Subgroups — #36448171

Order 22784.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020369	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151350	C₂	1	2	yes	maximal
4	C₂×#19151348	C₂	1	2	yes	maximal
5	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
8	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
10	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151353	—	2	2	no	—
14	#19151350	—	2	4	no	—
15	#19151348	—	2	4	no	—
16	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₄³×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	2	no	—
19	C₂²×#10623494	—	2	2	no	—
20	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
22	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₄×#10623494	C₈	1	4	yes	complemented
30	C₄×#10623494	D₄	1	2	yes	—
31	C₄×#10623494	D₄	1	2	yes	—
32	C₄×#10623494	Q₈	1	2	yes	—
33	C₄×#10623494	Q₈	1	2	yes	—
34	C₂²×C₈×C₈₉	—	2	2	no	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	2	yes	—
38	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	C₂×#10623494	—	2	1	no	—
43	C₂×#10623494	—	2	1	no	—
44	C₂×#10623494	—	2	1	no	—
45	C₂×#10623494	—	2	1	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	4	no	—
48	C₂×#10623494	C₄⋊C₄	1	2	yes	—
49	C₂×#10623494	C₄⋊C₄	1	2	yes	—
50	C₂×#10623494	C₄⋊C₄	1	2	yes	—
51	C₂×#10623494	C₄⋊C₄	1	2	yes	—
52	C₂×#10623494	C₂×C₈	1	2	yes	—
53	C₂×#10623494	C₂×C₈	1	2	yes	—
54	C₂×#10623494	M₁₆	1	2	yes	—
55	C₂×#10623494	M₁₆	1	2	yes	—
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₂×C₈×C₈₉	—	4	4	no	—
63	C₄²×C₈₉	—	2	4	no	—
64	C₄²×C₈₉	—	2	1	no	—
65	C₄²×C₈₉	—	2	1	no	—
66	C₄²×C₈₉	—	2	4	no	—
67	C₄²×C₈₉	C₂×C₈	1	2	yes	—
68	C₄²×C₈₉	M₁₆	1	2	yes	—
69	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₈₉	#108	1	4	yes	—
81	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#135	1	4	yes	—
92	C₈×C₈₉	—	8	2	no	—
93	#10623494	—	2	2	no	—
94	#10623494	—	2	2	no	—
95	#10623494	—	8	2	no	—
96	#10623494	C₄⋊C₈	1	4	yes	—
97	#10623494	C₄⋊C₈	1	4	yes	—
98	C₂²×C₈₉	#384	1	1	yes	characteristic
99	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₈₉	#423	1	1	yes	characteristic
101	C₂²×C₈₉	#427	1	2	yes	—
102	C₂²×C₈₉	C₂×#108	1	2	yes	—
103	C₄×C₈₉	—	2	4	no	—
104	C₄×C₈₉	—	2	4	no	—
105	C₄×C₈₉	—	2	1	no	—
106	C₄×C₈₉	—	2	1	no	—
107	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₈₉	#423	1	1	yes	characteristic
110	C₄×C₈₉	#423	1	1	yes	characteristic
111	C₄×C₈₉	#495	1	2	yes	—
112	C₄×C₈₉	#499	1	2	yes	—
113	#8807	—	89	1	no	maximal, Sylow
114	C₂×C₈₉	#1564	1	2	yes	—
115	C₂×C₈₉	#1568	1	2	yes	—
116	C₂×C₈₉	#1842	1	1	yes	characteristic
117	C₂×C₈₉	#1846	1	1	yes	characteristic
118	C₂×C₈₉	#1848	1	1	yes	characteristic
119	#1831	—	89	1	no	—
120	#1831	D₈₉	1	1	yes	characteristic
121	C₄²×C₈	—	89	1	no	—
122	C₂×#335	—	89	2	no	—
123	C₂×#334	—	89	2	no	—
124	C₈₉	#8807	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊C₈	—	89	2	no	—
126	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
127	C₂×C₄×C₈	—	89	1	no	—
128	C₂×C₄×C₈	—	89	1	no	—
129	C₂×C₄×C₈	—	89	1	no	—
