Subgroups — #36573259

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26023207	C₂	1	2	yes	maximal
3	#26021019	C₂	1	4	yes	maximal
4	#26021010	C₂	1	2	yes	maximal
5	#26020413	C₂	1	2	yes	maximal
6	#26020409	C₂	1	1	yes	maximal, characteristic
7	#106×Dic₈₉	C₂	1	2	yes	maximal
8	C₄×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₇₈	C₄	1	8	yes	complemented
11	C₂×#15667945	C₂²	1	4	yes	—
12	C₂×#15667944	C₂²	1	2	yes	—
13	C₂×#15667944	C₂²	1	2	yes	—
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×#15667941	C₂²	1	2	yes	—
16	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
17	#19151504	—	2	2	no	—
18	#19151485	—	2	4	no	—
19	#19151468	C₂²	1	2	yes	—
20	#19151463	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₈₉	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₈₉	—	2	2	no	—
23	#19151360	C₂²	1	4	yes	—
24	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
28	#340×C₈₉	C₂²	1	2	yes	—
29	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
30	C₂²×#10623494	C₂³	1	2	yes	—
31	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₇₈	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₈₉	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	—	2	2	no	—
35	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
37	C₂×#13083553	D₄	1	2	yes	—
38	C₂×C₄×Dic₈₉	—	2	4	no	—
39	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
40	#15667945	—	4	4	no	—
41	#15667944	—	2	4	no	—
42	#15667944	—	4	2	no	—
43	C₄⋊#10623494	C₂×C₄	1	4	yes	—
44	#15667941	C₂×C₄	1	8	yes	—
45	C₈×Dic₈₉	C₂×C₄	1	8	yes	—
46	#15667919	—	2	4	no	—
47	#15667919	—	2	4	no	—
48	#15667919	—	2	4	no	—
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄²×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	#106×C₈₉	—	2	2	no	—
56	#106×C₈₉	Q₈	1	4	yes	complemented
57	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
58	#95×C₈₉	—	2	2	no	—
59	C₂×Dic₁₇₈	—	2	4	no	—
60	C₂×Dic₁₇₈	C₄⋊C₄	1	4	yes	—
61	C₂²×Dic₈₉	—	2	4	no	—
62	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
63	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	C₂²×C₄	1	4	yes	—
67	C₄⋊Dic₈₉	—	2	4	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	C₄⋊Dic₈₉	—	4	4	no	—
70	C₄⋊Dic₈₉	—	8	2	no	—
71	C₄⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
72	#13083553	C₄⋊C₄	1	8	yes	—
73	C₄×Dic₈₉	—	8	4	no	—
74	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
75	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₈₉	—	2	2	no	—
77	C₂²×C₄×C₈₉	—	2	2	no	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	2	1	no	—
83	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₈₉	#41	1	2	yes	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
91	C₄²×C₈₉	—	4	4	no	—
92	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
93	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
94	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	4	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	—	4	2	no	—
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₈₉	#135	1	4	yes	—
105	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	4	no	—
112	C₂×Dic₈₉	—	2	4	no	—
113	C₂×Dic₈₉	—	4	4	no	—
114	C₂×Dic₈₉	—	4	4	no	—
115	C₂×Dic₈₉	—	4	4	no	—
116	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
118	Dic₁₇₈	—	2	8	no	—
119	C₈×C₈₉	—	2	2	no	—
120	C₈×C₈₉	C₄×Q₈	1	4	yes	—
121	#10623494	—	2	4	no	—
122	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
124	C₂²×C₈₉	#550	1	1	yes	characteristic
125	C₂²×C₈₉	C₂×#135	1	2	yes	—
126	C₂²×C₈₉	#577	1	2	yes	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	4	4	no	—
129	C₄×C₈₉	—	8	2	no	—
130	C₄×C₈₉	#427	1	4	yes	—
131	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₈₉	#520	1	1	yes	characteristic
133	C₄×C₈₉	#557	1	1	yes	characteristic
134	C₄×C₈₉	Q₈²	1	1	yes	characteristic
135	Dic₈₉	—	2	4	no	—
136	Dic₈₉	—	2	4	no	—
137	Dic₈₉	—	8	4	no	—
138	#22725	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
141	C₂×C₈₉	#2912	1	1	yes	characteristic
142	C₂×C₈₉	#3381	1	2	yes	—
143	C₂×C₈₉	#3394	1	2	yes	—
144	C₂×C₈×Q₈	—	89	1	no	—
145	#2355	—	89	2	no	—
146	#1949	—	89	4	no	—
147	#1947	—	89	2	no	—
148	#1846	—	89	2	no	—
149	#1844	—	89	1	no	—
