Subgroups — #36454430

Order 22784.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#26020105	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₇₈	—	2	1	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×#15667923	—	2	1	no	—
9	#19151504	—	2	1	no	—
10	#19151496	—	2	1	no	—
11	#19151496	C₄	1	2	yes	—
12	#19151468	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₈₉	—	2	1	no	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
15	#19151353	—	2	1	no	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
18	C₄³×C₈₉	C₂²	1	1	yes	characteristic
19	#336×C₈₉	—	2	1	no	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂²×#10623494	—	2	2	no	—
22	#15668012	—	2	4	no	—
23	C₄×Dic₁₇₈	—	4	2	no	—
24	C₄×Dic₁₇₈	—	4	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
30	C₂×#13083553	—	2	2	no	—
31	C₂×C₄×Dic₈₉	—	2	2	no	—
32	C₄⋊#10623494	—	2	2	no	—
33	C₄⋊#10623494	Q₈	1	4	yes	complemented
34	#15667923	—	4	2	no	—
35	#15667919	—	2	2	no	—
36	#15667919	—	2	2	no	—
37	C₂²×C₈×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	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	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₈₉	—	2	4	no	—
46	C₄×C₈×C₈₉	—	2	4	no	—
47	C₂×Dic₁₇₈	—	2	4	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂×Dic₁₇₈	—	4	2	no	—
50	C₂²×Dic₈₉	—	2	2	no	—
51	C₂²×Dic₈₉	—	2	2	no	—
52	C₂×#10623494	—	2	4	no	—
53	C₂×#10623494	—	4	2	no	—
54	C₂×#10623494	—	4	2	no	—
55	C₄⋊Dic₈₉	—	2	4	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	C₄⋊Dic₈₉	—	2	4	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
61	#13083553	—	4	4	no	—
62	C₄×Dic₈₉	—	4	2	no	—
63	C₄×Dic₈₉	—	4	2	no	—
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	—	2	4	no	—
72	C₂×C₈×C₈₉	—	4	2	no	—
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	4	1	no	—
79	C₄²×C₈₉	D₈	1	2	yes	—
80	C₄²×C₈₉	SD₁₆	1	2	yes	—
81	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
82	C₄²×C₈₉	#41	1	2	yes	—
83	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
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	#102	1	2	yes	—
96	C₂×C₄×C₈₉	#102	1	2	yes	—
97	C₂×C₄×C₈₉	#104	1	2	yes	—
98	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₈₉	#131	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#137	1	2	yes	—
107	C₂×Dic₈₉	—	2	4	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	Dic₁₇₈	—	4	8	no	—
114	C₈×C₈₉	—	4	4	no	—
115	#10623494	—	4	4	no	—
116	C₂²×C₈₉	#391	1	1	yes	characteristic
117	C₂²×C₈₉	#407	1	1	yes	characteristic
118	C₂²×C₈₉	#408	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
120	C₂²×C₈₉	#419	1	1	yes	characteristic
121	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
122	C₂²×C₈₉	#421	1	1	yes	characteristic
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	4	1	no	—
129	C₄×C₈₉	#446	1	4	yes	—
130	C₄×C₈₉	#474	1	2	yes	—
131	C₄×C₈₉	#475	1	2	yes	—
132	Dic₈₉	—	4	4	no	—
133	Dic₈₉	—	8	2	no	—
134	#9869	—	89	1	no	maximal, Sylow
135	C₂×C₈₉	#1486	1	1	yes	characteristic
136	C₂×C₈₉	#1491	1	1	yes	characteristic
137	C₂×C₈₉	#1492	1	1	yes	characteristic
138	C₂×C₈₉	#1497	1	1	yes	characteristic
139	C₂×C₈₉	#1985	1	1	yes	characteristic
140	C₂×C₈₉	#1986	1	1	yes	characteristic
141	C₂×C₈₉	#1988	1	1	yes	characteristic
142	C₄×C₄⋊Q₈	—	89	1	no	—
143	#1767	—	89	1	no	—
144	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
145	#1279	—	89	4	no	—
146	C₈₉	#9869	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄⋊Q₈	—	89	1	no	—
148	C₂×C₄×Q₈	—	178	1	no	—
