Subgroups — #36454447

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₇₈	—	2	1	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	#19151504	—	2	1	no	—
9	#19151496	—	2	1	no	—
10	#19151496	C₄	1	2	yes	—
11	#19151468	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₈₉	—	2	1	no	—
13	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	#19151353	—	2	1	no	—
15	#19151353	—	2	1	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
17	C₄³×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	1	no	—
19	#336×C₈₉	—	2	1	no	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂²×#10623494	—	2	2	no	—
22	#15668012	—	2	2	no	—
23	#15668012	C₈	1	4	yes	complemented
24	C₄×Dic₁₇₈	—	4	4	no	—
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
30	C₂×#13083553	—	2	2	no	—
31	C₂×C₄×Dic₈₉	—	2	2	no	—
32	C₄⋊#10623494	—	2	4	no	—
33	#15667919	—	2	2	no	—
34	#15667919	—	2	2	no	—
35	C₂²×C₈×C₈₉	—	2	2	no	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₈₉	—	2	2	no	—
44	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
45	C₂×Dic₁₇₈	—	2	4	no	—
46	C₂×Dic₁₇₈	—	4	2	no	—
47	C₂×Dic₁₇₈	—	4	2	no	—
48	C₂²×Dic₈₉	—	2	2	no	—
49	C₂²×Dic₈₉	—	2	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	4	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
59	#13083553	—	4	4	no	—
60	C₄×Dic₈₉	—	4	4	no	—
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₂×C₈×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₄²×C₈₉	—	2	2	no	—
72	C₄²×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	—	4	1	no	—
75	C₄²×C₈₉	—	4	1	no	—
76	C₄²×C₈₉	C₂×C₈	1	2	yes	—
77	C₄²×C₈₉	M₁₆	1	2	yes	—
78	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
79	C₄²×C₈₉	#41	1	2	yes	—
80	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#131	1	2	yes	—
102	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₈₉	#137	1	2	yes	—
104	C₂×Dic₈₉	—	2	4	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	Dic₁₇₈	—	4	8	no	—
111	C₈×C₈₉	—	4	4	no	—
112	#10623494	—	8	2	no	—
113	C₂²×C₈₉	#391	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₈₉	#408	1	1	yes	characteristic
116	C₂²×C₈₉	#418	1	1	yes	characteristic
117	C₂²×C₈₉	#419	1	1	yes	characteristic
118	C₂²×C₈₉	#421	1	1	yes	characteristic
119	C₂²×C₈₉	#421	1	1	yes	characteristic
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	—	4	1	no	—
124	C₄×C₈₉	—	4	1	no	—
125	C₄×C₈₉	C₈×Q₈	1	2	yes	—
126	C₄×C₈₉	#446	1	2	yes	—
127	C₄×C₈₉	#489	1	2	yes	—
128	C₄×C₈₉	#501	1	2	yes	—
129	Dic₈₉	—	4	4	no	—
130	Dic₈₉	—	8	2	no	—
131	#9873	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	#1478	1	1	yes	characteristic
133	C₂×C₈₉	#1479	1	1	yes	characteristic
134	C₂×C₈₉	#1496	1	1	yes	characteristic
135	C₂×C₈₉	#1497	1	1	yes	characteristic
136	C₂×C₈₉	#1984	1	1	yes	characteristic
137	C₂×C₈₉	#1993	1	1	yes	characteristic
138	C₂×C₈₉	#1995	1	1	yes	characteristic
139	C₄×C₄⋊Q₈	—	89	1	no	—
140	#1831	—	89	1	no	—
141	#1744	D₈₉	1	1	yes	characteristic
142	#1279	—	89	4	no	—
143	C₈₉	#9873	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	89	1	no	—
145	C₂×C₄×Q₈	—	178	1	no	—
146	C₂×C₄⋊C₈	—	89	2	no	—
147	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
148	#391	—	178	1	no	—
149	#389	—	89	2	no	—
150	#389	—	178	1	no	—
151	#384	—	89	1	no	—
152	C₄×C₄⋊C₄	—	89	2	no	—
153	C₄×C₄⋊C₄	—	178	1	no	—
154	C₄³	C₂×D₈₉	1	1	yes	characteristic
