Subgroups — #36453330

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26021188	C₂	1	2	yes	maximal
3	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151330	C₂	1	2	yes	maximal
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
8	C₄×Dic₃₅₆	—	2	2	no	—
9	C₄×Dic₃₅₆	—	2	2	no	—
10	C₂×#15667946	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×#15667937	C₄	1	2	yes	—
13	#19151352	—	2	4	no	—
14	#19151349	—	2	2	no	—
15	#19151330	—	2	4	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#322×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₅₆	—	2	4	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₄×Dic₁₇₈	—	2	2	no	—
25	C₄×Dic₁₇₈	—	2	2	no	—
26	C₄×Dic₁₇₈	—	2	2	no	—
27	C₄×Dic₁₇₈	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×#13083553	—	2	2	no	—
30	C₂×C₄×Dic₈₉	—	2	2	no	—
31	#15667946	—	2	2	no	—
32	C₄⋊#10623494	—	2	2	no	—
33	C₄⋊#10623494	—	2	2	no	—
34	#15667937	—	2	4	no	—
35	C₂²×C₈×C₈₉	—	2	1	no	—
36	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	D₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	—	2	1	no	—
41	C₄×C₈×C₈₉	—	2	1	no	—
42	C₄×C₈×C₈₉	—	2	1	no	—
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₄×C₈×C₈₉	D₄	1	2	yes	—
46	C₂×Dic₁₇₈	—	2	2	no	—
47	C₂×Dic₁₇₈	—	2	2	no	—
48	C₂×Dic₁₇₈	—	2	2	no	—
49	C₂×Dic₁₇₈	—	2	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	M₁₆	1	4	yes	—
52	C₂²×Dic₈₉	—	2	2	no	—
53	C₂²×Dic₈₉	—	2	2	no	—
54	Dic₃₅₆	—	2	8	no	—
55	C₂×#10623494	—	4	2	no	—
56	C₂×#10623494	—	4	4	no	—
57	C₄⋊Dic₈₉	—	2	2	no	—
58	C₄⋊Dic₈₉	—	2	2	no	—
59	#13083553	—	4	2	no	—
60	#13083553	—	4	2	no	—
61	C₄×Dic₈₉	—	4	2	no	—
62	C₄×Dic₈₉	—	4	2	no	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
66	C₂×C₈×C₈₉	—	2	4	no	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	2	4	no	—
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₉	#41	1	1	yes	characteristic
78	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₈₉	#131	1	2	yes	—
88	C₂×C₄×C₈₉	#137	1	4	yes	—
89	C₂×Dic₈₉	—	2	4	no	—
90	C₂×Dic₈₉	—	4	2	no	—
91	C₂×Dic₈₉	—	4	2	no	—
92	C₂×Dic₈₉	—	4	2	no	—
93	C₂×Dic₈₉	—	4	4	no	—
94	Dic₁₇₈	—	2	4	no	—
95	Dic₁₇₈	—	2	4	no	—
96	Dic₁₇₈	—	4	2	no	—
97	Dic₁₇₈	—	4	2	no	—
98	C₈×C₈₉	—	2	4	no	—
99	C₈×C₈₉	—	2	2	no	—
100	C₈×C₈₉	—	2	2	no	—
101	C₈×C₈₉	—	4	2	no	—
102	#10623494	—	8	2	no	—
103	C₂²×C₈₉	#390	1	1	yes	characteristic
104	C₂²×C₈₉	#435	1	2	yes	—
105	C₂²×C₈₉	#441	1	2	yes	—
106	C₂²×C₈₉	#443	1	2	yes	—
107	C₄×C₈₉	—	2	1	no	—
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	#407	1	1	yes	characteristic
110	C₄×C₈₉	#408	1	1	yes	characteristic
111	C₄×C₈₉	#435	1	2	yes	—
112	C₄×C₈₉	#470	1	2	yes	—
113	C₄×C₈₉	#497	1	2	yes	—
114	Dic₈₉	—	4	4	no	—
115	Dic₈₉	—	8	2	no	—
116	#9664	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	#1587	1	4	yes	—
118	C₂×C₈₉	#1955	1	1	yes	characteristic
119	C₂×C₈₉	#1971	1	1	yes	characteristic
120	C₂×C₈₉	#1974	1	1	yes	characteristic
121	C₂×C₄×Q₁₆	—	89	1	no	—
122	#1984	—	89	2	no	—
123	C₂×#334	—	89	1	no	—
124	C₂×Q₈⋊C₈	—	89	2	no	—
125	C₂×#322	D₈₉	1	1	yes	characteristic
126	C₈₉	#9664	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₁₆	—	89	1	no	—
128	C₂×C₄×Q₈	—	89	2	no	—
129	C₄×Q₁₆	—	178	2	no	—
130	C₄×Q₁₆	—	178	2	no	—
131	C₂×#107	—	89	1	no	—
132	C₂×C₄⋊C₈	—	89	2	no	—
