Subgroups — #36470530

Order 22784.

430 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151716	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151409	C₂	1	2	yes	maximal
3	#26020409	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018351	C₂	1	2	yes	maximal, complemented
5	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668034	C₂²	1	1	yes	characteristic
7	C₂×#15668018	—	2	1	no	—
8	#19151716	—	2	2	no	—
9	#19151716	C₄	1	4	yes	complemented
10	C₂×#15667948	C₂²	1	1	yes	characteristic
11	C₂×#15667944	—	2	1	no	—
12	C₂×#15667944	C₂²	1	2	yes	—
13	C₂×#15667940	—	2	1	no	—
14	#19151468	—	2	1	no	—
15	#19151409	—	2	4	no	—
16	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#364×C₈₉	—	2	2	no	—
19	C₂×#13083574	—	2	1	no	—
20	#15668034	—	2	1	no	—
21	#15668034	—	2	1	no	—
22	#15668034	C₂×C₄	1	2	yes	—
23	#15668034	D₄	1	2	yes	—
24	C₂×Dic₃₅₆	Q₈	1	1	yes	characteristic
25	C₂×#13083563	—	2	1	no	—
26	C₂×D₇₁₂	Q₈	1	1	yes	characteristic
27	#15668018	—	2	4	no	—
28	#15668018	—	4	2	no	—
29	C₂×#13083558	—	2	1	no	—
30	C₂×#13083557	—	4	1	no	—
31	C₂×C₈×D₈₉	—	4	1	no	—
32	C₂×C₄⋊Dic₈₉	—	2	2	no	—
33	C₂×C₄⋊Dic₈₉	—	4	2	no	—
34	#15667948	—	2	1	no	—
35	#15667948	C₂×C₄	1	2	yes	—
36	#15667944	—	2	2	no	—
37	#15667944	—	2	2	no	—
38	#15667944	—	4	2	no	—
39	#15667940	—	2	4	no	—
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	M₃₂×C₈₉	—	2	2	no	—
44	M₃₂×C₈₉	—	2	2	no	—
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₉	D₄	1	2	yes	—
48	#13083574	—	2	2	no	—
49	#13083574	—	2	2	no	—
50	#13083574	—	4	1	no	—
51	#13083574	—	4	1	no	—
52	C₂×Dic₁₇₈	—	2	1	no	—
53	C₂×D₃₅₆	—	2	1	no	—
54	C₂×C₂²⋊D₈₉	—	4	1	no	—
55	C₂²×Dic₈₉	—	4	2	no	—
56	C₂×C₄×D₈₉	—	4	1	no	—
57	Dic₃₅₆	—	2	1	no	—
58	Dic₃₅₆	C₄⋊C₄	1	2	yes	complemented
59	#13083563	—	2	2	no	—
60	#13083563	—	2	2	no	—
61	D₇₁₂	—	2	1	no	—
62	D₇₁₂	C₄⋊C₄	1	2	yes	complemented
63	#13083558	—	2	2	no	—
64	#13083558	—	4	1	no	—
65	#13083557	—	4	4	no	—
66	C₂×#10623494	—	4	1	no	—
67	C₈×D₈₉	—	4	4	no	—
68	C₄⋊Dic₈₉	—	4	2	no	—
69	C₄⋊Dic₈₉	—	4	2	no	—
70	C₄⋊Dic₈₉	—	8	4	no	—
71	C₂²×C₄×C₈₉	—	2	1	no	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₉	D₈	1	2	yes	—
75	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
76	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
83	C₄²×C₈₉	—	2	1	no	—
84	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
85	C₁₆×C₈₉	—	4	2	no	—
86	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂²×D₈₉	—	4	1	no	—
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
96	C₂²⋊D₈₉	—	4	2	no	—
97	C₂²⋊D₈₉	—	4	2	no	—
98	C₂×Dic₈₉	—	4	1	no	—
99	C₂×Dic₈₉	—	8	4	no	—
100	C₂×Dic₈₉	—	8	2	no	—
101	D₃₅₆	—	2	2	no	—
102	D₃₅₆	—	4	1	no	—
103	C₄×D₈₉	—	4	2	no	—
104	C₄×D₈₉	—	4	2	no	—
105	Dic₁₇₈	—	2	2	no	—
106	Dic₁₇₈	—	4	1	no	—
107	C₈×C₈₉	#102	1	2	yes	—
108	C₈×C₈₉	#102	1	2	yes	—
109	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₈×C₈₉	#117	1	1	yes	characteristic
111	C₈×C₈₉	#121	1	1	yes	characteristic
112	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
113	#10623494	—	4	2	no	—
114	C₂²×C₈₉	—	2	2	no	—
115	C₂²×C₈₉	#417	1	1	yes	characteristic
116	C₂²×C₈₉	#459	1	1	yes	characteristic
117	C₂²×C₈₉	#476	1	1	yes	characteristic
118	C₂×D₈₉	—	4	2	no	—
119	C₂×D₈₉	—	8	2	no	—
120	C₄×C₈₉	—	2	4	no	—
121	C₄×C₈₉	#380	1	1	yes	characteristic
122	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
123	C₄×C₈₉	#461	1	1	yes	characteristic
124	C₄×C₈₉	#474	1	1	yes	characteristic
125	Dic₈₉	—	4	2	no	—
126	Dic₈₉	—	16	2	no	—
127	#12823	—	89	1	no	maximal, Sylow
128	C₂×C₈₉	—	2	1	no	—
129	C₂×C₈₉	—	2	1	no	—
130	C₂×C₈₉	#1802	1	1	yes	characteristic
131	C₂×C₈₉	#2230	1	2	yes	—
132	D₈₉	—	8	2	no	—
133	C₂×#443	—	89	1	no	—
134	C₂×#364	D₈₉	1	1	yes	characteristic, complemented
135	C₂×#360	—	89	2	no	—
136	#1844	—	89	1	no	—
137	#1383	—	89	2	no	—
138	C₈₉	#12823	1	1	yes	characteristic, complemented, Sylow
139	C₂×#135	—	89	1	no	—
140	C₂×#131	—	178	1	no	—
141	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
