Subgroups — #117×#10623494 #36552633

Order 22784.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 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	1	yes	maximal, characteristic
2	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#10623494	C₂	1	4	yes	maximal, complemented
4	C₂²⋊C₄×#10623494	C₂	1	4	yes	maximal
5	#26019998	C₂	1	2	yes	maximal, complemented
6	C₄²×#10623494	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×#10623494	C₂²	1	2	yes	—
9	#19151651	C₄	1	8	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₂²⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₄×#10623494	C₄	1	8	yes	complemented
13	C₂×C₄×#10623494	C₂²	1	8	yes	—
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	2	yes	—
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	#19151353	C₂²	1	8	yes	—
19	C₂×#117×C₈₉	C₄	1	2	yes	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₈₉	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
25	#376×C₈₉	C₄	1	2	yes	—
26	C₄³×C₈₉	C₂²	1	2	yes	—
27	C₂²×#10623494	—	2	8	no	—
28	C₂²×#10623494	C₂×C₄	1	8	yes	—
29	C₂²×#10623494	C₂³	1	4	yes	—
30	C₂²×#10623494	C₂³	1	2	yes	—
31	C₂²×#10623494	C₂³	1	2	yes	—
32	C₄⋊#10623494	C₂×C₄	1	8	yes	—
33	C₂²⋊#10623494	C₂×C₄	1	8	yes	—
34	C₄×#10623494	—	2	16	no	—
35	C₄×#10623494	C₂×C₄	1	8	yes	—
36	C₄×#10623494	C₂×C₄	1	4	yes	—
37	C₄×#10623494	C₂×C₄	1	8	yes	—
38	C₄×#10623494	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	#117×C₈₉	C₈	1	16	yes	complemented
43	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	#95×C₈₉	C₂×C₄	1	4	yes	—
55	#95×C₈₉	C₂×C₄	1	4	yes	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	2	8	no	—
58	C₂×#10623494	—	2	16	no	—
59	C₂×#10623494	C₂²×C₄	1	4	yes	—
60	C₂×#10623494	C₂²×C₄	1	8	yes	—
61	C₂×#10623494	C₂²×C₄	1	4	yes	—
62	C₂×#10623494	#41	1	8	yes	—
63	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	—	2	4	no	—
65	C₂²×C₄×C₈₉	—	2	4	no	—
66	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
67	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
68	C₂²×C₄×C₈₉	C₂×C₈	1	8	yes	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₈₉	C₂×C₈	1	16	yes	complemented
79	C₂²⋊C₄×C₈₉	C₂×C₈	1	16	yes	—
80	C₄²×C₈₉	—	2	8	no	—
81	C₄²×C₈₉	—	2	8	no	—
82	C₄²×C₈₉	C₄²	1	4	yes	—
83	C₄²×C₈₉	C₄²	1	4	yes	—
84	C₄²×C₈₉	C₂×C₈	1	16	yes	complemented
85	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₈₉	—	2	2	no	—
90	C₂³×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
92	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
93	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	—	2	8	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	8	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	2	4	no	—
100	C₂×C₄×C₈₉	—	2	8	no	—
101	C₂×C₄×C₈₉	C₄×C₈	1	16	yes	complemented
102	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₈₉	C₂²×C₈	1	16	yes	—
108	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₈₉	#131	1	4	yes	—
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
114	#10623494	—	2	4	no	—
115	#10623494	—	2	8	no	—
116	#10623494	#117	1	8	yes	direct product factor
117	C₂²×C₈₉	—	2	2	no	—
118	C₂²×C₈₉	—	2	8	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	C₂×C₄×C₈	1	4	yes	—
121	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₈₉	C₄×#41	1	2	yes	—
123	C₂²×C₈₉	C₂³×C₈	1	2	yes	—
124	C₂²×C₈₉	C₂×#131	1	2	yes	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	16	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₈₉	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₈₉	#405	1	4	yes	—
131	C₄×C₈₉	C₂×#117	1	2	yes	—
132	C₄×C₈₉	C₄×#41	1	2	yes	—
133	C₈×#117	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	—	2	4	no	—
135	C₂×C₈₉	C₄×#117	1	1	yes	characteristic
136	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
137	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
138	C₂×C₈₉	C₈×#41	1	4	yes	—
139	C₂×#431	—	89	1	no	—
140	C₂²×C₄×C₈	—	89	1	no	—
141	C₄×#117	D₈₉	1	1	yes	characteristic
142	C₈×C₄⋊C₄	—	89	4	no	—
143	C₈×C₂²⋊C₄	—	89	4	no	—
144	#1744	—	89	2	no	—
145	C₄²×C₈	—	89	2	no	—
146	C₈₉	C₈×#117	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₈	—	89	2	no	—
148	C₂×#117	Dic₈₉	1	2	yes	—
149	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
