Subgroups — #36469721

Order 22784.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151721	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151409	C₂	1	1	yes	maximal, characteristic
3	#26021965	C₂	1	1	yes	maximal, characteristic
4	#26020149	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018361	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018351	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#347×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	1	yes	characteristic
9	C₂×#15668018	—	2	1	no	—
10	#19151721	—	2	2	no	—
11	#19151721	C₄	1	4	yes	complemented
12	C₂×#15667948	C₂²	1	1	yes	characteristic
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667944	C₂²	1	1	yes	characteristic
15	C₂×#15667940	—	2	1	no	—
16	C₄×C₄⋊Dic₈₉	—	2	1	no	—
17	#19151409	C₄	1	4	yes	complemented
18	#19151403	C₄	1	2	yes	complemented
19	#19151400	C₄	1	2	yes	complemented
20	#19151361	—	2	1	no	—
21	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
24	#347×C₈₉	—	2	2	no	—
25	C₂×#13083574	—	2	1	no	—
26	#15668034	—	2	1	no	—
27	#15668034	—	2	1	no	—
28	#15668034	C₂×C₄	1	2	yes	—
29	#15668034	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083563	—	2	1	no	—
32	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
33	#15668018	—	2	4	no	—
34	#15668018	—	4	2	no	—
35	C₂×#13083558	—	2	1	no	—
36	C₂×#13083557	—	4	1	no	—
37	C₂×C₈×D₈₉	—	4	1	no	—
38	C₂×C₄⋊Dic₈₉	—	2	1	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×C₄×Dic₈₉	—	4	1	no	—
41	#15667948	—	2	1	no	—
42	#15667948	C₂×C₄	1	2	yes	—
43	#15667946	—	2	1	no	—
44	#15667946	C₂×C₄	1	2	yes	—
45	#15667944	—	2	1	no	—
46	#15667944	C₂×C₄	1	2	yes	—
47	#15667940	—	2	4	no	—
48	#15667919	—	4	1	no	—
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	—	2	1	no	—
54	M₃₂×C₈₉	—	2	1	no	—
55	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
58	#97×C₈₉	C₂×C₄	1	2	yes	—
59	#97×C₈₉	D₄	1	2	yes	—
60	#13083574	—	2	2	no	—
61	#13083574	—	2	2	no	—
62	#13083574	—	4	1	no	—
63	#13083574	—	4	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	4	1	no	—
67	C₂²×Dic₈₉	—	4	1	no	—
68	C₂²×Dic₈₉	—	4	1	no	—
69	C₂×C₄×D₈₉	—	4	1	no	—
70	Dic₃₅₆	—	2	1	no	—
71	Dic₃₅₆	C₄²	1	2	yes	complemented
72	#13083563	—	2	2	no	—
73	#13083563	—	2	2	no	—
74	D₇₁₂	—	2	1	no	—
75	D₇₁₂	C₄²	1	2	yes	complemented
76	#13083558	—	2	2	no	—
77	#13083558	—	4	1	no	—
78	#13083557	—	4	4	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₈×D₈₉	—	4	4	no	—
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	4	1	no	—
84	C₄⋊Dic₈₉	—	4	1	no	—
85	C₄×Dic₈₉	—	8	2	no	—
86	C₂²×C₄×C₈₉	—	2	1	no	—
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
96	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
97	C₄²×C₈₉	—	2	1	no	—
98	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
99	C₁₆×C₈₉	—	2	2	no	—
100	C₁₆×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×D₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	4	1	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
112	C₂²⋊D₈₉	—	4	2	no	—
