Subgroups — #36554663

Order 22784.

778 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028816	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151554	C₂	1	1	yes	maximal, characteristic
3	#26020392	C₂	1	4	yes	maximal, complemented
4	#26020389	C₂	1	4	yes	maximal, complemented
5	#26019978	C₂	1	2	yes	maximal
6	C₂×#19151362	C₂	1	2	yes	maximal
7	C₈₉×#2905	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13083558	C₂²	1	2	yes	—
9	C₂²×#13083558	C₂²	1	1	yes	characteristic
10	C₂³×#10623494	C₂²	1	1	yes	characteristic
11	#19151625	—	2	4	no	—
12	#19151622	C₂²	1	4	yes	—
13	#19151554	C₄	1	8	yes	—
14	C₂×#15667923	C₂²	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	2	yes	—
16	#19151362	C₂²	1	4	yes	—
17	#19151362	C₂²	1	8	yes	—
18	#19151362	C₂²	1	4	yes	—
19	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
21	#424×C₈₉	—	2	4	no	—
22	#423×C₈₉	C₂²	1	4	yes	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	—	2	4	no	—
27	C₂×#13083558	—	2	4	no	—
28	C₂×#13083558	C₂×C₄	1	4	yes	—
29	C₂×#13083558	C₂³	1	4	yes	—
30	C₂²×#10623494	—	2	2	no	—
31	C₂²×#10623494	—	2	2	no	—
32	C₂²×#10623494	—	2	2	no	—
33	C₂²×#10623494	C₂×C₄	1	4	yes	—
34	C₂²×#10623494	C₂³	1	1	yes	characteristic
35	C₂²×#10623494	C₂³	1	2	yes	—
36	C₂²×#10623494	C₂³	1	1	yes	characteristic
37	C₄⋊#10623494	—	2	8	no	—
38	#15667923	C₂×C₄	1	8	yes	—
39	C₄×#10623494	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	4	no	—
43	C₂×M₁₆×C₈₉	—	2	4	no	—
44	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	#117×C₈₉	C₂×C₄	1	4	yes	—
47	#117×C₈₉	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₉	Q₈	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	C₄⋊C₈×C₈₉	—	2	8	no	—
52	#13083558	—	2	8	no	—
53	#13083558	—	2	4	no	—
54	#13083558	—	2	4	no	—
55	#13083558	—	4	4	no	—
56	C₂×#10623494	—	2	8	no	—
57	C₂×#10623494	—	2	1	no	—
58	C₂×#10623494	—	2	1	no	—
59	C₂×#10623494	—	2	1	no	—
60	C₂×#10623494	—	2	1	no	—
61	C₂×#10623494	—	2	2	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	—	4	4	no	—
65	C₂×#10623494	C₂²×C₄	1	2	yes	—
66	C₂×#10623494	C₂²×C₄	1	2	yes	—
67	C₂×#10623494	C₂²×C₄	1	4	yes	—
68	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	#41	1	2	yes	—
74	C₂²×C₄×C₈₉	#41	1	2	yes	—
75	C₂²×C₄×C₈₉	#41	1	2	yes	—
76	C₂²×C₄×C₈₉	#41	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₈₉	—	2	4	no	—
79	M₁₆×C₈₉	—	2	4	no	—
80	M₁₆×C₈₉	—	4	4	no	—
81	C₂×C₈×C₈₉	—	2	4	no	—
82	C₂×C₈×C₈₉	—	4	4	no	—
83	C₄⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
84	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
86	C₄²×C₈₉	C₂×D₄	1	4	yes	—
87	C₂³×C₈₉	—	4	2	no	—
88	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₈₉	#125	1	2	yes	—
90	C₂³×C₈₉	#125	1	2	yes	—
91	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	4	4	no	—
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₈₉	#117	1	4	yes	—
96	C₂×C₄×C₈₉	#117	1	4	yes	—
97	C₂×C₄×C₈₉	#124	1	4	yes	—
98	C₂×C₄×C₈₉	#124	1	4	yes	—
99	C₂×C₄×C₈₉	#125	1	2	yes	—
100	C₂×C₄×C₈₉	#125	1	2	yes	—
101	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₈×C₈₉	—	4	4	no	—
108	#10623494	—	2	2	no	—
109	#10623494	—	2	2	no	—
110	#10623494	—	2	4	no	—
111	#10623494	—	4	4	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	4	no	—
114	C₂²×C₈₉	—	4	2	no	—
115	C₂²×C₈₉	—	4	2	no	—
116	C₂²×C₈₉	#382	1	4	yes	—
117	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×#125	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#125	1	1	yes	characteristic
120	C₄×C₈₉	—	2	8	no	—
121	C₄×C₈₉	#382	1	4	yes	—
122	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
123	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
124	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
125	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
126	#20768	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	—	2	1	no	—
128	C₂×C₈₉	—	2	1	no	—
129	C₂×C₈₉	—	2	4	no	—
130	C₂×C₈₉	C₂×#382	1	1	yes	characteristic
131	C₂×C₈₉	#2916	1	2	yes	—
132	#2905	—	89	1	no	—
133	#2905	D₈₉	1	1	yes	characteristic
134	C₂×#405	—	89	1	no	—