150	C₄×#106	—	89	2	no	—
151	C₄×#106	D₈₉	1	1	yes	characteristic
152	C₈₉	#22725	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄×Q₈	—	89	1	no	—
154	C₈×Q₈	—	89	8	no	—
155	C₂×#107	—	89	4	no	—
156	C₂×#106	—	89	2	no	—
157	C₂×#106	—	89	2	no	—
158	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
159	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	89	2	no	—
161	C₂×C₄⋊C₈	—	89	1	no	—
162	C₂×C₄×C₈	—	89	2	no	—
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	#391	—	178	2	no	—
165	#387	—	178	4	no	—
166	#384	—	89	2	no	—
167	#382	—	89	4	no	—
168	C₄×C₄⋊C₄	—	89	4	no	—
169	C₄×C₄⋊C₄	—	178	2	no	—
170	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
171	#340	—	89	4	no	—
172	#340	C₂×D₈₉	1	2	yes	—
173	C₂²×Q₈	—	89	1	no	—
174	C₂²×C₈	—	89	2	no	—
175	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
177	C₄×Q₈	—	89	4	no	—
178	C₄×Q₈	—	89	4	no	—
179	C₂×C₄⋊C₄	—	89	1	no	—
180	C₂×C₄⋊C₄	—	89	2	no	—
181	C₂×C₄⋊C₄	—	89	4	no	—
182	C₂×C₄⋊C₄	—	178	2	no	—
183	C₂×C₄⋊C₄	—	178	2	no	—
184	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	89	2	no	—
188	C₂×C₄²	—	178	4	no	—
189	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
190	#107	—	356	4	no	—
191	#106	—	2	2	no	—
192	#106	—	356	2	no	—
193	#106	—	178	4	no	—
194	#106	Dic₁₇₈	1	4	yes	complemented
195	C₄⋊C₈	—	89	8	no	—
196	C₄⋊C₈	—	89	4	no	—
197	C₄×C₈	—	89	8	no	—
198	C₄×C₈	C₄×D₈₉	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	178	4	no	—
201	#95	—	178	4	no	—
202	#95	—	178	4	no	—
203	C₂×Q₈	—	89	4	no	—
204	C₂×Q₈	—	178	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	89	2	no	—
208	C₂²×C₄	—	89	2	no	—
209	C₂²×C₄	—	178	4	no	—
210	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	89	4	no	—
216	C₂×C₈	—	178	2	no	—
217	C₂×C₈	—	178	2	no	—
218	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
220	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
221	C₂×C₈	#13083574	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	89	8	no	—
226	C₄⋊C₄	—	89	4	no	—
227	C₄⋊C₄	—	356	4	no	—
228	C₄⋊C₄	—	178	4	no	—
229	C₄⋊C₄	—	178	4	no	—
230	C₄⋊C₄	—	712	2	no	—
231	C₄⋊C₄	C₂×Dic₁₇₈	1	4	yes	—
232	C₄²	—	4	4	no	—
233	C₄²	—	89	4	no	—
234	C₄²	—	89	4	no	—
235	C₄²	—	712	4	no	—
236	C₄²	C₂×C₄×D₈₉	1	2	yes	—
237	C₄²	Q₈×D₈₉	1	2	yes	—
238	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
239	Q₈	—	178	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	89	4	no	—
248	C₂×C₄	—	89	4	no	—
249	C₂×C₄	—	356	4	no	—
250	C₂×C₄	—	356	4	no	—
251	C₂×C₄	—	356	4	no	—
252	C₂×C₄	—	178	4	no	—
253	C₂×C₄	—	178	4	no	—
254	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
255	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
256	C₂×C₄	#15668033	1	4	yes	—
257	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
260	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
261	C₂×C₄	#15668058	1	1	yes	characteristic
262	C₂×C₄	#15668059	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	178	4	no	—
265	C₈	C₄×Dic₁₇₈	1	4	yes	—
266	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
267	C₂²	D₄×Dic₁₇₈	1	1	yes	characteristic
268	C₂²	#19152269	1	1	yes	characteristic
269	C₂²	C₂×#15668033	1	2	yes	—
270	C₂²	#19152405	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	8	2	no	—
274	C₄	—	178	4	no	—
275	C₄	—	178	4	no	—
276	C₄	—	712	4	no	—
277	C₄	#19151633	1	4	yes	—
278	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
279	C₄	#19152078	1	1	yes	characteristic
280	C₄	#19152326	1	1	yes	characteristic
281	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
282	C₂	C₄⋊C₄×Dic₁₇₈	1	1	yes	characteristic
283	C₂	C₂×#19151633	1	1	yes	characteristic
284	C₂	#26028844	1	1	yes	characteristic
285	C₂	#26033125	1	2	yes	—
286	C₂	#26033246	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#106×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#106	D₈₉
hypocenter	C₈₉	#22725
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₇₈	C₄	semidirect
56. #106×C₈₉	Q₈	semidirect
152. C₈₉	#22725	semidirect
194. #106	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₇₈	2
#26023207	2
#26021019	2
#26021010	2
#26020413	2
#26020409	2
#106×Dic₈₉	2
C₄×#106×C₈₉	2
#22725	89