Subgroups — #36523573

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	2	yes	maximal
2	#26019023	C₂	1	8	yes	maximal, complemented
3	#26019008	C₂	1	4	yes	maximal, complemented
4	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	—	2	2	no	—
6	#19151622	—	2	2	no	—
7	#19151622	—	2	2	no	—
8	#19151622	C₂²	1	4	yes	—
9	#19151622	C₂²	1	4	yes	—
10	C₂×C₄⋊#10623494	—	2	2	no	—
11	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
12	#19151349	C₂²	1	16	yes	complemented
13	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	#431×C₈₉	C₄	1	4	yes	—
15	#431×C₈₉	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
19	C₂×#13083558	—	2	4	no	—
20	C₂×#13083558	—	2	4	no	—
21	C₂×#13083558	—	4	8	no	—
22	C₂²×#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₄⋊#10623494	—	2	4	no	—
25	C₄⋊#10623494	—	2	4	no	—
26	C₄⋊#10623494	C₂³	1	8	yes	—
27	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₈₉	D₄	1	2	yes	—
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
40	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₈₉	C₂³	1	4	yes	—
43	#13083558	—	4	8	no	—
44	#13083558	—	4	8	no	—
45	C₂×#10623494	—	2	8	no	—
46	C₂×#10623494	—	4	8	no	—
47	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	—	4	2	no	—
49	C₂²×C₄×C₈₉	—	4	2	no	—
50	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	8	yes	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₈₉	—	4	4	no	—
64	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
65	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₈₉	C₂×D₄	1	2	yes	—
71	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
73	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₂³×C₈₉	—	4	1	no	—
76	C₂³×C₈₉	—	4	1	no	—
77	C₂³×C₈₉	#106	1	4	yes	—
78	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	4	no	—
82	C₂×C₄×C₈₉	—	2	4	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	4	2	no	—
85	C₂×C₄×C₈₉	—	4	1	no	—
86	C₂×C₄×C₈₉	—	4	1	no	—
87	C₂×C₄×C₈₉	—	4	2	no	—
88	C₂×C₄×C₈₉	#106	1	4	yes	—
89	C₂×C₄×C₈₉	#106	1	8	yes	—
90	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
97	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
102	C₈×C₈₉	—	2	8	no	—
103	#10623494	—	4	8	no	—
104	C₂²×C₈₉	—	2	1	no	—
105	C₂²×C₈₉	—	2	2	no	—
106	C₂²×C₈₉	—	2	2	no	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	—	4	4	no	—
109	C₂²×C₈₉	C₂×#106	1	4	yes	—
110	C₂²×C₈₉	C₂×#106	1	2	yes	—
111	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₄×C₈₉	—	2	1	no	—
113	C₄×C₈₉	—	2	2	no	—
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	C₂×#106	1	4	yes	—
117	C₄×C₈₉	C₂×#106	1	2	yes	—
118	C₄×C₈₉	#430	1	2	yes	—
119	C₄×C₈₉	#568	1	2	yes	—
120	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
121	#17813	—	89	1	no	maximal, Sylow
122	C₂×C₈₉	—	2	2	no	—
123	C₂×C₈₉	—	2	4	no	—
124	C₂×C₈₉	#2905	1	1	yes	characteristic
125	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
126	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
127	C₂×#431	D₈₉	1	1	yes	characteristic
128	C₂×#423	—	89	2	no	—
129	#1570	—	89	8	no	—
130	#1566	—	89	4	no	—
131	C₈₉	#17813	1	1	yes	characteristic, complemented, Sylow
132	C₂²×M₁₆	—	178	2	no	—
133	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
134	#431	Dic₈₉	1	4	yes	—
135	#431	C₂×D₈₉	1	4	yes	—
136	#423	—	89	4	no	—
137	#423	—	89	4	no	—
138	#423	—	178	2	no	—
139	#423	—	178	2	no	—
140	C₂×C₄⋊C₈	—	89	4	no	—
141	C₂×C₄⋊C₈	—	178	2	no	—
142	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
143	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
144	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
145	#334	—	89	16	no	—
146	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
147	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
148	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
149	C₂×M₁₆	—	356	8	no	—
150	C₂×M₁₆	—	178	4	no	—
151	C₂×M₁₆	—	178	4	no	—
152	C₂²×C₈	—	178	4	no	—
153	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
154	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
155	C₂×C₄²	Dic₁₇₈	1	2	yes	—
156	C₂×C₄²	D₃₅₆	1	2	yes	—
157	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
160	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
162	C₄⋊C₈	—	89	8	no	—
163	C₄⋊C₈	—	178	4	no	—
164	C₄⋊C₈	—	178	4	no	—
165	C₄⋊C₈	—	178	4	no	—
166	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
167	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
168	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
169	C₄×C₈	C₂²×D₈₉	1	4	yes	—
170	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
175	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
179	C₂²×C₄	#13083563	1	8	yes	—
180	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
183	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
186	M₁₆	—	356	8	no	—
187	M₁₆	—	356	8	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	356	8	no	—
190	C₂×C₈	—	178	8	no	—
191	C₂×C₈	C₂²×Dic₈₉	1	8	yes	—
192	C₄²	C₄⋊Dic₈₉	1	2	yes	—
193	C₄²	C₄⋊Dic₈₉	1	2	yes	—
194	C₄²	C₄⋊Dic₈₉	1	2	yes	—
195	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
196	C₄²	C₂²×Dic₈₉	1	2	yes	—
197	C₄²	C₂×D₃₅₆	1	2	yes	—
198	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
199	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
200	C₄²	C₂×Dic₁₇₈	1	2	yes	—
201	C₄²	C₂³×D₈₉	1	1	yes	characteristic
202	C₂³	—	4	1	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#15667944	1	4	yes	—
205	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
206	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	#15667944	1	4	yes	—
216	C₂×C₄	#15667944	1	8	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
219	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
223	C₂×C₄	C₂×#13083563	1	4	yes	—
224	C₂×C₄	C₂×#13083563	1	4	yes	—
225	C₂×C₄	C₂×#13083563	1	4	yes	—
226	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
228	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
229	C₈	—	2	8	no	—
230	C₈	—	356	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂×#15667944	1	4	yes	—
237	C₂²	C₂×#15667944	1	2	yes	—
238	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	C₂×#15667944	1	4	yes	—
244	C₄	C₂×#15667944	1	2	yes	—
245	C₄	#19151647	1	2	yes	—
246	C₄	#19152365	1	2	yes	—
247	C₄	C₂²×#13083563	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	4	no	—
250	C₂	#26028816	1	1	yes	characteristic
251	C₂	C₂²×#15667944	1	1	yes	characteristic
252	C₂	C₂×#19151647	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#431×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#431	D₈₉
hypocenter	C₈₉	#17813
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #26019023	C₂	semidirect
3. #26019008	C₂	semidirect
12. #19151349	C₂²	semidirect
131. C₈₉	#17813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
#26019023	2
#26019008	2
C₂×#431×C₈₉	2
#17813	89