Subgroups — #36447574

Order 22784.

374 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	#26018680	C₂	1	2	yes	maximal
4	#26018147	C₂	1	2	yes	maximal
5	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	#19151728	—	2	1	no	—
8	#19151728	—	2	1	no	—
9	#19151728	C₄	1	2	yes	—
10	#19151728	C₄	1	2	yes	—
11	#19151651	C₂²	1	2	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×#15667941	—	2	1	no	—
14	C₂×#15667922	—	2	1	no	—
15	#19151351	—	2	1	no	—
16	#19151349	—	2	1	no	—
17	#19151328	—	2	4	no	—
18	C₂×#108×C₈₉	—	2	1	no	—
19	C₂×#108×C₈₉	—	2	1	no	—
20	#423×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₇₈	Q₈	1	1	yes	characteristic
23	C₂²×#10623494	—	2	1	no	—
24	C₄×Dic₁₇₈	—	2	1	no	—
25	C₄×Dic₁₇₈	—	2	1	no	—
26	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₇₈	D₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	Q₈	1	1	yes	characteristic
29	C₂×#13083553	—	2	1	no	—
30	C₂×C₄×Dic₈₉	—	2	1	no	—
31	C₄⋊#10623494	—	2	1	no	—
32	C₄⋊#10623494	—	2	2	no	—
33	C₄⋊#10623494	C₂×C₄	1	2	yes	—
34	#15667941	—	2	2	no	—
35	#15667941	—	2	2	no	—
36	C₂²⋊#10623494	—	2	2	no	—
37	#15667922	—	2	2	no	—
38	#15667922	—	2	2	no	—
39	C₄×#10623494	—	2	2	no	—
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	#108×C₈₉	—	4	2	no	—
45	#108×C₈₉	—	4	2	no	—
46	C₄⋊C₈×C₈₉	—	2	2	no	—
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂×Dic₁₇₈	—	2	1	no	—
53	C₂×Dic₁₇₈	—	4	2	no	—
54	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₈₉	—	2	1	no	—
56	C₂²×Dic₈₉	—	2	1	no	—
57	C₂×#10623494	—	2	2	no	—
58	C₂×#10623494	—	4	1	no	—
59	C₂×#10623494	—	4	1	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₄⋊Dic₈₉	—	2	1	no	—
62	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
63	#13083553	—	2	2	no	—
64	#13083553	—	2	2	no	—
65	C₄×Dic₈₉	—	2	2	no	—
66	C₄×Dic₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
70	M₁₆×C₈₉	—	4	2	no	—
71	M₁₆×C₈₉	—	4	2	no	—
72	C₂×C₈×C₈₉	—	2	1	no	—
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	—	4	2	no	—
75	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
80	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
81	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₉	#41	1	1	yes	characteristic
85	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	4	1	no	—
88	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
98	C₂×Dic₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	2	2	no	—
100	C₂×Dic₈₉	—	4	2	no	—
101	C₂×Dic₈₉	—	4	2	no	—
102	Dic₁₇₈	—	2	4	no	—
103	Dic₁₇₈	—	4	2	no	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	—	2	1	no	—
106	C₈×C₈₉	—	8	2	no	—
107	C₈×C₈₉	#103	1	2	yes	—
108	C₈×C₈₉	#103	1	2	yes	—
109	#10623494	—	4	2	no	—
110	#10623494	—	8	2	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	#380	1	1	yes	characteristic
114	C₂²×C₈₉	#475	1	1	yes	characteristic
115	C₂²×C₈₉	#477	1	1	yes	characteristic
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
120	C₄×C₈₉	#417	1	1	yes	characteristic
121	C₄×C₈₉	#430	1	2	yes	—
122	C₄×C₈₉	#460	1	1	yes	characteristic
123	C₄×C₈₉	#462	1	1	yes	characteristic
124	Dic₈₉	—	4	2	no	—
125	Dic₈₉	—	4	2	no	—
126	#8663	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	—	2	1	no	—
128	C₂×C₈₉	—	2	1	no	—
129	C₂×C₈₉	#1803	1	1	yes	characteristic
130	C₂×C₈₉	#1811	1	1	yes	characteristic
131	C₂×C₈₉	#1814	1	1	yes	characteristic
132	C₂×#446	—	89	1	no	—
133	#1848	D₈₉	1	1	yes	characteristic
134	#1564	—	89	1	no	—
135	#1487	—	89	2	no	—
136	#1301	—	89	2	no	—
137	C₈₉	#8663	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄×Q₈	—	89	1	no	—
139	#446	—	89	2	no	—
140	#446	—	89	2	no	—
141	#446	—	178	1	no	—
142	#446	—	178	1	no	—
143	#431	—	89	2	no	—