142	#443	—	89	4	no	—
143	#443	—	178	2	no	—
144	C₂×#108	—	89	1	no	—
145	C₂×#106	—	89	2	no	—
146	C₂×#106	—	178	1	no	—
147	C₂×#104	—	178	1	no	—
148	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
149	#384	—	178	1	no	—
150	#364	—	2	2	no	—
151	#360	—	178	4	no	—
152	C₂×#41	—	178	1	no	—
153	#135	—	89	2	no	—
154	#135	—	89	2	no	—
155	#135	—	178	1	no	—
156	#135	—	178	1	no	—
157	C₂×Q₁₆	—	89	1	no	—
158	C₂×SD₁₆	—	178	1	no	—
159	C₂×D₈	—	89	1	no	—
160	#131	—	356	2	no	—
161	#131	—	178	4	no	—
162	C₂×M₁₆	—	356	1	no	—
163	C₂×M₁₆	—	178	1	no	—
164	C₂²×C₈	—	356	1	no	—
165	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	356	2	no	—
168	C₂×C₄⋊C₄	—	178	2	no	—
169	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	#108	—	89	2	no	—
174	#108	—	178	1	no	—
175	#106	—	356	2	no	—
176	#106	—	178	2	no	—
177	#106	—	178	2	no	—
178	#104	—	178	4	no	—
179	C₄×C₈	C₄×D₈₉	1	2	yes	—
180	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
181	#41	—	356	1	no	—
182	#41	—	356	1	no	—
183	#41	—	178	2	no	—
184	#41	—	178	2	no	—
185	C₂×Q₈	—	178	1	no	—
186	C₂×D₄	—	356	1	no	—
187	C₂×D₄	—	178	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	356	1	no	—
190	C₂²×C₄	—	356	2	no	—
191	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
192	Q₁₆	—	89	2	no	—
193	Q₁₆	—	178	1	no	—
194	SD₁₆	—	178	2	no	—
195	SD₁₆	—	178	2	no	—
196	D₈	—	89	2	no	—
197	D₈	—	178	1	no	—
198	M₁₆	—	356	1	no	—
199	M₁₆	—	356	4	no	—
200	M₁₆	—	178	2	no	—
201	C₂×C₈	—	356	1	no	—
202	C₂×C₈	—	356	4	no	—
203	C₂×C₈	#13083551	1	2	yes	—
204	C₂×C₈	D₇₁₂	1	2	yes	—
205	C₂×C₈	#13083563	1	1	yes	characteristic
206	C₂×C₈	#13083563	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
208	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
210	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
211	C₂×C₈	#13083573	1	1	yes	characteristic
212	C₂×C₈	#13083575	1	1	yes	characteristic
213	C₄⋊C₄	—	356	2	no	—
214	C₄⋊C₄	—	356	2	no	—
215	C₄⋊C₄	—	712	4	no	—
216	C₄²	—	2	1	no	—
217	C₄²	#13083551	1	2	yes	—
218	C₁₆	—	4	2	no	—
219	C₂³	—	356	1	no	—
220	C₂³	#15668040	1	1	yes	characteristic
221	Q₈	—	356	1	no	—
222	Q₈	—	178	2	no	—
223	D₄	—	356	2	no	—
224	D₄	—	356	2	no	—
225	D₄	—	356	1	no	—
226	D₄	—	178	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	356	1	no	—
230	C₂×C₄	—	356	2	no	—
231	C₂×C₄	—	356	2	no	—
232	C₂×C₄	—	712	4	no	—
233	C₂×C₄	—	712	2	no	—
234	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
235	C₂×C₄	#15667987	1	1	yes	characteristic
236	C₂×C₄	#15667992	1	1	yes	characteristic
237	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
238	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
239	C₂×C₄	#15668045	1	1	yes	characteristic
240	C₈	—	356	2	no	—
241	C₈	#15667934	1	2	yes	—
242	C₈	#15667934	1	2	yes	—
243	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
244	C₈	#15667970	1	1	yes	characteristic
245	C₈	#15667983	1	1	yes	characteristic
246	C₈	#15667988	1	1	yes	characteristic
247	C₂²	—	2	2	no	—
248	C₂²	—	356	2	no	—
249	C₂²	—	712	2	no	—
250	C₂²	#19151600	1	1	yes	characteristic
251	C₂²	#19151794	1	1	yes	characteristic
252	C₂²	#19151889	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	356	2	no	—
255	C₄	—	1424	2	no	—
256	C₄	#19151455	1	1	yes	characteristic
257	C₄	C₂×#15667934	1	1	yes	characteristic
258	C₄	#19151804	1	1	yes	characteristic
259	C₄	#19151879	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	712	2	no	—
263	C₂	#26020253	1	1	yes	characteristic
264	C₂	#26022424	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151889
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#12823
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #26020409	C₂	semidirect
4. #26018351	C₂	semidirect
5. C₂×#364×C₈₉	C₂	semidirect
9. #19151716	C₄	semidirect
58. Dic₃₅₆	C₄⋊C₄	semidirect
62. D₇₁₂	C₄⋊C₄	semidirect
134. C₂×#364	D₈₉	semidirect
138. C₈₉	#12823	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151716	2
C₂×#19151409	2
#26020409	2
#26018351	2
C₂×#364×C₈₉	2
#12823	89