Subgroups — #36445412

Order 22784.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151343	C₂	1	2	yes	maximal
4	#26018136	C₂	1	2	yes	maximal
5	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	—	2	1	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₄×#10623494	—	2	1	no	—
10	#19151504	—	2	1	no	—
11	#19151496	—	2	1	no	—
12	#19151496	C₄	1	2	yes	—
13	#19151468	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Dic₈₉	—	2	1	no	—
15	C₄×C₄⋊Dic₈₉	C₄	1	2	yes	—
16	#19151353	—	2	1	no	—
17	#19151343	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
19	C₂×#97×C₈₉	—	2	1	no	—
20	C₄³×C₈₉	C₂²	1	1	yes	characteristic
21	#336×C₈₉	—	2	1	no	—
22	C₂²×Dic₁₇₈	—	2	1	no	—
23	C₂²×#10623494	—	2	2	no	—
24	#15668012	—	2	2	no	—
25	#15668012	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₇₈	—	4	4	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
32	C₂×#13083553	—	2	2	no	—
33	C₂×C₄×Dic₈₉	—	2	2	no	—
34	C₄⋊#10623494	—	2	2	no	—
35	C₄⋊#10623494	C₂×C₄	1	4	yes	—
36	C₄×#10623494	—	4	2	no	—
37	#15667919	—	2	2	no	—
38	#15667919	—	2	2	no	—
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
49	#97×C₈₉	—	4	2	no	—
50	C₂×Dic₁₇₈	—	2	4	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂×Dic₁₇₈	—	4	1	no	—
54	C₂²×Dic₈₉	—	2	2	no	—
55	C₂²×Dic₈₉	—	2	2	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₄⋊Dic₈₉	—	2	2	no	—
60	C₄⋊Dic₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₄⋊Dic₈₉	—	4	2	no	—
63	C₄⋊Dic₈₉	—	4	2	no	—
64	C₄⋊Dic₈₉	C₄²	1	4	yes	—
65	#13083553	—	4	4	no	—
66	C₄×Dic₈₉	—	4	4	no	—
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	4	no	—
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₂×C₈×C₈₉	—	4	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	—	2	1	no	—
81	C₄²×C₈₉	—	2	1	no	—
82	C₄²×C₈₉	—	4	1	no	—
83	C₄²×C₈₉	SD₁₆	1	2	yes	—
84	C₄²×C₈₉	Q₁₆	1	2	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₈₉	#41	1	2	yes	—
87	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	#101	1	2	yes	—
100	C₂×C₄×C₈₉	#103	1	2	yes	—
101	C₂×C₄×C₈₉	#103	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#135	1	2	yes	—
109	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
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	Dic₁₇₈	—	4	4	no	—
119	Dic₁₇₈	—	4	4	no	—
120	C₈×C₈₉	—	4	4	no	—
121	#10623494	—	4	4	no	—
122	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
124	C₂²×C₈₉	#413	1	1	yes	characteristic
125	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
126	C₂²×C₈₉	#416	1	1	yes	characteristic
127	C₂²×C₈₉	#417	1	1	yes	characteristic
128	C₂²×C₈₉	#418	1	1	yes	characteristic
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	4	1	no	—
135	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
136	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
137	C₄×C₈₉	#440	1	2	yes	—
138	C₄×C₈₉	#442	1	2	yes	—
139	Dic₈₉	—	4	4	no	—
140	Dic₈₉	—	8	2	no	—
141	#8237	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	#1549	1	1	yes	characteristic
143	C₂×C₈₉	#1550	1	1	yes	characteristic
144	C₂×C₈₉	#1552	1	1	yes	characteristic
145	C₂×C₈₉	#1553	1	1	yes	characteristic
146	C₂×C₈₉	C₄×#101	1	1	yes	characteristic
147	C₂×C₈₉	C₄×#103	1	1	yes	characteristic
148	C₂×C₈₉	#1762	1	1	yes	characteristic
149	C₄×C₄⋊Q₈	—	89	1	no	—
150	#1767	D₈₉	1	1	yes	characteristic
151	C₄×C₄⋊C₈	—	89	1	no	—
152	C₂×#332	—	89	2	no	—
153	#1296	—	89	2	no	—
154	C₈₉	#8237	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄⋊Q₈	—	89	1	no	—
156	C₂×C₄×Q₈	—	178	1	no	—
157	C₂×C₄⋊C₈	—	89	2	no	—
158	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
159	C₂×#97	—	2	1	no	—
160	C₂×C₄×C₈	—	178	1	no	—
161	#391	—	178	1	no	—
162	#389	—	89	2	no	—
