Subgroups — #36523412

Order 22784.

778 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	2	yes	maximal
2	#26019004	C₂	1	8	yes	maximal, complemented
3	#26019000	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	—	2	2	no	—
6	#19151651	—	2	2	no	—
7	#19151651	C₂²	1	4	yes	—
8	#19151651	C₂²	1	4	yes	—
9	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
10	C₂×C₂²⋊#10623494	—	2	2	no	—
11	C₂×C₄×#10623494	—	2	2	no	—
12	#19151351	C₂²	1	8	yes	complemented
13	#19151349	C₂²	1	8	yes	complemented
14	C₂²×M₁₆×C₈₉	C₄	1	1	yes	characteristic
15	C₂²×M₁₆×C₈₉	C₄	1	1	yes	characteristic
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₈₉	C₄	1	4	yes	—
18	C₄×M₁₆×C₈₉	C₂²	1	4	yes	—
19	C₂×#97×C₈₉	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
21	C₂²×#10623494	—	2	4	no	—
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	C₂³	1	8	yes	—
24	C₂²⋊#10623494	—	2	4	no	—
25	C₂²⋊#10623494	—	2	4	no	—
26	C₄×#10623494	—	2	4	no	—
27	C₄×#10623494	—	2	4	no	—
28	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₈₉	—	2	4	no	—
32	C₂×M₁₆×C₈₉	C₈	1	8	yes	complemented
33	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
34	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	2	yes	—
41	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	#97×C₈₉	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₈₉	C₂³	1	4	yes	—
48	C₂×#10623494	—	2	8	no	—
49	C₂×#10623494	—	4	8	no	—
50	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	—	4	2	no	—
52	C₂²×C₄×C₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
61	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
67	M₁₆×C₈₉	—	2	8	no	—
68	M₁₆×C₈₉	C₂×C₈	1	16	yes	complemented
69	C₂×C₈×C₈₉	—	2	4	no	—
70	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	complemented
71	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
73	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
74	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
78	C₄²×C₈₉	C₂×D₄	1	2	yes	—
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
81	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₂³×C₈₉	—	4	1	no	—
84	C₂³×C₈₉	—	4	1	no	—
85	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
86	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	4	1	no	—
94	C₂×C₄×C₈₉	—	4	1	no	—
95	C₂×C₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	C₄⋊C₈	1	8	yes	—
97	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₈₉	—	2	4	no	—
112	C₈×C₈₉	C₂²×C₈	1	8	yes	—
113	#10623494	—	4	8	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	2	no	—
116	C₂²×C₈₉	—	2	2	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	4	4	no	—
119	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
120	C₂²×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
121	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	—	2	2	no	—
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
127	C₄×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
128	C₄×C₈₉	#430	1	2	yes	—
129	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
131	C₄×C₈₉	#575	1	2	yes	—
132	#17787	—	89	1	no	maximal, Sylow
133	C₂×C₈₉	—	2	2	no	—
134	C₂×C₈₉	—	2	4	no	—
135	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₈₉	#2911	1	1	yes	characteristic
137	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
138	C₂×#431	—	89	2	no	—
139	C₂×C₄×M₁₆	D₈₉	1	1	yes	characteristic
140	#1565	—	89	8	no	—
141	#1564	—	89	4	no	—
142	C₈₉	#17787	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	Dic₈₉	1	1	yes	characteristic
144	C₂²×M₁₆	Dic₈₉	1	1	yes	characteristic
145	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
146	#431	—	89	4	no	—
147	#431	—	89	4	no	—
148	#431	—	178	2	no	—
149	#431	—	178	2	no	—
150	C₂×C₄⋊C₈	—	89	4	no	—
151	C₂×C₂²⋊C₈	—	178	2	no	—
152	C₄×M₁₆	Dic₈₉	1	4	yes	—
153	C₄×M₁₆	C₂×D₈₉	1	4	yes	—
154	C₂×#97	Dic₈₉	1	2	yes	—
155	C₂×C₄×C₈	—	178	2	no	—
156	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
