Subgroups — #36573251

Order 22784.

838 conjugacy classes of subgroups.

The subgroup lattice has 294 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	#26023164	C₂	1	2	yes	maximal
3	#26021013	C₂	1	2	yes	maximal
4	#26021012	C₂	1	4	yes	maximal
5	#26020407	C₂	1	2	yes	maximal
6	#106×Dic₈₉	C₂	1	2	yes	maximal
7	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	C₂×#15667977	—	2	2	no	—
11	C₈×Dic₁₇₈	C₄	1	8	yes	complemented
12	C₂×#15667944	C₂²	1	1	yes	characteristic
13	C₂×#15667944	C₂²	1	1	yes	characteristic
14	C₂×#15667943	C₂²	1	4	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×#15667941	C₂²	1	2	yes	—
17	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
18	#19151494	—	2	2	no	—
19	#19151493	—	2	4	no	—
20	#19151467	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₈₉	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
23	#19151361	C₂²	1	2	yes	—
24	#19151360	C₂²	1	4	yes	—
25	C₂×#106×C₈₉	C₂²	1	2	yes	—
26	C₂×#106×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	#384×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	#15667977	—	4	8	no	—
34	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
37	C₂×#13083553	—	2	4	no	—
38	C₂×#13083553	—	2	4	no	—
39	C₂×#13083553	D₄	1	2	yes	—
40	C₂×C₄×Dic₈₉	—	2	4	no	—
41	C₂×C₄×Dic₈₉	—	2	2	no	—
42	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
43	#15667944	—	2	2	no	—
44	#15667944	Q₈	1	4	yes	complemented
45	#15667943	—	4	4	no	—
46	C₄⋊#10623494	C₂×C₄	1	4	yes	—
47	#15667941	C₂×C₄	1	8	yes	—
48	C₈×Dic₈₉	C₂×C₄	1	8	yes	—
49	#15667919	—	2	4	no	—
50	#15667919	—	2	2	no	—
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	#106×C₈₉	—	2	4	no	—
58	#106×C₈₉	—	4	2	no	—
59	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
60	C₂×Dic₁₇₈	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₁₇₈	SD₁₆	1	8	yes	—
62	C₂²×Dic₈₉	—	2	2	no	—
63	C₂²×Dic₈₉	—	2	2	no	—
64	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
65	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	C₂²×C₄	1	4	yes	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
73	C₄⋊Dic₈₉	C₂×Q₈	1	4	yes	—
74	#13083553	—	4	8	no	—
75	#13083553	—	8	4	no	—
76	#13083553	C₄⋊C₄	1	8	yes	—
77	C₄×Dic₈₉	—	4	4	no	—
78	C₄×Dic₈₉	—	4	8	no	—
79	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
80	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₈₉	—	2	4	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₈₉	#41	1	2	yes	—
91	C₄⋊C₄×C₈₉	—	2	4	no	—
92	C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₄⋊C₄×C₈₉	—	4	4	no	—
94	C₄⋊C₄×C₈₉	—	8	2	no	—
95	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
96	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
97	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	4	4	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	4	4	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	C₂×SD₁₆	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	2	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×Dic₈₉	C₂×SD₁₆	1	8	yes	—
121	Dic₁₇₈	#106	1	16	yes	complemented
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
124	#10623494	—	2	4	no	—
125	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
127	C₂²×C₈₉	#550	1	1	yes	characteristic
128	C₂²×C₈₉	C₂²×SD₁₆	1	2	yes	—
129	C₂²×C₈₉	#577	1	2	yes	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	8	4	no	—
132	C₄×C₈₉	C₂×#106	1	4	yes	—
133	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₈₉	#520	1	1	yes	characteristic
135	C₄×C₈₉	#557	1	1	yes	characteristic
136	C₄×C₈₉	Q₈²	1	1	yes	characteristic
137	Dic₈₉	—	2	4	no	—
138	Dic₈₉	—	4	4	no	—
139	Dic₈₉	—	8	2	no	—
140	Dic₈₉	C₂×#106	1	8	yes	—
141	Q₈×#106	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
144	C₂×C₈₉	#2912	1	1	yes	characteristic
145	C₂×C₈₉	Q₈×SD₁₆	1	2	yes	—
146	C₂×C₈₉	#3392	1	2	yes	—
147	C₂×C₈×Q₈	—	89	1	no	—
148	Q₈×C₄⋊C₄	—	89	2	no	—
149	#1948	—	89	4	no	—
150	#1948	—	89	2	no	—
151	#1844	—	89	2	no	—
152	#1844	D₈₉	1	1	yes	characteristic
153	C₄×#106	—	89	2	no	—
154	C₄×#106	—	89	1	no	—
