Subgroups — #36563081

Order 22784.

1014 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030698	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151721	C₂	1	2	yes	maximal, complemented
3	C₂×#19151716	C₂	1	2	yes	maximal, complemented
4	#26028971	C₂	1	2	yes	maximal, complemented
5	#26021145	C₂	1	2	yes	maximal, complemented
6	#26021140	C₂	1	2	yes	maximal, complemented
7	#26021080	C₂	1	2	yes	maximal, complemented
8	#26020427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#97×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	C₂²	1	2	yes	—
11	C₂×#15668034	C₂²	1	2	yes	—
12	C₂×#15668018	C₂²	1	4	yes	—
13	#19152148	C₂²	1	2	yes	—
14	#19151992	C₄	1	4	yes	complemented
15	#19151987	C₄	1	4	yes	complemented
16	#19151721	—	2	8	no	—
17	#19151716	—	2	8	no	—
18	#19151671	—	2	8	no	—
19	C₂×#15667948	C₂²	1	2	yes	—
20	C₂×#15667948	C₂²	1	2	yes	—
21	#19151622	C₂²	1	2	yes	—
22	C₂×#15667940	C₂²	1	4	yes	—
23	C₂×#15667940	C₂²	1	4	yes	—
24	#19151564	C₂²	1	4	yes	—
25	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₄³×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#13083574	C₂³	1	4	yes	—
33	#15668034	—	2	4	no	—
34	#15668034	—	2	4	no	—
35	#15668034	—	2	4	no	—
36	#15668034	—	2	4	no	—
37	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
38	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
39	C₂×#13083563	C₂×C₄	1	4	yes	—
40	C₂×#13083563	C₂×C₄	1	4	yes	—
41	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
42	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
43	#15668018	—	2	8	no	—
44	#15668018	—	2	8	no	—
45	C₂×#13083558	C₂³	1	4	yes	—
46	C₂×#13083557	—	2	4	no	—
47	C₂×C₈×D₈₉	—	2	4	no	—
48	#15668012	C₂×C₄	1	2	yes	—
49	#15668009	C₂×C₄	1	2	yes	—
50	#15668000	C₂×C₄	1	4	yes	—
51	#15667986	—	2	4	no	—
52	C₄×D₃₅₆	—	2	4	no	—
53	#15667948	—	2	4	no	—
54	#15667948	—	2	4	no	—
55	C₄⋊#10623494	—	2	4	no	—
56	#15667940	—	2	8	no	—
57	#15667940	—	2	8	no	—
58	#15667924	—	2	8	no	—
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	#97×C₈₉	—	2	2	no	—
68	#97×C₈₉	C₂×C₄	1	4	yes	—
69	#97×C₈₉	D₄	1	4	yes	—
70	#97×C₈₉	D₄	1	4	yes	complemented
71	C₄×C₈×C₈₉	—	2	2	no	—
72	C₄×C₈×C₈₉	D₄	1	4	yes	complemented
73	#13083574	—	2	4	no	—
74	#13083574	—	2	4	no	—
75	#13083574	—	2	4	no	—
76	#13083574	—	2	4	no	—
77	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
78	C₂×D₃₅₆	—	2	4	no	—
79	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
80	C₂×C₂²⋊D₈₉	—	2	4	no	—
81	C₂×C₄×D₈₉	—	2	4	no	—
82	Dic₃₅₆	—	2	4	no	—
83	Dic₃₅₆	—	2	4	no	—
84	#13083563	—	2	8	no	—
85	#13083563	—	2	8	no	—
86	D₇₁₂	—	2	4	no	—
87	D₇₁₂	—	2	4	no	—
88	#13083558	—	2	4	no	—
89	#13083558	—	2	4	no	—
90	#13083557	—	4	8	no	—
91	C₂×#10623494	—	2	4	no	—
92	C₈×D₈₉	—	4	8	no	—
93	C₄⋊Dic₈₉	—	2	4	no	—
94	#13083551	—	2	8	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	#41	1	2	yes	—
98	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	#41	1	2	yes	—
110	C₂×C₈×C₈₉	#41	1	2	yes	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	—	4	4	no	—
114	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	2	yes	—
116	C₄²×C₈₉	C₂×D₄	1	2	yes	—
117	C₄²×C₈₉	C₂×D₄	1	2	yes	—