163	#389	—	178	1	no	—
164	#384	—	89	1	no	—
165	C₄×C₄⋊C₄	—	89	2	no	—
166	C₄×C₄⋊C₄	—	178	1	no	—
167	C₄³	C₂×D₈₉	1	1	yes	characteristic
168	#336	—	2	1	no	—
169	#336	—	178	1	no	—
170	#332	—	89	8	no	—
171	C₂²×Q₈	—	178	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	178	2	no	—
174	C₄⋊Q₈	—	89	4	no	—
175	C₄⋊Q₈	—	178	2	no	—
176	C₄×Q₈	—	356	4	no	—
177	C₂×C₄⋊C₄	—	89	2	no	—
178	C₂×C₄⋊C₄	—	89	2	no	—
179	C₂×C₄⋊C₄	—	178	2	no	—
180	C₂×C₄⋊C₄	—	178	1	no	—
181	C₂×C₄⋊C₄	—	178	1	no	—
182	C₂×C₄⋊C₄	—	178	1	no	—
183	C₂×C₄²	—	178	2	no	—
184	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
186	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
187	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	89	4	no	—
193	C₄⋊C₈	—	178	2	no	—
194	C₄⋊C₈	C₄×D₈₉	1	4	yes	—
195	#97	—	4	2	no	—
196	C₄×C₈	—	356	2	no	—
197	#95	—	178	2	no	—
198	#95	—	178	2	no	—
199	C₂×Q₈	—	356	2	no	—
200	C₂×Q₈	—	356	1	no	—
201	C₂×Q₈	—	356	1	no	—
202	C₂×Q₈	—	178	4	no	—
203	C₂²×C₄	—	178	2	no	—
204	C₂²×C₄	—	178	2	no	—
205	C₂²×C₄	#13083551	1	1	yes	characteristic
206	C₂²×C₄	#13083551	1	1	yes	characteristic
207	C₂²×C₄	#13083551	1	1	yes	characteristic
208	C₂²×C₄	#13083551	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	356	2	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	—	178	4	no	—
218	C₄⋊C₄	—	89	4	no	—
219	C₄⋊C₄	—	356	2	no	—
220	C₄⋊C₄	—	356	4	no	—
221	C₄⋊C₄	—	356	2	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	178	2	no	—
224	C₄⋊C₄	—	178	4	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	356	4	no	—
232	C₄²	#13083562	1	2	yes	—
233	C₄²	#13083564	1	2	yes	—
234	C₄²	C₂×C₄×D₈₉	1	2	yes	—
235	C₄²	#13083574	1	2	yes	—
236	C₂³	C₂×#13083551	1	1	yes	characteristic
237	Q₈	—	356	4	no	—
238	Q₈	—	356	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	356	2	no	—
252	C₂×C₄	—	356	2	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	178	4	no	—
256	C₂×C₄	#15667930	1	2	yes	—
257	C₂×C₄	#15667935	1	2	yes	—
258	C₂×C₄	#15667935	1	2	yes	—
259	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
260	C₂×C₄	#15667969	1	1	yes	characteristic
261	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
262	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
263	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
264	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
265	C₂×C₄	#15668032	1	2	yes	—
266	C₂×C₄	#15668040	1	2	yes	—
267	C₂×C₄	#15668040	1	1	yes	characteristic
268	C₂×C₄	#15668045	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	—	356	4	no	—
271	C₂²	C₄×#13083551	1	1	yes	characteristic
272	C₂²	C₂×#15667930	1	1	yes	characteristic
273	C₂²	#19151580	1	1	yes	characteristic
274	C₂²	C₂×#15667935	1	1	yes	characteristic
275	C₂²	#19151592	1	1	yes	characteristic
276	C₂²	#19151600	1	1	yes	characteristic
277	C₂²	#19151604	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	356	4	no	—
285	C₄	—	712	2	no	—
286	C₄	C₄×#13083562	1	2	yes	—
287	C₄	C₄×#13083564	1	2	yes	—
288	C₄	#19151700	1	2	yes	—
289	C₄	#19151711	1	2	yes	—
290	C₂	#26018938	1	1	yes	characteristic
291	C₂	#26018943	1	1	yes	characteristic
292	C₂	#26018953	1	1	yes	characteristic
293	C₂	#26018958	1	1	yes	characteristic
294	C₂	C₄×#15667930	1	1	yes	characteristic
295	C₂	C₄×#15667935	1	1	yes	characteristic
296	C₂	#26020078	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
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	#8237
LCS	G ▷ C₂×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
17. #19151343	C₄	semidirect
154. C₈₉	#8237	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
C₄×C₄⋊#10623494	2
C₂×#19151343	2
#26018136	2
C₈₉×#1767	2
#8237	89