155	C₈₉	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄×Q₈	—	89	1	no	—
157	C₂×C₄×Q₈	—	178	2	no	—
158	C₈×Q₈	—	89	8	no	—
159	C₂×#106	—	89	1	no	—
160	C₂×#106	—	89	4	no	—
161	C₂×#106	—	89	1	no	—
162	C₂×#106	C₂×D₈₉	1	2	yes	—
163	C₂×#106	C₂×D₈₉	1	2	yes	—
164	C₂×C₄⋊C₈	—	89	2	no	—
165	C₂×C₄⋊C₈	—	89	1	no	—
166	C₂×C₄×C₈	—	89	2	no	—
167	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
168	#389	—	178	4	no	—
169	#389	—	178	2	no	—
170	#384	—	89	4	no	—
171	#384	C₂×D₈₉	1	2	yes	—
172	C₄×C₄⋊C₄	—	89	4	no	—
173	C₄×C₄⋊C₄	—	89	2	no	—
174	#340	—	89	2	no	—
175	#340	—	89	4	no	—
176	C₂²×Q₈	—	89	1	no	—
177	C₂²×C₈	—	89	2	no	—
178	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
180	C₄×Q₈	—	89	4	no	—
181	C₄×Q₈	—	89	4	no	—
182	C₄×Q₈	—	356	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	89	2	no	—
186	C₂×C₄⋊C₄	—	89	1	no	—
187	C₂×C₄⋊C₄	—	89	2	no	—
188	C₂×C₄⋊C₄	—	89	2	no	—
189	C₂×C₄⋊C₄	—	178	4	no	—
190	C₂×C₄⋊C₄	—	178	4	no	—
191	C₂×C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
192	C₂×C₄²	—	89	2	no	—
193	C₂×C₄²	—	178	4	no	—
194	C₂×C₄²	—	178	2	no	—
195	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	#106	—	2	4	no	—
197	#106	—	4	2	no	—
198	#106	—	89	4	no	—
199	#106	—	356	4	no	—
200	#106	—	178	2	no	—
201	C₄⋊C₈	—	89	8	no	—
202	C₄⋊C₈	—	89	4	no	—
203	C₄×C₈	—	89	8	no	—
204	C₄×C₈	C₄×D₈₉	1	4	yes	—
205	#95	—	178	4	no	—
206	#95	—	178	2	no	—
207	C₂×Q₈	—	89	8	no	—
208	C₂×Q₈	—	89	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	89	2	no	—
211	C₂²×C₄	—	89	2	no	—
212	C₂²×C₄	—	178	2	no	—
213	C₂²×C₄	—	178	2	no	—
214	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	89	4	no	—
220	C₂×C₈	—	178	2	no	—
221	C₂×C₈	—	178	2	no	—
222	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
224	C₂×C₈	Q₈×D₈₉	1	2	yes	—
225	C₂×C₈	#13083573	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄⋊C₄	—	89	8	no	—
231	C₄⋊C₄	—	89	4	no	—
232	C₄⋊C₄	—	89	4	no	—
233	C₄⋊C₄	—	356	8	no	—
234	C₄⋊C₄	—	178	2	no	—
235	C₄⋊C₄	—	178	2	no	—
236	C₄⋊C₄	—	178	2	no	—
237	C₄⋊C₄	—	712	4	no	—
238	C₄²	—	89	4	no	—
239	C₄²	—	89	4	no	—
240	C₄²	—	356	8	no	—
241	C₄²	—	356	4	no	—
242	C₄²	C₂×C₄×D₈₉	1	2	yes	—
243	C₄²	Q₈×D₈₉	1	2	yes	—
244	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
245	Q₈	—	89	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	89	4	no	—
251	C₂×C₄	—	89	8	no	—
252	C₂×C₄	—	89	4	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	356	2	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	178	4	no	—
259	C₂×C₄	—	178	4	no	—
260	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
262	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
267	C₂×C₄	#15668057	1	1	yes	characteristic
268	C₂×C₄	#15668060	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	178	4	no	—
271	C₈	#15667977	1	4	yes	complemented
272	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
273	C₂²	#19152263	1	1	yes	characteristic
274	C₂²	#19152268	1	1	yes	characteristic
275	C₂²	C₂×SD₁₆×D₈₉	1	2	yes	—
276	C₂²	#19152404	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	8	4	no	—
279	C₄	—	89	8	no	—
280	C₄	—	356	4	no	—
281	C₄	—	178	4	no	—
282	C₄	—	712	2	no	—
283	C₄	#106×D₈₉	1	4	yes	—
284	C₄	C₂×#15667977	1	1	yes	characteristic
285	C₄	#19152079	1	1	yes	characteristic
286	C₄	#19152323	1	1	yes	characteristic
287	C₄	#19152328	1	1	yes	characteristic
288	C₂	#26023164	1	1	yes	characteristic
289	C₂	C₂×#106×D₈₉	1	1	yes	characteristic
290	C₂	#26028843	1	1	yes	characteristic
291	C₂	#26033109	1	2	yes	—
292	C₂	#26033230	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1844	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1844	D₈₉
hypocenter	C₈₉	Q₈×#106
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₇₈	C₄	semidirect
44. #15667944	Q₈	semidirect
121. Dic₁₇₈	#106	semidirect
123. C₈×C₈₉	C₄×Q₈	semidirect
155. C₈₉	Q₈×#106	semidirect
271. C₈	#15667977	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
#26023164	2
#26021013	2
#26021012	2
#26020407	2
#106×Dic₈₉	2
C₄×#15667944	2
C₈₉×#1844	2
Q₈×#106	89