118	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
119	C₂²×D₈₉	—	2	4	no	—
120	C₂×C₄×C₈₉	—	2	4	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	4	4	no	—
133	C₂²⋊D₈₉	—	4	4	no	—
134	C₂×Dic₈₉	—	2	4	no	—
135	D₃₅₆	—	2	4	no	—
136	D₃₅₆	—	2	4	no	—
137	D₃₅₆	—	8	4	no	—
138	C₄×D₈₉	—	4	4	no	—
139	C₄×D₈₉	—	4	4	no	—
140	Dic₁₇₈	—	2	4	no	—
141	Dic₁₇₈	—	2	4	no	—
142	C₈×C₈₉	—	2	2	no	—
143	C₈×C₈₉	C₄×D₄	1	4	yes	—
144	C₈×C₈₉	C₄×D₄	1	4	yes	—
145	C₈×C₈₉	#127	1	4	yes	—
146	#10623494	—	4	4	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₈₉	#532	1	1	yes	characteristic
150	C₂²×C₈₉	#532	1	1	yes	characteristic
151	C₂×D₈₉	—	4	4	no	—
152	C₂×D₈₉	—	4	8	no	—
153	C₄×C₈₉	—	2	4	no	—
154	C₄×C₈₉	—	2	4	no	—
155	C₄×C₈₉	—	4	2	no	—
156	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
159	C₄×C₈₉	#532	1	1	yes	characteristic
160	Dic₈₉	—	4	4	no	—
161	#21710	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
165	C₂×C₈₉	#2956	1	2	yes	—
166	D₈₉	—	8	4	no	—
167	#3150	—	89	1	no	—
168	C₂×#444	—	89	2	no	—
169	C₂×#443	—	89	2	no	—
170	#2932	—	89	2	no	—
171	#1975	—	89	2	no	—
172	#1974	—	89	2	no	—
173	#1962	—	89	2	no	—
174	#1849	—	89	1	no	—
175	C₄×#97	D₈₉	1	1	yes	characteristic, complemented
176	C₈₉	#21710	1	1	yes	characteristic, complemented, Sylow
177	C₂×#135	—	89	2	no	—
178	C₂×#135	—	89	2	no	—
179	C₂×#131	—	89	4	no	—
180	#532	—	89	2	no	—
181	#497	—	89	4	no	—
182	#496	—	89	4	no	—
183	#444	—	178	8	no	—
184	#443	—	178	8	no	—
185	#435	—	178	8	no	—
186	C₂×#108	—	89	2	no	—
187	C₂×#108	—	89	2	no	—
188	#423	—	89	2	no	—
189	C₂×#104	—	89	4	no	—
190	C₂×#104	—	89	4	no	—
191	#408	—	89	4	no	—
192	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
193	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
194	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
195	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	C₄³	C₂×D₈₉	1	1	yes	characteristic
199	C₂×#41	—	89	4	no	—
200	#135	—	178	4	no	—
201	#135	—	178	4	no	—
202	#135	—	178	4	no	—
203	#135	—	178	4	no	—
204	C₂×Q₁₆	—	89	2	no	—
205	C₂×Q₁₆	—	89	2	no	—
206	C₂×SD₁₆	—	89	4	no	—
207	C₂×SD₁₆	—	89	4	no	—
208	C₂×D₈	—	89	2	no	—
209	C₂×D₈	—	89	2	no	—
210	#131	—	178	8	no	—
211	#131	—	178	8	no	—
212	C₂×M₁₆	—	89	4	no	—
213	C₂×M₁₆	—	178	4	no	—
214	C₂²×C₈	—	178	4	no	—
215	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
219	C₄⋊Q₈	—	89	2	no	—
220	#127	—	89	2	no	—
221	#124	—	89	4	no	—
222	#121	—	178	4	no	—
223	C₄×D₄	—	178	4	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
228	#108	—	178	4	no	—
229	#108	—	178	4	no	—
230	C₄⋊C₈	—	178	4	no	—
231	#104	—	178	8	no	—
232	#104	—	178	8	no	—
233	C₂²⋊C₈	—	178	8	no	—
234	#97	—	2	2	no	—
235	#97	C₄×D₈₉	1	4	yes	—
236	#97	D₃₅₆	1	4	yes	—
237	#97	D₃₅₆	1	4	yes	complemented
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	D₃₅₆	1	4	yes	complemented
240	#41	—	178	4	no	—
241	#41	—	178	4	no	—
242	#41	—	178	4	no	—
243	#41	—	178	4	no	—
244	C₂×Q₈	—	89	4	no	—
245	C₂×D₄	—	89	4	no	—
246	C₂×D₄	—	178	4	no	—
247	C₂×D₄	—	178	4	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	178	4	no	—
251	C₂²×C₄	#13083574	1	2	yes	—
252	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
