Subgroups — #36444065

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	2	yes	maximal
2	#26018097	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151655	—	2	2	no	—
5	#19151655	C₄	1	4	yes	—
6	#19151651	—	2	2	no	—
7	#19151651	C₂²	1	4	yes	—
8	#19151622	—	2	2	no	—
9	#19151562	C₄	1	4	yes	—
10	C₄×#13083558	—	2	2	no	—
11	C₂²×M₁₆×C₈₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₈₉	—	2	2	no	—
14	C₄×M₁₆×C₈₉	—	2	2	no	—
15	C₂×#97×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
17	C₂×#13083558	—	2	4	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄⋊#10623494	—	2	4	no	—
20	C₂²⋊#10623494	—	2	4	no	—
21	C₂²⋊#10623494	C₂×C₄	1	8	yes	—
22	#15667923	—	2	4	no	—
23	C₄×#10623494	—	2	4	no	—
24	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₈₉	—	2	2	no	—
27	C₂×M₁₆×C₈₉	—	2	2	no	—
28	C₂×M₁₆×C₈₉	—	2	8	no	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	2	yes	—
35	C₂×C₄²×C₈₉	D₄	1	2	yes	—
36	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	#97×C₈₉	—	2	4	no	—
40	C₄×C₈×C₈₉	—	2	4	no	—
41	#13083558	—	4	8	no	—
42	C₂×#10623494	—	2	8	no	—
43	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	—	2	4	no	—
45	C₂²×C₄×C₈₉	—	4	4	no	—
46	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	M₁₆×C₈₉	—	2	8	no	—
59	M₁₆×C₈₉	—	2	8	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	2	4	no	—
62	C₂×C₈×C₈₉	M₁₆	1	8	yes	complemented
63	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×D₄	1	2	yes	—
71	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₈₉	—	2	2	no	—
74	C₂³×C₈₉	—	4	2	no	—
75	C₂³×C₈₉	#95	1	1	yes	characteristic
76	C₂³×C₈₉	#95	1	1	yes	characteristic
77	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	4	no	—
80	C₂×C₄×C₈₉	—	2	4	no	—
81	C₂×C₄×C₈₉	—	2	4	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	4	2	no	—
87	C₂×C₄×C₈₉	—	4	2	no	—
88	C₂×C₄×C₈₉	—	4	2	no	—
89	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#95	1	2	yes	—
92	C₂×C₄×C₈₉	#95	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
98	C₈×C₈₉	—	2	8	no	—
99	#10623494	—	4	8	no	—
100	C₂²×C₈₉	—	2	2	no	—
101	C₂²×C₈₉	—	2	1	no	—
102	C₂²×C₈₉	—	2	1	no	—
103	C₂²×C₈₉	—	2	2	no	—
104	C₂²×C₈₉	—	2	2	no	—
105	C₂²×C₈₉	—	4	4	no	—
106	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
107	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	—	2	2	no	—
111	C₄×C₈₉	—	2	2	no	—
112	C₄×C₈₉	—	2	2	no	—
113	C₄×C₈₉	#344	1	4	yes	—
114	C₄×C₈₉	#407	1	2	yes	—
115	C₄×C₈₉	#423	1	2	yes	—
116	#7920	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	2	no	—
118	C₂×C₈₉	—	4	1	no	—
119	C₂×C₈₉	—	4	1	no	—
120	C₂×C₈₉	#1728	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#344	1	2	yes	—
122	#2920	—	89	2	no	—
123	C₂×C₄×M₁₆	D₈₉	1	1	yes	characteristic
124	#1281	—	89	4	no	—
125	C₈₉	#7920	1	1	yes	characteristic, complemented, Sylow
126	C₂²×M₁₆	Dic₈₉	1	2	yes	—
127	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
128	#432	—	89	4	no	—
129	#432	—	178	2	no	—
130	#431	—	89	4	no	—
131	#431	—	178	2	no	—
132	#423	—	178	2	no	—
133	#407	—	89	4	no	—
134	C₄×M₁₆	—	2	2	no	—
135	C₄×M₁₆	—	2	2	no	—
136	C₄×M₁₆	—	178	2	no	—
137	C₂×#97	C₂×D₈₉	1	2	yes	—
138	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
139	C₂³×C₄	C₄×D₈₉	1	2	yes	—
140	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	8	no	—
144	C₂×M₁₆	—	178	4	no	—
145	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
146	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
147	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₈₉	1	2	yes	—
149	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
150	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
154	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
155	C₄⋊C₈	—	178	4	no	—
156	C₄⋊C₈	—	178	4	no	—
157	C₂²⋊C₈	—	89	8	no	—
158	C₂²⋊C₈	—	178	4	no	—
159	#97	—	2	4	no	—
160	#97	—	178	4	no	—
161	C₄×C₈	—	2	4	no	—
162	C₄×C₈	—	178	4	no	—
163	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
167	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
168	C₂²×C₄	#13083551	1	2	yes	—
169	C₂²×C₄	#13083551	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
172	C₂²×C₄	#13083553	1	2	yes	—
173	C₂²×C₄	#13083553	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	—	356	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	178	8	no	—
184	C₂×C₈	#13083558	1	8	yes	complemented
185	C₄²	#13083551	1	2	yes	—
186	C₄²	#13083551	1	2	yes	—
187	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
188	C₄²	#13083553	1	2	yes	—
189	C₄²	#13083553	1	2	yes	—
190	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
191	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
193	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#15667919	1	1	yes	characteristic
198	C₂³	#15667919	1	1	yes	characteristic
199	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	#15667919	1	1	yes	characteristic
212	C₂×C₄	#15667919	1	1	yes	characteristic
213	C₂×C₄	#15667919	1	2	yes	—
214	C₂×C₄	#15667919	1	2	yes	—
215	C₂×C₄	C₂×#13083551	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
217	C₂×C₄	C₂×#13083553	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
219	C₂×C₄	C₂×#13083558	1	4	yes	—
220	C₈	—	2	8	no	—
221	C₈	—	356	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₂×#15667919	1	1	yes	characteristic
229	C₂²	C₄×#13083558	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#19151369	1	4	yes	—
236	C₄	#19151562	1	2	yes	—
237	C₄	#19151622	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	#26019946	1	1	yes	characteristic
242	C₂	C₂×#19151369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₄×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₈₉
hypocenter	C₈₉	#7920
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #26018097	C₂	semidirect
62. C₂×C₈×C₈₉	M₁₆	semidirect
125. C₈₉	#7920	semidirect
184. C₂×C₈	#13083558	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26018097	2
C₂×C₄×M₁₆×C₈₉	2
#7920	89