157	#335	—	89	8	no	—
158	#334	—	89	8	no	—
159	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
160	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	#10623494	1	8	yes	complemented
164	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
165	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
166	C₂²×C₈	—	178	4	no	—
167	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
168	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
169	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
170	C₂×C₄²	Dic₁₇₈	1	2	yes	—
171	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
172	C₂×C₄²	D₃₅₆	1	2	yes	—
173	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
175	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
176	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
177	C₄⋊C₈	—	89	8	no	—
178	C₄⋊C₈	—	178	4	no	—
179	C₂²⋊C₈	—	178	4	no	—
180	C₂²⋊C₈	—	178	4	no	—
181	#97	C₂×Dic₈₉	1	8	yes	—
182	C₄×C₈	—	178	4	no	—
183	C₄×C₈	—	178	4	no	—
184	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
185	C₄×C₈	C₂²×D₈₉	1	4	yes	—
186	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
190	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
192	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
195	C₂²×C₄	C₂×#10623494	1	4	yes	—
196	C₂²×C₄	#13083558	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
199	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
201	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
203	M₁₆	—	2	8	no	—
204	M₁₆	C₂×#10623494	1	16	yes	complemented
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	356	8	no	—
207	C₂×C₈	—	178	8	no	—
208	C₂×C₈	C₂×#10623494	1	8	yes	complemented
209	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
210	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
211	C₄²	C₄⋊Dic₈₉	1	2	yes	—
212	C₄²	C₄⋊Dic₈₉	1	2	yes	—
213	C₄²	C₄⋊Dic₈₉	1	2	yes	—
214	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
215	C₄²	C₂²×Dic₈₉	1	2	yes	—
216	C₄²	C₂×D₃₅₆	1	2	yes	—
217	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
218	C₄²	C₂×Dic₁₇₈	1	2	yes	—
219	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
220	C₄²	C₂³×D₈₉	1	1	yes	characteristic
221	C₂³	—	4	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₄⋊#10623494	1	4	yes	—
224	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
225	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₄⋊#10623494	1	8	yes	—
235	C₂×C₄	C₄⋊#10623494	1	4	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂²×#10623494	1	2	yes	—
243	C₂×C₄	C₂²×#10623494	1	4	yes	—
244	C₂×C₄	C₂×#13083558	1	2	yes	—
245	C₂×C₄	C₂×#13083558	1	4	yes	—
246	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	356	8	no	—
251	C₈	C₂²×#10623494	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂×C₄⋊#10623494	1	2	yes	—
258	C₂²	C₂×C₄⋊#10623494	1	4	yes	—
259	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	C₂×C₄⋊#10623494	1	2	yes	—
265	C₄	C₂×C₄⋊#10623494	1	4	yes	—
266	C₄	#19151647	1	2	yes	—
267	C₄	C₂³×#10623494	1	1	yes	characteristic
268	C₄	C₂²×#13083558	1	1	yes	characteristic
269	C₄	#19152398	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	C₂²×C₄⋊#10623494	1	1	yes	characteristic
273	C₂	#26028842	1	1	yes	characteristic
274	C₂	C₂×#19151647	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₄×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₈₉
hypocenter	C₈₉	#17787
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #26019004	C₂	semidirect
3. #26019000	C₂	semidirect
12. #19151351	C₂²	semidirect
13. #19151349	C₂²	semidirect
32. C₂×M₁₆×C₈₉	C₈	semidirect
68. M₁₆×C₈₉	C₂×C₈	semidirect
70. C₂×C₈×C₈₉	C₂×C₈	semidirect
142. C₈₉	#17787	semidirect
163. C₂×M₁₆	#10623494	semidirect
204. M₁₆	C₂×#10623494	semidirect
208. C₂×C₈	C₂×#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151651	2
#26019004	2
#26019000	2
C₂×C₄×M₁₆×C₈₉	2
#17787	89