Subgroups — #36454583

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028650	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal
4	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	C₂×#15668012	—	2	1	no	—
7	C₂×#15668012	C₄	1	2	yes	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151468	—	2	1	no	—
10	#19151468	—	2	1	no	—
11	#19151468	C₄	1	2	yes	—
12	#19151468	C₂²	1	2	yes	—
13	#19151353	—	2	1	no	—
14	#19151353	—	2	1	no	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×#97×C₈₉	—	2	1	no	—
17	C₄³×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	1	no	—
19	C₂²×Dic₁₇₈	—	2	2	no	—
20	C₂²×#10623494	—	2	2	no	—
21	#15668012	—	4	4	no	—
22	#15668012	—	4	4	no	—
23	#15668012	—	4	2	no	—
24	#15668012	—	4	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
31	C₄⋊#10623494	—	2	4	no	—
32	C₂²×C₈×C₈₉	—	2	2	no	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₄⋊C₈×C₈₉	Q₈	1	4	yes	complemented
42	#97×C₈₉	—	4	2	no	—
43	C₂×Dic₁₇₈	—	4	4	no	—
44	C₂×Dic₁₇₈	—	4	4	no	—
45	C₂×Dic₁₇₈	—	4	4	no	—
46	C₂²×Dic₈₉	—	2	4	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	4	no	—
51	C₄⋊Dic₈₉	—	4	4	no	—
52	C₄⋊Dic₈₉	—	4	4	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	C₄⋊Dic₈₉	—	4	4	no	—
56	C₄⋊Dic₈₉	—	4	4	no	—
57	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₂×C₈×C₈₉	—	4	2	no	—
67	C₄²×C₈₉	—	2	2	no	—
68	C₄²×C₈₉	—	2	2	no	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₄²×C₈₉	—	2	2	no	—
71	C₄²×C₈₉	—	2	2	no	—
72	C₄²×C₈₉	D₈	1	2	yes	—
73	C₄²×C₈₉	SD₁₆	1	2	yes	—
74	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
75	C₄²×C₈₉	#41	1	2	yes	—
76	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
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	#100	1	2	yes	—
88	C₂×C₄×C₈₉	#101	1	2	yes	—
89	C₂×C₄×C₈₉	#102	1	2	yes	—
90	C₂×C₄×C₈₉	#102	1	2	yes	—
91	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₄×C₈₉	#137	1	2	yes	—
99	C₂×C₄×C₈₉	#137	1	2	yes	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	4	4	no	—
102	C₂×Dic₈₉	—	4	4	no	—
103	Dic₁₇₈	—	8	8	no	—
104	C₈×C₈₉	—	4	4	no	—
105	#10623494	—	8	2	no	—
106	C₂²×C₈₉	#391	1	1	yes	characteristic
107	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
108	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
109	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
110	C₂²×C₈₉	#418	1	1	yes	characteristic
111	C₂²×C₈₉	#419	1	1	yes	characteristic
112	C₂²×C₈₉	#419	1	1	yes	characteristic
113	C₄×C₈₉	—	2	2	no	—
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	—	2	2	no	—
116	C₄×C₈₉	—	2	2	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	#474	1	2	yes	—
119	C₄×C₈₉	#475	1	2	yes	—
120	C₄×C₈₉	#489	1	2	yes	—
121	C₄×C₈₉	#501	1	2	yes	—
122	Dic₈₉	—	8	4	no	—
123	#9905	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	#1545	1	1	yes	characteristic
125	C₂×C₈₉	#1546	1	1	yes	characteristic
126	C₂×C₈₉	#1556	1	1	yes	characteristic
127	C₂×C₈₉	#1558	1	1	yes	characteristic
128	C₂×C₈₉	#1988	1	1	yes	characteristic
129	C₂×C₈₉	#1993	1	1	yes	characteristic
130	C₂×C₈₉	#1997	1	1	yes	characteristic
131	#2868	—	89	1	no	—
132	#1831	—	89	1	no	—
133	#1767	D₈₉	1	1	yes	characteristic
134	#1296	—	89	4	no	—
135	C₈₉	#9905	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄⋊Q₈	—	89	2	no	—
137	C₂×C₄⋊Q₈	—	178	1	no	—
138	C₂×C₄⋊Q₈	—	178	1	no	—