155	#336	—	2	1	no	—
156	#336	—	2	1	no	—
157	#336	—	178	1	no	—
158	#336	—	178	1	no	—
159	C₂²×Q₈	—	178	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	178	2	no	—
162	C₄⋊Q₈	—	89	4	no	—
163	C₄⋊Q₈	—	178	2	no	—
164	C₄×Q₈	—	356	4	no	—
165	C₂×C₄⋊C₄	—	89	2	no	—
166	C₂×C₄⋊C₄	—	89	2	no	—
167	C₂×C₄⋊C₄	—	178	1	no	—
168	C₂×C₄⋊C₄	—	178	2	no	—
169	C₂×C₄⋊C₄	—	178	1	no	—
170	C₂×C₄⋊C₄	—	178	1	no	—
171	C₂×C₄²	—	178	2	no	—
172	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
174	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
175	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
179	C₄⋊C₈	—	178	4	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
182	#95	—	178	2	no	—
183	#95	—	178	2	no	—
184	C₂×Q₈	—	356	2	no	—
185	C₂×Q₈	—	356	2	no	—
186	C₂×Q₈	—	178	4	no	—
187	C₂²×C₄	—	178	2	no	—
188	C₂²×C₄	—	178	2	no	—
189	C₂²×C₄	#13083551	1	1	yes	characteristic
190	C₂²×C₄	#13083551	1	1	yes	characteristic
191	C₂²×C₄	#13083551	1	1	yes	characteristic
192	C₂²×C₄	#13083551	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	356	2	no	—
200	C₂×C₈	—	356	2	no	—
201	C₂×C₈	—	356	2	no	—
202	C₄⋊C₄	—	89	4	no	—
203	C₄⋊C₄	—	356	4	no	—
204	C₄⋊C₄	—	356	2	no	—
205	C₄⋊C₄	—	356	2	no	—
206	C₄⋊C₄	—	178	4	no	—
207	C₄⋊C₄	—	178	2	no	—
208	C₄⋊C₄	—	178	4	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	356	4	no	—
215	C₄²	C₈×D₈₉	1	2	yes	—
216	C₄²	#13083557	1	2	yes	—
217	C₄²	C₂×Dic₁₇₈	1	2	yes	—
218	C₄²	#13083574	1	2	yes	—
219	C₂³	C₂×#13083551	1	1	yes	characteristic
220	Q₈	—	356	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	356	2	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	—	356	2	no	—
236	C₂×C₄	—	356	2	no	—
237	C₂×C₄	—	356	2	no	—
238	C₂×C₄	—	178	4	no	—
239	C₂×C₄	#15667924	1	2	yes	—
240	C₂×C₄	#15667924	1	2	yes	—
241	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
242	C₂×C₄	#15667991	1	2	yes	—
243	C₂×C₄	#15668000	1	1	yes	characteristic
244	C₂×C₄	#15668012	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
246	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
247	C₂×C₄	#15668018	1	2	yes	—
248	C₂×C₄	#15668040	1	2	yes	—
249	C₂×C₄	#15668045	1	2	yes	—
250	C₈	—	4	4	no	—
251	C₈	—	712	2	no	—
252	C₂²	#19151504	1	1	yes	characteristic
253	C₂²	C₂×#15667924	1	1	yes	characteristic
254	C₂²	#19151564	1	1	yes	characteristic
255	C₂²	#19151604	1	1	yes	characteristic
256	C₂²	#19151608	1	1	yes	characteristic
257	C₂²	#19151616	1	1	yes	characteristic
258	C₂²	#19151616	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	356	4	no	—
265	C₄	—	712	2	no	—
266	C₄	C₈×Dic₁₇₈	1	2	yes	—
267	C₄	#19151728	1	2	yes	—
268	C₄	#19151958	1	2	yes	—
269	C₄	#19152007	1	2	yes	—
270	C₂	#26018644	1	1	yes	characteristic
271	C₂	#26018648	1	1	yes	characteristic
272	C₂	#26018722	1	1	yes	characteristic
273	C₂	#26018726	1	1	yes	characteristic
274	C₂	#26021188	1	1	yes	characteristic
275	C₂	#26021225	1	1	yes	characteristic
276	C₂	#26021233	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1744	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1744	D₈₉
hypocenter	C₈₉	#9873
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
23. #15668012	C₈	semidirect
44. C₄×C₈×C₈₉	Q₈	semidirect
143. C₈₉	#9873	semidirect
181. C₄×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668012	2
#26020369	2
#26018092	2
C₈₉×#1744	2
#9873	89