150	#431	—	89	8	no	—
151	C₂×C₄⋊C₈	—	89	2	no	—
152	C₂×C₂²⋊C₈	—	89	2	no	—
153	C₂×C₄×C₈	—	89	8	no	—
154	C₂×C₄×C₈	—	89	2	no	—
155	C₂×C₄×C₈	—	89	1	no	—
156	C₂×C₄×C₈	—	89	1	no	—
157	C₂×C₄×C₈	—	89	2	no	—
158	C₂×C₄×C₈	—	89	8	no	—
159	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
160	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
161	C₄×C₂²⋊C₄	Dic₈₉	1	2	yes	—
162	C₄×C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
163	#376	Dic₈₉	1	2	yes	—
164	C₄³	C₂×D₈₉	1	2	yes	—
165	#336	—	89	8	no	—
166	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
169	C₂²×C₈	—	89	2	no	—
170	C₂²×C₈	—	89	8	no	—
171	C₂²×C₈	—	89	2	no	—
172	C₂²×C₈	—	89	4	no	—
173	C₂²×C₈	—	178	8	no	—
174	#117	#10623494	1	16	yes	direct product factor
175	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
176	C₂×C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
177	C₂×C₄²	C₄×D₈₉	1	4	yes	—
178	C₂×C₄²	C₄×D₈₉	1	4	yes	—
179	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
180	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
183	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
184	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	C₄⋊C₈	—	89	8	no	—
187	C₂²⋊C₈	—	89	8	no	—
188	C₄×C₈	—	89	8	no	—
189	C₄×C₈	—	89	4	no	—
190	C₄×C₈	—	89	4	no	—
191	C₄×C₈	—	89	8	no	—
192	C₄×C₈	—	178	16	no	—
193	#95	C₂×Dic₈₉	1	4	yes	—
194	#95	C₂×Dic₈₉	1	4	yes	—
195	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
199	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
200	C₂²×C₄	C₂×#10623494	1	8	yes	—
201	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
204	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
210	C₂×C₈	—	89	4	no	—
211	C₂×C₈	—	89	8	no	—
212	C₂×C₈	—	89	4	no	—
213	C₂×C₈	—	89	8	no	—
214	C₂×C₈	—	178	16	no	—
215	C₂×C₈	—	178	8	no	—
216	C₂×C₈	—	178	4	no	—
217	C₄⋊C₄	C₂×#10623494	1	16	yes	complemented
218	C₂²⋊C₄	C₂×#10623494	1	16	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	C₄×Dic₈₉	1	4	yes	—
222	C₄²	C₄×Dic₈₉	1	4	yes	—
223	C₄²	C₂×#10623494	1	16	yes	complemented
224	C₄²	C₂×C₄×D₈₉	1	2	yes	—
225	C₄²	C₂×C₄×D₈₉	1	2	yes	—
226	C₄²	C₂×C₄×D₈₉	1	2	yes	—
227	C₄²	C₂×C₄×D₈₉	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
231	C₂³	C₂²×#10623494	1	4	yes	—
232	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₄×#10623494	1	16	yes	complemented
241	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
242	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
246	C₂×C₄	C₂²×#10623494	1	16	yes	—
247	C₂×C₄	C₂²×#10623494	1	4	yes	—
248	C₂×C₄	C₂²×#10623494	1	4	yes	—
249	C₂×C₄	#15668019	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
252	C₂×C₄	#41×D₈₉	1	4	yes	—
253	C₈	—	89	8	no	—
254	C₈	—	178	4	no	—
255	C₈	—	178	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×C₄×#10623494	1	4	yes	—
260	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
261	C₂²	#41×Dic₈₉	1	2	yes	—
262	C₂²	C₂³×#10623494	1	2	yes	—
263	C₂²	C₂×#15668019	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	16	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×C₄×#10623494	1	4	yes	—
268	C₄	C₂×C₄×#10623494	1	4	yes	—
269	C₄	#19151554	1	4	yes	—
270	C₄	#117×D₈₉	1	2	yes	—
271	C₄	#19152066	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	#117×Dic₈₉	1	1	yes	characteristic
274	C₂	C₂²×C₄×#10623494	1	1	yes	characteristic
275	C₂	C₂×#19151554	1	1	yes	characteristic
276	C₂	#41×#10623494	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#117×C₈₉	C₂
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#117	D₈₉
hypocenter	C₈₉	C₈×#117
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×#10623494	C₂	semidirect
5. #26019998	C₂	semidirect
6. C₄²×#10623494	C₂	semidirect
9. #19151651	C₄	semidirect
12. C₂×C₄×#10623494	C₄	semidirect
42. #117×C₈₉	C₈	semidirect
78. C₄⋊C₄×C₈₉	C₂×C₈	semidirect
84. C₄²×C₈₉	C₂×C₈	semidirect
101. C₂×C₄×C₈₉	C₄×C₈	semidirect
116. #10623494	#117	direct (N × H)
146. C₈₉	C₈×#117	semidirect
174. #117	#10623494	direct (N × H)
217. C₄⋊C₄	C₂×#10623494	semidirect
223. C₄²	C₂×#10623494	semidirect
240. C₂×C₄	C₄×#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
C₂²×C₄×#10623494	2
C₄⋊C₄×#10623494	2
C₂²⋊C₄×#10623494	2
#26019998	2
C₄²×#10623494	2
C₄×#117×C₈₉	2
C₈×#117	89