139	C₂×C₄⋊C₈	—	89	2	no	—
140	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
141	C₂×#97	—	2	1	no	—
142	#384	—	89	2	no	—
143	#384	—	89	2	no	—
144	#384	—	178	1	no	—
145	#384	—	178	1	no	—
146	C₄³	C₂×D₈₉	1	1	yes	characteristic
147	#336	—	2	1	no	—
148	#336	—	178	1	no	—
149	#336	—	178	1	no	—
150	C₂²×Q₈	—	178	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	178	2	no	—
153	C₄⋊Q₈	—	356	4	no	—
154	C₄⋊Q₈	—	356	4	no	—
155	C₄⋊Q₈	—	356	2	no	—
156	C₄⋊Q₈	—	356	2	no	—
157	C₂×C₄⋊C₄	—	89	4	no	—
158	C₂×C₄⋊C₄	—	178	2	no	—
159	C₂×C₄⋊C₄	—	178	2	no	—
160	C₂×C₄⋊C₄	—	178	2	no	—
161	C₂×C₄⋊C₄	—	178	2	no	—
162	C₂×C₄⋊C₄	—	178	2	no	—
163	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
165	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
166	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	178	4	no	—
172	C₄⋊C₈	Dic₁₇₈	1	4	yes	complemented
173	#97	—	4	2	no	—
174	C₂×Q₈	—	356	4	no	—
175	C₂×Q₈	—	356	4	no	—
176	C₂×Q₈	—	356	4	no	—
177	C₂²×C₄	—	178	4	no	—
178	C₂²×C₄	#13083551	1	1	yes	characteristic
179	C₂²×C₄	#13083551	1	1	yes	characteristic
180	C₂²×C₄	#13083551	1	1	yes	characteristic
181	C₂²×C₄	#13083551	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	356	2	no	—
189	C₂×C₈	—	356	2	no	—
190	C₂×C₈	—	356	2	no	—
191	C₄⋊C₄	—	356	4	no	—
192	C₄⋊C₄	—	356	4	no	—
193	C₄⋊C₄	—	356	2	no	—
194	C₄⋊C₄	—	356	2	no	—
195	C₄⋊C₄	—	356	4	no	—
196	C₄⋊C₄	—	356	4	no	—
197	C₄⋊C₄	—	356	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	#13083559	1	2	yes	—
204	C₄²	#13083561	1	2	yes	—
205	C₄²	C₂×Dic₁₇₈	1	2	yes	—
206	C₄²	#13083574	1	2	yes	—
207	C₂³	C₂×#13083551	1	1	yes	characteristic
208	Q₈	—	712	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	356	4	no	—
220	C₂×C₄	—	356	4	no	—
221	C₂×C₄	—	356	4	no	—
222	C₂×C₄	#15667928	1	2	yes	—
223	C₂×C₄	#15667930	1	2	yes	—
224	C₂×C₄	#15667932	1	2	yes	—
225	C₂×C₄	#15667932	1	2	yes	—
226	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
227	C₂×C₄	#15667991	1	2	yes	—
228	C₂×C₄	#15668000	1	1	yes	characteristic
229	C₂×C₄	#15668012	1	1	yes	characteristic
230	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
231	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
232	C₂×C₄	#15668040	1	2	yes	—
233	C₂×C₄	#15668042	1	2	yes	—
234	C₂×C₄	#15668045	1	2	yes	—
235	C₈	—	4	4	no	—
236	C₈	—	712	2	no	—
237	C₂²	#19151504	1	1	yes	characteristic
238	C₂²	C₂×#15667928	1	1	yes	characteristic
239	C₂²	C₂×#15667930	1	1	yes	characteristic
240	C₂²	C₂×#15667932	1	1	yes	characteristic
241	C₂²	#19151604	1	1	yes	characteristic
242	C₂²	#19151606	1	1	yes	characteristic
243	C₂²	#19151608	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	712	4	no	—
250	C₄	#19151878	1	2	yes	—
251	C₄	#19151883	1	2	yes	—
252	C₄	#19151958	1	2	yes	—
253	C₄	#19152007	1	2	yes	—
254	C₂	#26018921	1	1	yes	characteristic
255	C₂	#26018925	1	1	yes	characteristic
256	C₂	#26018968	1	1	yes	characteristic
257	C₂	#26018977	1	1	yes	characteristic
258	C₂	#26021204	1	1	yes	characteristic
259	C₂	#26021225	1	1	yes	characteristic
260	C₂	#26021241	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#9905
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
41. C₄⋊C₈×C₈₉	Q₈	semidirect
135. C₈₉	#9905	semidirect
172. 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
#26028650	2
#26020369	2
#26018136	2
C₈₉×#1767	2
#9905	89