C₈₉	#430	1	2	yes	—
127	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
128	C₄×C₈₉	#519	1	1	yes	characteristic
129	C₄×C₈₉	#548	1	2	yes	—
130	#22747	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	—	2	2	no	—
132	C₂×C₈₉	#2351	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
134	C₂×C₈₉	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	—	89	2	no	—
136	C₂×#446	D₈₉	1	1	yes	characteristic
137	#1950	—	89	4	no	—
138	#1950	—	89	2	no	—
139	#1838	—	89	2	no	—
140	#1837	—	89	1	no	—
141	#1778	—	89	2	no	—
142	C₄×#108	—	89	1	no	—
143	C₈₉	#22747	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
145	#446	—	178	2	no	—
146	#446	—	178	4	no	—
147	#446	Dic₈₉	1	4	yes	—
148	#446	Dic₈₉	1	4	yes	—
149	C₈×Q₈	—	178	2	no	—
150	C₂×#108	—	89	2	no	—
151	C₂×#108	—	89	4	no	—
152	#423	—	89	2	no	—
153	#423	—	89	4	no	—
154	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
156	C₄×M₁₆	—	89	2	no	—
157	C₄×M₁₆	—	89	4	no	—
158	C₂×#97	C₂×D₈₉	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	#341	—	89	2	no	—
161	#341	—	89	2	no	—
162	#341	—	89	1	no	—
163	#341	—	89	4	no	—
164	#341	—	89	1	no	—
165	C₂²×Q₈	Dic₁₇₈	1	1	yes	characteristic
166	C₂×M₁₆	—	89	2	no	—
167	C₂×M₁₆	—	89	4	no	—
168	C₂×M₁₆	—	89	2	no	—
169	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
172	C₄×Q₈	D₃₅₆	1	2	yes	—
173	C₄×Q₈	D₃₅₆	1	2	yes	—
174	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
175	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
176	C₂×C₄⋊C₄	Dic₁₇₈	1	1	yes	characteristic
177	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
178	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
180	#108	—	178	4	no	—
181	#108	—	178	8	no	—
182	C₄⋊C₈	—	178	2	no	—
183	C₄⋊C₈	—	178	4	no	—
184	C₄⋊C₈	—	178	4	no	—
185	C₄⋊C₈	—	178	2	no	—
186	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
187	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
188	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
189	#97	—	178	4	no	—
190	#97	—	178	2	no	—
191	#97	C₂×Dic₈₉	1	8	yes	—
192	C₄×C₈	—	178	2	no	—
193	C₄×C₈	—	178	4	no	—
194	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
195	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₄⋊Dic₈₉	1	2	yes	—
198	C₂×Q₈	C₄⋊Dic₈₉	1	2	yes	—
199	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
202	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
203	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
204	M₁₆	—	89	4	no	—
205	M₁₆	—	178	2	no	—
206	M₁₆	—	178	4	no	—
207	M₁₆	—	178	4	no	—
208	M₁₆	—	178	2	no	—
209	M₁₆	—	178	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	178	2	no	—
214	C₂×C₈	—	178	4	no	—
215	C₂×C₈	—	178	2	no	—
216	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
217	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
218	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
219	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
220	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
221	C₄⋊C₄	C₄⋊Dic₈₉	1	2	yes	—
222	C₄⋊C₄	C₄⋊Dic₈₉	1	2	yes	—
223	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
224	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
225	C₄²	C₂²×Dic₈₉	1	4	yes	—
226	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
227	C₄²	C₂×D₃₅₆	1	4	yes	—
228	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
229	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
244	C₂×C₄	#41×D₈₉	1	2	yes	—
245	C₂×C₄	#15668060	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	2	2	no	—
249	C₈	—	356	2	no	—
250	C₈	—	356	4	no	—
251	C₈	—	178	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
255	C₂²	#19152324	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#19151647	1	2	yes	—
262	C₄	#41×Dic₈₉	1	1	yes	characteristic
263	C₄	#19152076	1	1	yes	characteristic
264	C₄	#19152260	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#26023173	1	1	yes	characteristic
267	C₂	C₂×#19151647	1	1	yes	characteristic
268	C₂	#26028863	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#446×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₉	C₂²×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#446	D₈₉
hypocenter	C₈₉	#22747
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
143. C₈₉	#22747	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13083558	2
#26021028	2
#26020393	2
#26020390	2
#26020158	2
C₄×#15667948	2
C₂×#446×C₈₉	2
#22747	89