133	C₂×#103	—	89	2	no	—
134	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
137	#336	—	178	4	no	—
138	#334	—	178	2	no	—
139	Q₈⋊C₈	—	178	4	no	—
140	#322	—	2	2	no	—
141	C₂²×Q₈	—	89	2	no	—
142	C₂×Q₁₆	—	178	4	no	—
143	C₂×Q₁₆	—	178	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	178	2	no	—
146	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
148	C₂²×C₈	D₃₅₆	1	2	yes	—
149	C₄×Q₈	—	178	2	no	—
150	C₄×Q₈	—	178	2	no	—
151	C₄×Q₈	—	178	2	no	—
152	C₄×Q₈	—	178	2	no	—
153	C₂×C₄⋊C₄	—	89	2	no	—
154	C₂×C₄⋊C₄	—	178	2	no	—
155	C₂×C₄²	—	178	2	no	—
156	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
157	#107	—	178	2	no	—
158	C₄⋊C₈	—	178	2	no	—
159	C₄⋊C₈	—	178	2	no	—
160	#103	—	178	4	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	D₃₅₆	1	2	yes	—
165	C₄×C₈	D₃₅₆	1	2	yes	—
166	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
167	C₂×Q₈	—	89	4	no	—
168	C₂×Q₈	—	178	2	no	—
169	C₂×Q₈	—	178	2	no	—
170	C₂×Q₈	—	178	2	no	—
171	C₂×Q₈	—	178	2	no	—
172	C₂×Q₈	—	178	2	no	—
173	C₂²×C₄	—	178	2	no	—
174	C₂²×C₄	—	178	2	no	—
175	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
177	C₂²×C₄	#13083574	1	1	yes	characteristic
178	Q₁₆	—	178	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	356	2	no	—
186	C₂×C₈	—	356	4	no	—
187	C₂×C₈	#13083551	1	2	yes	—
188	C₂×C₈	#13083551	1	2	yes	—
189	C₄⋊C₄	—	356	2	no	—
190	C₄⋊C₄	—	356	2	no	—
191	C₄⋊C₄	—	178	2	no	—
192	C₄⋊C₄	—	178	2	no	—
193	C₄²	—	356	2	no	—
194	C₄²	—	356	2	no	—
195	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
196	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
197	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
198	C₄²	#13083574	1	1	yes	characteristic
199	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
200	Q₈	—	356	2	no	—
201	Q₈	—	356	2	no	—
202	Q₈	—	178	4	no	—
203	Q₈	—	178	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	356	2	no	—
206	C₂×C₄	—	356	2	no	—
207	C₂×C₄	—	356	4	no	—
208	C₂×C₄	—	356	2	no	—
209	C₂×C₄	—	178	4	no	—
210	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
211	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
212	C₂×C₄	#15667986	1	1	yes	characteristic
213	C₂×C₄	#15667986	1	1	yes	characteristic
214	C₂×C₄	#15668000	1	1	yes	characteristic
215	C₂×C₄	#15668009	1	1	yes	characteristic
216	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
217	C₂×C₄	#15668020	1	2	yes	—
218	C₂×C₄	#15668045	1	4	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	712	2	no	—
224	C₂²	#19151500	1	1	yes	characteristic
225	C₂²	#19151671	1	2	yes	—
226	C₂²	#19151706	1	2	yes	—
227	C₂²	#19151716	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	356	4	no	—
231	C₄	—	712	2	no	—
232	C₄	#19151561	1	1	yes	characteristic
233	C₄	#19151566	1	1	yes	characteristic
234	C₄	#19151671	1	2	yes	—
235	C₄	#19151856	1	2	yes	—
236	C₄	#19151992	1	2	yes	—
237	C₂	#26019125	1	4	yes	—
238	C₂	#26021050	1	1	yes	characteristic
239	C₂	#26021123	1	1	yes	characteristic
240	C₂	#26021140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9664
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
126. C₈₉	#9664	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₅₆	2
#26021188	2
C₂×#19151349	2
C₂×#19151330	2
C₂×#322×C₈₉	2
#9664	89