113	C₂²⋊D₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	4	1	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	8	1	no	—
118	C₂×Dic₈₉	—	8	1	no	—
119	C₂×Dic₈₉	—	8	1	no	—
120	C₂×Dic₈₉	—	8	1	no	—
121	D₃₅₆	—	2	2	no	—
122	D₃₅₆	—	4	1	no	—
123	C₄×D₈₉	—	4	2	no	—
124	C₄×D₈₉	—	4	2	no	—
125	Dic₁₇₈	—	2	2	no	—
126	Dic₁₇₈	—	4	1	no	—
127	C₈×C₈₉	—	2	2	no	—
128	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₈₉	#117	1	1	yes	characteristic
130	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
131	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
132	#10623494	—	4	2	no	—
133	C₂²×C₈₉	—	2	1	no	—
134	C₂²×C₈₉	—	2	1	no	—
135	C₂²×C₈₉	#417	1	1	yes	characteristic
136	C₂²×C₈₉	#442	1	1	yes	characteristic
137	C₂²×C₈₉	#442	1	1	yes	characteristic
138	C₂×D₈₉	—	4	2	no	—
139	C₂×D₈₉	—	8	1	no	—
140	C₂×D₈₉	—	8	1	no	—
141	C₄×C₈₉	—	4	2	no	—
142	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₈₉	#418	1	1	yes	characteristic
144	C₄×C₈₉	#440	1	1	yes	characteristic
145	C₄×C₈₉	#440	1	1	yes	characteristic
146	Dic₈₉	—	4	2	no	—
147	Dic₈₉	—	8	2	no	—
148	Dic₈₉	—	8	2	no	—
149	#12650	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	#1762	1	1	yes	characteristic
153	C₂×C₈₉	#2180	1	1	yes	characteristic
154	C₂×C₈₉	AGL(1,ℤ/16)	1	1	yes	characteristic
155	D₈₉	—	8	2	no	—
156	C₂×#444	—	89	1	no	—
157	C₂×#360	—	89	1	no	—
158	#2139	—	89	1	no	—
159	C₂×#347	D₈₉	1	1	yes	characteristic, complemented
160	#1776	—	89	1	no	—
161	#1387	—	89	1	no	—
162	#1383	—	89	1	no	—
163	C₈₉	#12650	1	1	yes	characteristic, complemented, Sylow
164	C₂×#135	—	89	1	no	—
165	C₂×#131	—	178	1	no	—
166	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
167	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
168	#444	—	89	4	no	—
169	#444	—	178	2	no	—
170	C₂×#108	—	89	1	no	—
171	C₂×#107	—	89	1	no	—
172	C₂×#106	—	89	1	no	—
173	C₂×#104	—	178	1	no	—
174	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
175	C₄×C₄⋊C₄	—	178	1	no	—
176	#360	—	89	4	no	—
177	#358	—	89	2	no	—
178	#357	—	89	2	no	—
179	#347	—	2	2	no	—
180	#340	—	178	1	no	—
181	C₂×#41	—	178	1	no	—
182	#135	—	89	2	no	—
183	#135	—	89	2	no	—
184	#135	—	178	1	no	—
185	#135	—	178	1	no	—
186	C₂×Q₁₆	—	89	1	no	—
187	C₂×SD₁₆	—	178	1	no	—
188	C₂×D₈	—	89	1	no	—
189	#131	—	356	2	no	—
190	#131	—	178	4	no	—
191	C₂×M₁₆	—	356	1	no	—
192	C₂×M₁₆	—	178	1	no	—
193	C₂²×C₈	—	356	1	no	—
194	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	178	1	no	—
197	C₂×C₄⋊C₄	—	178	1	no	—
198	C₂×C₄²	—	356	1	no	—
199	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	C₄×D₈₉	1	2	yes	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
206	#108	—	89	2	no	—
207	#108	—	178	1	no	—
208	#107	—	89	2	no	—
209	#107	—	178	1	no	—
210	#106	—	89	2	no	—
211	#106	—	178	1	no	—
212	#104	—	178	4	no	—
213	#97	C₄×D₈₉	1	2	yes	—
214	#97	C₂²⋊D₈₉	1	2	yes	—
215	#95	—	356	1	no	—
216	#41	—	356	1	no	—
217	#41	—	356	1	no	—
218	#41	—	178	2	no	—
219	#41	—	178	2	no	—
220	C₂×Q₈	—	178	1	no	—
221	C₂×D₄	—	356	1	no	—
222	C₂×D₄	—	178	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	356	1	no	—
225	C₂²×C₄	—	356	1	no	—