144	C₂×#108	—	2	1	no	—
145	C₂×#108	—	2	1	no	—
146	#423	C₂×D₈₉	1	2	yes	—
147	C₂×C₄⋊C₈	—	89	1	no	—
148	C₂×C₄⋊C₈	—	178	1	no	—
149	C₂×#97	—	178	1	no	—
150	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
151	#335	—	178	1	no	—
152	#334	—	178	1	no	—
153	Q₈⋊C₈	—	178	4	no	—
154	C₂²×Q₈	—	89	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	—	178	1	no	—
157	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
159	C₄×Q₈	—	89	2	no	—
160	C₄×Q₈	—	89	2	no	—
161	C₄×Q₈	—	178	1	no	—
162	C₄×Q₈	—	178	1	no	—
163	C₂×C₄⋊C₄	—	89	1	no	—
164	C₂×C₄⋊C₄	—	178	1	no	—
165	C₂×C₄²	—	178	1	no	—
166	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
167	#108	—	4	2	no	—
168	#108	—	4	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	89	2	no	—
172	C₄⋊C₈	—	178	2	no	—
173	C₄⋊C₈	—	178	2	no	—
174	C₄⋊C₈	—	178	2	no	—
175	C₄⋊C₈	—	178	1	no	—
176	C₂²⋊C₈	—	178	2	no	—
177	#97	—	178	2	no	—
178	#97	—	178	2	no	—
179	C₄×C₈	—	178	2	no	—
180	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
181	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
182	C₄×C₈	D₃₅₆	1	1	yes	characteristic
183	C₄×C₈	D₃₅₆	1	1	yes	characteristic
184	C₂×Q₈	—	89	2	no	—
185	C₂×Q₈	—	356	2	no	—
186	C₂×Q₈	—	178	1	no	—
187	C₂²×C₄	—	178	1	no	—
188	C₂²×C₄	—	178	1	no	—
189	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
191	C₂²×C₄	#13083575	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	356	1	no	—
198	C₂×C₈	—	356	1	no	—
199	C₂×C₈	—	356	2	no	—
200	C₂×C₈	—	178	2	no	—
201	C₂×C₈	#13083551	1	1	yes	characteristic
202	C₂×C₈	#13083551	1	1	yes	characteristic
203	C₂×C₈	#13083551	1	1	yes	characteristic
204	C₂×C₈	#13083551	1	1	yes	characteristic
205	C₂×C₈	#13083562	1	2	yes	—
206	C₂×C₈	#13083564	1	2	yes	—
207	C₄⋊C₄	—	89	2	no	—
208	C₄⋊C₄	—	178	2	no	—
209	C₄⋊C₄	—	178	2	no	—
210	C₄⋊C₄	—	178	1	no	—
211	C₄²	—	178	2	no	—
212	C₄²	—	178	2	no	—
213	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
214	C₄²	D₄×D₈₉	1	1	yes	characteristic
215	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
216	C₄²	#13083573	1	1	yes	characteristic
217	C₂³	#15667988	1	1	yes	characteristic
218	Q₈	—	356	2	no	—
219	Q₈	—	178	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	—	178	2	no	—
225	C₂×C₄	—	178	2	no	—
226	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
228	C₂×C₄	#15667970	1	1	yes	characteristic
229	C₂×C₄	#15667983	1	1	yes	characteristic
230	C₂×C₄	#15667987	1	1	yes	characteristic
231	C₂×C₄	#15667992	1	1	yes	characteristic
232	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
233	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
234	C₂×C₄	#15668038	1	1	yes	characteristic
235	C₂×C₄	#15668043	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	356	2	no	—
240	C₈	—	712	2	no	—
241	C₈	#15667935	1	2	yes	—
242	C₈	#15667935	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#19151455	1	1	yes	characteristic
246	C₂²	#19151880	1	1	yes	characteristic
247	C₂²	#19151890	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	356	2	no	—
252	C₄	—	356	2	no	—
253	C₄	C₂×#15667935	1	1	yes	characteristic
254	C₄	#19151597	1	1	yes	characteristic
255	C₄	#19151645	1	2	yes	—
256	C₄	#19151795	1	1	yes	characteristic
257	C₄	#19151805	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#26020254	1	1	yes	characteristic
261	C₂	#26020291	1	1	yes	characteristic
262	C₂	#26020307	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667983
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#8663
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
137. C₈₉	#8663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151728	2
#26019000	2
#26018680	2
#26018147	2
C₈₉×#1848	2
#8663	89