Subgroups — #36445045

Order 22784.

610 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	#26020105	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10623494	C₂	1	2	yes	maximal
4	#26018093	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×#15667923	—	2	2	no	—
8	C₂×#15667923	—	2	1	no	—
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	#19151353	—	2	1	no	—
11	Q₈⋊#10623494	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₈₉	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
15	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
16	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₄³×C₈₉	—	2	1	no	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	—	2	2	no	—
21	C₂²×#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	C₂×C₄	1	4	yes	—
24	#15667923	—	2	8	no	—
25	#15667923	—	4	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₄×#10623494	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₈₉	—	2	4	no	—
30	C₄×Q₈×C₈₉	—	2	4	no	—
31	C₄×Q₈×C₈₉	—	2	2	no	—
32	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	—	2	1	no	—
39	C₂×C₄²×C₈₉	—	2	2	no	—
40	C₂×C₄²×C₈₉	—	2	1	no	—
41	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	2	yes	—
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	#95×C₈₉	—	2	2	no	—
47	#95×C₈₉	—	2	1	no	—
48	C₂×#10623494	—	2	8	no	—
49	C₂×#10623494	—	2	4	no	—
50	C₂×#10623494	—	2	4	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₂×Q₈×C₈₉	—	2	1	no	—
54	C₂×Q₈×C₈₉	—	2	2	no	—
55	C₂×Q₈×C₈₉	C₄²	1	2	yes	—
56	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
57	C₂²×C₄×C₈₉	—	2	1	no	—
58	C₂²×C₄×C₈₉	—	2	1	no	—
59	C₂²×C₄×C₈₉	—	2	2	no	—
60	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₈₉	—	2	8	no	—
66	C₄⋊C₄×C₈₉	—	2	1	no	—
67	C₄⋊C₄×C₈₉	—	2	4	no	—
68	C₄⋊C₄×C₈₉	—	2	2	no	—
69	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
70	C₄⋊C₄×C₈₉	M₁₆	1	4	yes	—
71	C₄²×C₈₉	—	2	8	no	—
72	C₄²×C₈₉	—	2	4	no	—
73	C₄²×C₈₉	—	2	4	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	1	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	1	no	—
79	C₄²×C₈₉	—	4	4	no	—
80	C₄²×C₈₉	—	4	1	no	—
81	C₄²×C₈₉	—	4	1	no	—
82	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
83	C₄²×C₈₉	#41	1	2	yes	—
84	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₈₉	—	2	2	no	—
86	Q₈×C₈₉	—	2	4	no	—
87	Q₈×C₈₉	#97	1	4	yes	complemented
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	4	2	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	#104	1	4	yes	—
102	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#137	1	2	yes	—
111	#10623494	—	2	8	no	—
112	#10623494	—	4	4	no	—
113	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₈₉	#407	1	2	yes	—
115	C₂²×C₈₉	#417	1	1	yes	characteristic
116	C₂²×C₈₉	#419	1	1	yes	characteristic
117	C₂²×C₈₉	C₂×#104	1	2	yes	—
118	C₄×C₈₉	—	2	4	no	—
119	C₄×C₈₉	—	2	4	no	—
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	—	4	2	no	—
122	C₄×C₈₉	—	4	4	no	—
123	C₄×C₈₉	—	4	2	no	—
124	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
125	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₈₉	#432	1	2	yes	—
127	C₄×C₈₉	#440	1	2	yes	—
128	C₄×C₈₉	#441	1	2	yes	—
129	#8173	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	#1489	1	2	yes	—
131	C₂×C₈₉	#1491	1	2	yes	—
132	C₂×C₈₉	#1749	1	1	yes	characteristic
133	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
134	C₂×C₈₉	#1763	1	1	yes	characteristic
135	C₄²×Q₈	D₈₉	1	1	yes	characteristic
136	#1767	—	89	1	no	—
137	C₄×#97	—	89	1	no	—
138	C₂×Q₈⋊C₈	—	89	2	no	—
139	#1279	—	89	2	no	—
140	C₈₉	#8173	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	Dic₈₉	1	2	yes	—
142	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
143	C₂×C₄⋊C₈	—	89	2	no	—
144	C₂×#97	—	178	1	no	—
145	C₂×#97	—	178	2	no	—
146	C₂×C₄×C₈	—	89	2	no	—
147	C₄×C₄⋊C₄	—	2	1	no	—
148	C₄×C₄⋊C₄	—	2	2	no	—
149	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
150	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
151	C₄³	—	2	1	no	—
152	C₄³	C₂×D₈₉	1	1	yes	characteristic
153	#336	—	178	1	no	—
154	Q₈⋊C₈	—	89	8	no	—
155	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
156	C₂²×C₈	—	178	2	no	—
157	C₂²×C₈	—	178	2	no	—
158	C₄×Q₈	—	2	4	no	—
159	C₄×Q₈	—	2	4	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	C₄×D₈₉	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
171	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
175	C₄⋊C₈	—	89	4	no	—
176	C₄⋊C₈	—	178	2	no	—
177	#97	—	356	2	no	—
178	#97	—	178	8	no	—
179	C₄×C₈	—	89	4	no	—
180	C₄×C₈	—	178	2	no	—
181	#95	—	2	2	no	—
182	#95	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₄×Dic₈₉	1	2	yes	—
186	C₂×Q₈	#13083558	1	4	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
191	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
195	C₂×C₈	—	356	2	no	—
196	C₂×C₈	—	356	2	no	—
197	C₂×C₈	—	178	4	no	—
198	C₂×C₈	—	178	8	no	—
199	C₂×C₈	—	178	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
205	C₄⋊C₄	#13083558	1	4	yes	—
206	C₄²	—	2	8	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂×C₄×D₈₉	1	2	yes	—
218	C₄²	#13083574	1	2	yes	—
219	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	2	no	—
222	Q₈	#15667923	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#15667939	1	4	yes	—
237	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
238	C₂×C₄	#15667971	1	1	yes	characteristic
239	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
240	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
241	C₂×C₄	C₂×#13083558	1	2	yes	—
242	C₂×C₄	C₂×#13083558	1	2	yes	—
243	C₂×C₄	#15668037	1	1	yes	characteristic
244	C₂×C₄	#15668042	1	1	yes	characteristic
245	C₂×C₄	#15668043	1	2	yes	—
246	C₈	—	356	4	no	—
247	C₈	—	178	8	no	—
248	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
249	C₂²	#19151562	1	2	yes	—
250	C₂²	#19151599	1	1	yes	characteristic
251	C₂²	#19151607	1	1	yes	characteristic
252	C₂²	C₂×#15667939	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	2	no	—
259	C₄	C₂×#15667923	1	1	yes	characteristic
260	C₄	C₄×#13083558	1	1	yes	characteristic
261	C₄	#19151655	1	2	yes	—
262	C₄	#19151696	1	2	yes	—
263	C₄	#19151703	1	2	yes	—
264	C₂	#26018691	1	2	yes	—
265	C₂	#26018699	1	2	yes	—
266	C₂	#26020014	1	1	yes	characteristic
267	C₂	C₄×#15667939	1	1	yes	characteristic
268	C₂	#26020082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×Q₈	D₈₉
hypocenter	C₈₉	#8173
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
11. Q₈⋊#10623494	C₄	semidirect
87. Q₈×C₈₉	#97	semidirect
140. C₈₉	#8173	semidirect
222. Q₈	#15667923	semidirect

Maximal subgroups

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

subgroup	index
#26020105	2
C₄×#15667923	2
C₂×Q₈⋊#10623494	2
#26018093	2
C₄²×Q₈×C₈₉	2
#8173	89