226	C₂²×C₄	—	356	1	no	—
227	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
228	Q₁₆	—	89	2	no	—
229	Q₁₆	—	178	1	no	—
230	SD₁₆	—	178	2	no	—
231	SD₁₆	—	178	2	no	—
232	D₈	—	89	2	no	—
233	D₈	—	178	1	no	—
234	M₁₆	—	356	4	no	—
235	M₁₆	—	356	1	no	—
236	M₁₆	—	178	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	356	4	no	—
240	C₂×C₈	—	356	1	no	—
241	C₂×C₈	#13083551	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂×C₈	#13083574	1	1	yes	characteristic
247	C₂×C₈	#13083574	1	1	yes	characteristic
248	C₄⋊C₄	—	356	1	no	—
249	C₄⋊C₄	—	356	1	no	—
250	C₄⋊C₄	—	178	2	no	—
251	C₄⋊C₄	—	178	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	712	2	no	—
254	C₄²	#13083551	1	2	yes	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	356	1	no	—
258	C₂³	#15668040	1	1	yes	characteristic
259	Q₈	—	356	1	no	—
260	Q₈	—	178	2	no	—
261	D₄	—	356	2	no	—
262	D₄	—	356	2	no	—
263	D₄	—	356	1	no	—
264	D₄	—	178	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	356	1	no	—
269	C₂×C₄	—	356	2	no	—
270	C₂×C₄	—	356	2	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	356	2	no	—
273	C₂×C₄	—	712	1	no	—
274	C₂×C₄	—	712	1	no	—
275	C₂×C₄	—	712	1	no	—
276	C₂×C₄	—	712	1	no	—
277	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
278	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
279	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
280	C₂×C₄	#15668040	1	1	yes	characteristic
281	C₂×C₄	#15668040	1	1	yes	characteristic
282	C₂×C₄	#15668045	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	356	2	no	—
285	C₈	C₄²×D₈₉	1	1	yes	characteristic
286	C₈	#15667969	1	1	yes	characteristic
287	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	356	2	no	—
292	C₂²	—	712	1	no	—
293	C₂²	—	712	1	no	—
294	C₂²	#19151600	1	1	yes	characteristic
295	C₂²	#19151711	1	1	yes	characteristic
296	C₂²	#19151711	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	356	2	no	—
299	C₄	—	712	2	no	—
300	C₄	—	712	2	no	—
301	C₄	C₄×#13083551	1	1	yes	characteristic
302	C₄	#19151604	1	1	yes	characteristic
303	C₄	#19151700	1	1	yes	characteristic
304	C₄	#19151700	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	712	2	no	—
308	C₂	#26020078	1	1	yes	characteristic
309	C₂	#26022145	1	1	yes	characteristic
310	C₂	#26022152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151711
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#347×C₈₉	C₂
socle	C₂²×C₈₉	#442
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#347	D₈₉
hypocenter	C₈₉	#12650
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #26020149	C₂	semidirect
5. #26018361	C₂	semidirect
6. #26018351	C₂	semidirect
7. C₂×#347×C₈₉	C₂	semidirect
11. #19151721	C₄	semidirect
17. #19151409	C₄	semidirect
18. #19151403	C₄	semidirect
19. #19151400	C₄	semidirect
71. Dic₃₅₆	C₄²	semidirect
75. D₇₁₂	C₄²	semidirect
159. C₂×#347	D₈₉	semidirect
163. C₈₉	#12650	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151721	2
C₂×#19151409	2
#26021965	2
#26020149	2
#26018361	2
#26018351	2
C₂×#347×C₈₉	2
#12650	89