130	C₂×C₄×C₈	—	89	1	no	—
131	C₂×C₄×C₈	—	89	1	no	—
132	C₂×C₄×C₈	—	89	1	no	—
133	C₄³	C₂×D₈₉	1	1	yes	characteristic
134	#336	—	2	2	no	—
135	#336	—	178	2	no	—
136	#335	—	178	4	no	—
137	#334	—	178	4	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	89	1	no	—
140	C₂²×C₈	—	89	1	no	—
141	C₂²×C₈	—	89	1	no	—
142	C₂²×C₈	—	89	1	no	—
143	C₂²×C₈	—	178	2	no	—
144	C₂×C₄²	Dic₁₇₈	1	2	yes	—
145	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
148	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	178	2	no	—
152	C₄⋊C₈	—	178	2	no	—
153	C₄×C₈	—	89	4	no	—
154	C₄×C₈	—	89	2	no	—
155	C₄×C₈	—	89	2	no	—
156	C₄×C₈	—	89	2	no	—
157	C₄×C₈	—	89	2	no	—
158	C₄×C₈	—	178	2	no	—
159	C₄×C₈	—	178	2	no	—
160	C₄×C₈	—	178	2	no	—
161	C₂²×C₄	#13083553	1	2	yes	—
162	C₂²×C₄	#13083553	1	2	yes	—
163	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	89	2	no	—
169	C₂×C₈	—	89	2	no	—
170	C₂×C₈	—	89	2	no	—
171	C₂×C₈	—	89	2	no	—
172	C₂×C₈	—	89	2	no	—
173	C₂×C₈	—	89	2	no	—
174	C₂×C₈	—	89	2	no	—
175	C₂×C₈	—	89	2	no	—
176	C₂×C₈	—	356	2	no	—
177	C₂×C₈	—	356	4	no	—
178	C₂×C₈	—	178	1	no	—
179	C₂×C₈	—	178	1	no	—
180	C₂×C₈	—	178	1	no	—
181	C₂×C₈	—	178	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	C₈×D₈₉	1	2	yes	—
187	C₄²	#13083557	1	2	yes	—
188	C₄²	D₄×D₈₉	1	1	yes	characteristic
189	C₄²	D₄×D₈₉	1	1	yes	characteristic
190	C₄²	Q₈×D₈₉	1	1	yes	characteristic
191	C₄²	Q₈×D₈₉	1	1	yes	characteristic
192	C₂³	C₂×#13083553	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	#15667941	1	4	yes	—
199	C₂×C₄	#15667947	1	4	yes	—
200	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
201	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
202	C₂×C₄	#15668008	1	1	yes	characteristic
203	C₂×C₄	#15668010	1	1	yes	characteristic
204	C₂×C₄	#15668011	1	1	yes	characteristic
205	C₂×C₄	#15668011	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
209	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
210	C₂×C₄	#15668032	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	89	4	no	—
213	C₈	—	89	4	no	—
214	C₈	—	178	2	no	—
215	C₈	—	178	2	no	—
216	C₈	—	712	2	no	—
217	C₂²	#19151467	1	1	yes	characteristic
218	C₂²	C₂×#15667941	1	1	yes	characteristic
219	C₂²	#19151621	1	1	yes	characteristic
220	C₂²	#19151634	1	2	yes	—
221	C₂²	C₂×#15667947	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
227	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
228	C₄	#19151620	1	1	yes	characteristic
229	C₄	#19151620	1	1	yes	characteristic
230	C₄	#19151980	1	2	yes	—
231	C₄	#19151997	1	2	yes	—
232	C₂	#26018998	1	2	yes	—
233	C₂	#26019013	1	2	yes	—
234	C₂	#26020403	1	1	yes	characteristic
235	C₂	#26020414	1	1	yes	characteristic
236	C₂	#26020423	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#8807
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
29. C₄×#10623494	C₈	semidirect
124. C₈₉	#8807	semidirect

Maximal subgroups

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

subgroup	index
#26020369	2
C₄²×#10623494	2
C₂×#19151350	2
C₂×#19151348	2
C₈₉×#1831	2
#8807	89