Subgroups — #36701331

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021830	C₂	1	2	yes	maximal, complemented
2	#26021425	C₂	1	2	yes	maximal, complemented
3	#26021421	C₂	1	2	yes	maximal, complemented
4	#26021229	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021080	C₂	1	1	yes	maximal, characteristic
6	#26020634	C₂	1	2	yes	maximal
7	#26020493	C₂	1	2	yes	maximal, complemented
8	#26020491	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1994	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152364	C₂²	1	2	yes	—
11	#19152155	C₂²	1	1	yes	characteristic
12	#19152148	C₂²	1	1	yes	characteristic
13	#19151646	C₂²	1	2	yes	—
14	#19151622	C₂²	1	1	yes	characteristic
15	#19151622	C₂²	1	1	yes	characteristic
16	#19151616	C₂²	1	2	yes	—
17	#19151614	C₂²	1	2	yes	—
18	C₂×#15667940	C₂²	1	2	yes	—
19	C₂×#15667940	C₂²	1	2	yes	—
20	#19151571	C₂²	1	2	yes	—
21	#19151369	C₂²	1	2	yes	—
22	#19151368	C₂²	1	2	yes	—
23	#19151358	C₂²	1	2	yes	—
24	#19151358	C₂²	1	2	yes	—
25	#19151354	C₂²	1	2	yes	—
26	#533×C₈₉	C₂²	1	1	yes	characteristic
27	#423×C₈₉	C₂²	1	1	yes	characteristic
28	C₄³×C₈₉	C₂²	1	1	yes	characteristic
29	#339×C₈₉	C₂²	1	2	yes	—
30	#339×C₈₉	C₂²	1	2	yes	—
31	C₂×#13083574	C₂³	1	2	yes	—
32	#15668020	—	4	2	no	—
33	#15668018	—	4	2	no	—
34	C₂×#13083558	C₂³	1	2	yes	—
35	C₂×#13083558	C₂³	1	2	yes	—
36	M₁₆×D₈₉	—	4	2	no	—
37	#15668012	—	2	1	no	—
38	#15668012	Q₈	1	1	yes	characteristic, complemented
39	#15668009	Q₈	1	1	yes	characteristic, complemented
40	#15668004	—	2	1	no	—
41	#15668000	Q₈	1	2	yes	complemented
42	#15667991	—	4	1	no	—
43	#15667986	—	4	1	no	—
44	C₄×Dic₁₇₈	—	4	1	no	—
45	C₄×D₃₅₆	—	4	1	no	—
46	#15667971	C₂³	1	2	yes	—
47	#15667947	—	4	2	no	—
48	C₄⋊#10623494	—	4	1	no	—
49	C₄⋊#10623494	—	4	1	no	—
50	#15667940	—	2	2	no	—
51	#15667940	—	2	2	no	—
52	#15667940	—	2	2	no	—
53	#15667940	—	2	2	no	—
54	#15667940	—	4	2	no	—
55	#15667938	—	4	2	no	—
56	#15667926	—	4	2	no	—
57	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₈₉	—	2	1	no	—
59	#125×C₈₉	—	2	1	no	—
60	C₂²⋊Q₈×C₈₉	—	4	1	no	—
61	C₄×Q₈×C₈₉	—	4	1	no	—
62	#117×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₈₉	—	4	1	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	C₄⋊C₈×C₈₉	—	4	1	no	—
68	#13083574	—	2	2	no	—
69	#13083574	—	2	2	no	—
70	#13083574	—	2	2	no	—
71	#13083574	—	2	2	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂×Dic₁₇₈	C₂×Q₈	1	2	yes	—
74	C₂×D₃₅₆	—	4	1	no	—
75	C₂×D₃₅₆	C₂×Q₈	1	2	yes	—
76	C₂×C₂²⋊D₈₉	—	2	2	no	—
77	C₂×C₄×D₈₉	—	2	2	no	—
78	#13083558	—	2	2	no	—
79	#13083558	—	2	2	no	—
80	#13083558	—	2	2	no	—
81	#13083558	—	2	2	no	—
82	#13083557	—	4	4	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₈×D₈₉	—	4	4	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	1	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	#13083553	—	4	2	no	—
90	C₂²⋊Dic₈₉	—	2	2	no	—
91	#13083551	—	4	2	no	—
92	C₄×Dic₈₉	—	2	2	no	—
93	C₂×Q₈×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	—	2	1	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₈₉	—	2	2	no	—
99	M₁₆×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	4	2	no	—
103	C₄⋊C₄×C₈₉	—	4	1	no	—
104	C₂²⋊C₄×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	—	2	1	no	—
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	—	2	2	no	—
109	C₄²×C₈₉	—	4	4	no	—
110	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₄²×C₈₉	#41	1	1	yes	characteristic
114	C₄²×C₈₉	#41	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×D₈₉	—	2	2	no	—
118	Q₈×C₈₉	—	8	2	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	C₂²×Q₈	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	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²⋊D₈₉	—	4	2	no	—
136	C₂²⋊D₈₉	—	4	2	no	—
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	D₃₅₆	—	2	2	no	—