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	89	2	no	—
151	C₂×#97	—	178	1	no	—
152	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
153	#391	—	178	1	no	—
154	#389	—	89	2	no	—
155	#389	—	178	1	no	—
156	#384	—	89	1	no	—
157	C₄×C₄⋊C₄	—	89	2	no	—
158	C₄×C₄⋊C₄	—	178	1	no	—
159	C₄³	C₂×D₈₉	1	1	yes	characteristic
160	#336	—	2	1	no	—
161	#336	—	178	1	no	—
162	C₂²×Q₈	—	178	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	178	2	no	—
165	C₄⋊Q₈	—	178	4	no	—
166	C₄×Q₈	—	356	2	no	—
167	C₄×Q₈	—	356	2	no	—
168	C₂×C₄⋊C₄	—	89	2	no	—
169	C₂×C₄⋊C₄	—	89	2	no	—
170	C₂×C₄⋊C₄	—	178	1	no	—
171	C₂×C₄⋊C₄	—	178	2	no	—
172	C₂×C₄⋊C₄	—	178	1	no	—
173	C₂×C₄⋊C₄	—	178	1	no	—
174	C₂×C₄²	—	178	2	no	—
175	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
178	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	89	4	no	—
184	C₄⋊C₈	—	178	2	no	—
185	#97	—	356	2	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	—	178	2	no	—
188	#95	—	178	2	no	—
189	C₂×Q₈	—	356	2	no	—
190	C₂×Q₈	—	356	2	no	—
191	C₂×Q₈	—	178	4	no	—
192	C₂²×C₄	—	178	2	no	—
193	C₂²×C₄	—	178	2	no	—
194	C₂²×C₄	#13083551	1	1	yes	characteristic
195	C₂²×C₄	#13083551	1	1	yes	characteristic
196	C₂²×C₄	#13083551	1	1	yes	characteristic
197	C₂²×C₄	#13083551	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	356	2	no	—
205	C₂×C₈	—	356	2	no	—
206	C₂×C₈	—	178	4	no	—
207	C₄⋊C₄	—	89	4	no	—
208	C₄⋊C₄	—	356	4	no	—
209	C₄⋊C₄	—	356	2	no	—
210	C₄⋊C₄	—	356	2	no	—
211	C₄⋊C₄	—	178	4	no	—
212	C₄⋊C₄	—	178	4	no	—
213	C₄⋊C₄	—	178	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	356	2	no	—
220	C₄²	—	356	2	no	—
221	C₄²	D₇₁₂	1	2	yes	—
222	C₄²	#13083563	1	2	yes	—
223	C₄²	Q₈×D₈₉	1	2	yes	—
224	C₄²	#13083573	1	2	yes	—
225	C₂³	C₂×#13083551	1	1	yes	characteristic
226	Q₈	—	356	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	356	2	no	—
239	C₂×C₄	—	356	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	356	2	no	—
242	C₂×C₄	—	356	2	no	—
243	C₂×C₄	—	178	4	no	—
244	C₂×C₄	#15667934	1	2	yes	—
245	C₂×C₄	#15667934	1	2	yes	—
246	C₂×C₄	#15667940	1	2	yes	—
247	C₂×C₄	#15667977	1	2	yes	—
248	C₂×C₄	#15667992	1	2	yes	—
249	C₂×C₄	#15667999	1	1	yes	characteristic
250	C₂×C₄	#15668010	1	1	yes	characteristic
251	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
252	C₂×C₄	#15668018	1	2	yes	—
253	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
254	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
255	C₂×C₄	#15668045	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	356	4	no	—
258	C₂²	#19151503	1	1	yes	characteristic
259	C₂²	#19151561	1	1	yes	characteristic
260	C₂²	#19151564	1	1	yes	characteristic
261	C₂²	C₂×#15667934	1	1	yes	characteristic
262	C₂²	#19151608	1	1	yes	characteristic
263	C₂²	C₂×#15667940	1	1	yes	characteristic
264	C₂²	#19151616	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	356	4	no	—
272	C₄	—	712	2	no	—
273	C₄	#19151727	1	4	yes	—
274	C₄	#19151879	1	2	yes	—
275	C₄	#19151884	1	2	yes	—
276	C₂	#26018678	1	1	yes	characteristic
277	C₂	#26018700	1	1	yes	characteristic
278	C₂	#26018704	1	1	yes	characteristic
279	C₂	#26018726	1	1	yes	characteristic
280	C₂	#26021191	1	1	yes	characteristic
281	C₂	#26021195	1	1	yes	characteristic
282	C₂	#26021203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#9869
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₄⋊#10623494	Q₈	semidirect
146. C₈₉	#9869	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
#26020105	2
#26018092	2
C₄×C₄⋊C₈×C₈₉	2
#9869	89