253	Q₁₆	—	178	4	no	—
254	Q₁₆	—	178	4	no	—
255	SD₁₆	—	178	8	no	—
256	SD₁₆	—	178	8	no	—
257	D₈	—	178	4	no	—
258	D₈	—	178	4	no	—
259	M₁₆	—	356	8	no	—
260	M₁₆	—	178	4	no	—
261	M₁₆	—	178	4	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	356	8	no	—
265	C₂×C₈	—	178	4	no	—
266	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
267	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
268	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
269	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
270	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
271	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
272	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
273	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
274	C₂×C₈	#13083574	1	2	yes	—
275	C₂×C₈	#13083574	1	2	yes	—
276	C₄⋊C₄	—	178	4	no	—
277	C₂²⋊C₄	—	178	8	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	C₂×C₄×D₈₉	1	2	yes	—
282	C₄²	C₂×D₃₅₆	1	2	yes	—
283	C₄²	C₂×D₃₅₆	1	2	yes	—
284	C₄²	C₂×D₃₅₆	1	2	yes	—
285	C₂³	—	178	4	no	—
286	C₂³	C₂×#13083574	1	1	yes	characteristic
287	Q₈	—	178	4	no	—
288	Q₈	—	178	4	no	—
289	D₄	—	356	4	no	—
290	D₄	—	356	4	no	—
291	D₄	—	178	4	no	—
292	D₄	—	178	4	no	—
293	D₄	—	712	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	356	4	no	—
300	C₂×C₄	—	356	4	no	—
301	C₂×C₄	—	178	4	no	—
302	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
303	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
307	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
308	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	356	4	no	—
311	C₈	C₄×D₃₅₆	1	4	yes	—
312	C₈	C₄×D₃₅₆	1	4	yes	—
313	C₈	#15668009	1	4	yes	—
314	C₂²	—	2	2	no	—
315	C₂²	—	356	8	no	—
316	C₂²	—	356	4	no	—
317	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
318	C₂²	#19152148	1	1	yes	characteristic
319	C₂²	#19152148	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	356	4	no	—
324	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
325	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
326	C₄	C₂×#15668009	1	1	yes	characteristic
327	C₄	#19152148	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	712	4	no	—
331	C₂	C₄×#15668009	1	1	yes	characteristic
332	C₂	#26029163	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#97×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#26030698	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#97	D₈₉
hypocenter	C₈₉	#21710
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#19151721	C₂	semidirect
3. C₂×#19151716	C₂	semidirect
4. #26028971	C₂	semidirect
5. #26021145	C₂	semidirect
6. #26021140	C₂	semidirect
7. #26021080	C₂	semidirect
8. #26020427	C₂	semidirect
9. C₄×#97×C₈₉	C₂	semidirect
14. #19151992	C₄	semidirect
15. #19151987	C₄	semidirect
70. #97×C₈₉	D₄	semidirect
72. C₄×C₈×C₈₉	D₄	semidirect
175. C₄×#97	D₈₉	semidirect
176. C₈₉	#21710	semidirect
237. #97	D₃₅₆	semidirect
239. C₄×C₈	D₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26030698	2
C₂×#19151721	2
C₂×#19151716	2
#26028971	2
#26021145	2
#26021140	2
#26021080	2
#26020427	2
C₄×#97×C₈₉	2
#21710	89