135	#1838	—	89	4	no	—
136	#1837	—	89	4	no	—
137	#1738	—	89	2	no	—
138	C₂×#341	—	89	2	no	—
139	C₈₉	#20768	1	1	yes	characteristic, complemented, Sylow
140	C₂²×M₁₆	—	89	1	no	—
141	C₂²×M₁₆	—	89	2	no	—
142	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
143	C₂³×C₈	—	89	1	no	—
144	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
145	#424	—	2	4	no	—
146	#424	—	178	4	no	—
147	#423	—	89	4	no	—
148	#423	C₂×D₈₉	1	4	yes	—
149	#405	—	89	8	no	—
150	C₂×#97	—	89	2	no	—
151	C₂×C₄×C₈	—	89	2	no	—
152	#341	—	89	4	no	—
153	#341	—	89	8	no	—
154	#341	—	89	4	no	—
155	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	89	4	no	—
160	C₂×M₁₆	—	89	4	no	—
161	C₂×M₁₆	—	178	4	no	—
162	C₂×M₁₆	—	178	4	no	—
163	C₂×M₁₆	—	178	2	no	—
164	C₂×M₁₆	—	178	2	no	—
165	C₂×M₁₆	—	178	2	no	—
166	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	89	4	no	—
169	C₂²×C₈	—	89	1	no	—
170	C₂²×C₈	—	89	1	no	—
171	C₂²×C₈	—	89	2	no	—
172	C₂²×C₈	—	178	2	no	—
173	C₂²×C₈	—	178	2	no	—
174	C₂²×C₈	—	178	2	no	—
175	#117	C₄×D₈₉	1	4	yes	—
176	#117	C₂²⋊D₈₉	1	4	yes	—
177	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
178	C₂×C₂²⋊C₄	Dic₁₇₈	1	2	yes	—
179	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
180	C₄⋊C₈	—	2	8	no	—
181	C₄⋊C₈	—	178	8	no	—
182	#97	—	89	8	no	—
183	C₄×C₈	—	89	8	no	—
184	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
187	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
188	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
189	C₂²×C₄	#13083573	1	2	yes	—
190	C₂²×C₄	#13083573	1	2	yes	—
191	C₂²×C₄	#13083575	1	2	yes	—
192	C₂²×C₄	#13083575	1	2	yes	—
193	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	356	4	no	—
198	M₁₆	—	178	4	no	—
199	M₁₆	—	178	4	no	—
200	M₁₆	—	178	8	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	89	2	no	—
204	C₂×C₈	—	89	4	no	—
205	C₂×C₈	—	89	2	no	—
206	C₂×C₈	—	356	4	no	—
207	C₂×C₈	—	178	4	no	—
208	C₂×C₈	—	178	8	no	—
209	C₂×C₈	—	178	1	no	—
210	C₂×C₈	—	178	1	no	—
211	C₂×C₈	—	178	2	no	—
212	C₂×C₈	—	178	1	no	—
213	C₂×C₈	—	178	1	no	—
214	C₂×C₈	—	178	2	no	—
215	C₄⋊C₄	#13083553	1	8	yes	—
216	C₂²⋊C₄	#13083553	1	8	yes	—
217	C₄²	C₂×C₄×D₈₉	1	4	yes	—
218	C₄²	C₂×C₂²⋊D₈₉	1	4	yes	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×#13083553	1	2	yes	—
221	C₂³	#15668003	1	2	yes	—
222	C₂³	#15668003	1	2	yes	—
223	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₂×#13083553	1	8	yes	—
226	C₂×C₄	C₂×#13083553	1	2	yes	—
227	C₂×C₄	#15667970	1	4	yes	—
228	C₂×C₄	#15667970	1	4	yes	—
229	C₂×C₄	#15667999	1	4	yes	—
230	C₂×C₄	#15668001	1	4	yes	—
231	C₂×C₄	#15668003	1	2	yes	—
232	C₂×C₄	#15668003	1	2	yes	—
233	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
236	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
237	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
238	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
239	C₈	—	4	4	no	—
240	C₈	—	356	4	no	—
241	C₈	—	178	2	no	—
242	C₈	—	178	2	no	—
243	C₈	—	178	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#19151463	1	4	yes	—
249	C₂²	C₂²×#13083553	1	1	yes	characteristic
250	C₂²	C₂×#15668003	1	1	yes	characteristic
251	C₂²	C₂×#15668003	1	1	yes	characteristic
252	C₄	—	2	8	no	—
253	C₄	#19151463	1	4	yes	—
254	C₄	C₂×#15667970	1	1	yes	characteristic
255	C₄	C₂×#15667970	1	1	yes	characteristic
256	C₄	C₂×#15667999	1	1	yes	characteristic
257	C₄	C₂×#15668001	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	C₂×#19151463	1	1	yes	characteristic
262	C₂	#26028861	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#2905	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2905	D₈₉
hypocenter	C₈₉	#20768
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2905

Complements

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

N	H ≅ G/N	type
3. #26020392	C₂	semidirect
4. #26020389	C₂	semidirect
139. C₈₉	#20768	semidirect

Maximal subgroups

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

subgroup	index
#26028816	2
C₂×#19151554	2
#26020392	2
#26020389	2
#26019978	2
C₂×#19151362	2
C₈₉×#2905	2
#20768	89