140	D₃₅₆	—	2	2	no	—
141	D₃₅₆	—	8	2	no	—
142	C₄×D₈₉	—	4	2	no	—
143	C₄×D₈₉	—	4	2	no	—
144	Dic₁₇₈	—	2	2	no	—
145	Dic₁₇₈	—	2	2	no	—
146	Dic₁₇₈	—	8	2	no	—
147	C₈×C₈₉	—	4	2	no	—
148	#10623494	—	4	2	no	—
149	#10623494	—	4	2	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	#538	1	1	yes	characteristic
152	C₂²×C₈₉	#538	1	1	yes	characteristic
153	C₂²×C₈₉	#543	1	1	yes	characteristic
154	C₂×D₈₉	—	4	2	no	—
155	C₂×D₈₉	—	8	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	8	2	no	—
163	C₄×C₈₉	#554	1	1	yes	characteristic
164	C₄×C₈₉	#554	1	1	yes	characteristic
165	C₄×C₈₉	#559	1	1	yes	characteristic
166	C₄×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
167	Dic₈₉	—	4	2	no	—
168	Dic₈₉	—	8	2	no	—
169	#32704	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	#2804	1	1	yes	characteristic
174	D₈₉	—	8	2	no	—
175	#2102	—	89	2	no	—
176	#2031	—	89	2	no	—
177	#2031	—	89	2	no	—
178	#1994	—	89	1	no	—
179	#1994	D₈₉	1	1	yes	characteristic, complemented
180	#1962	—	89	1	no	—
181	#1884	—	89	2	no	—
182	#1861	—	89	2	no	—
183	#1861	—	89	2	no	—
184	C₈₉	#32704	1	1	yes	characteristic, complemented, Sylow
185	#568	—	89	2	no	—
186	#533	—	89	1	no	—
187	#533	C₂×D₈₉	1	1	yes	characteristic
188	#532	—	89	1	no	—
189	#430	—	89	2	no	—
190	#423	—	89	1	no	—
191	#423	—	89	1	no	—
192	#423	C₂×D₈₉	1	1	yes	characteristic
193	#421	—	89	2	no	—
194	#421	—	89	2	no	—
195	C₂×#104	—	89	2	no	—
196	C₂×#104	—	89	2	no	—
197	#410	—	89	2	no	—
198	C₄³	C₂×D₈₉	1	1	yes	characteristic
199	#344	—	89	2	no	—
200	#344	—	89	2	no	—
201	#339	—	89	2	no	—
202	#339	—	89	2	no	—
203	#339	C₂×D₈₉	1	2	yes	—
204	#339	C₂×D₈₉	1	2	yes	—
205	H(3,ℤ/4)	—	89	2	no	—
206	C₂×#41	—	89	2	no	—
207	#131	—	356	2	no	—
208	#131	—	356	2	no	—
209	C₂×M₁₆	—	89	2	no	—
210	C₂×M₁₆	—	89	2	no	—
211	C₂×M₁₆	—	356	2	no	—
212	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	89	1	no	—
215	C₄⋊Q₈	—	178	1	no	—
216	#127	—	89	1	no	—
217	#125	—	2	1	no	—
218	#125	—	178	1	no	—
219	#124	—	89	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	356	1	no	—
222	#121	—	356	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	—	356	1	no	—
225	C₄×D₄	—	356	1	no	—
226	#117	—	89	2	no	—
227	#117	C₂²×D₈₉	1	2	yes	—
228	C₂×C₄²	—	4	1	no	—
229	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
232	#108	—	356	2	no	—
233	C₄⋊C₈	—	4	1	no	—
234	C₄⋊C₈	—	356	1	no	—
235	C₄⋊C₈	—	356	1	no	—
236	#104	—	356	2	no	—
237	#104	—	356	2	no	—
238	#104	—	178	2	no	—
239	#104	—	178	2	no	—
240	#104	—	178	2	no	—
241	#104	—	178	2	no	—
242	#99	—	356	2	no	—
243	#41	—	178	2	no	—
244	#41	—	178	2	no	—
245	#41	—	178	2	no	—
246	#41	—	178	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	89	2	no	—
249	C₂×Q₈	—	356	1	no	—
250	C₂×D₄	—	89	2	no	—
251	C₂×D₄	—	356	1	no	—
252	C₂×D₄	—	178	2	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	178	2	no	—
257	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	356	4	no	—
261	M₁₆	—	178	2	no	—
262	M₁₆	—	178	2	no	—
263	M₁₆	—	178	2	no	—
264	M₁₆	—	178	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	356	4	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	—	178	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	356	2	no	—
273	C₄⋊C₄	—	356	1	no	—
274	C₄⋊C₄	—	356	1	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₂²⋊C₄	—	178	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	4	no	—
284	C₄²	—	178	2	no	—
285	C₄²	Q₈×D₈₉	1	1	yes	characteristic
286	C₄²	Q₈×D₈₉	1	1	yes	characteristic
287	C₄²	#13083574	1	1	yes	characteristic
288	C₄²	#13083574	1	1	yes	characteristic
289	C₄²	#13083575	1	1	yes	characteristic
290	C₄²	#13083575	1	1	yes	characteristic
291	C₂³	—	178	2	no	—
292	C₂³	#15668057	1	1	yes	characteristic
293	Q₈	—	8	2	no	—
294	Q₈	—	178	2	no	—
295	Q₈	—	178	2	no	—
296	Q₈	—	712	2	no	—
297	D₄	—	356	2	no	—
298	D₄	—	356	2	no	—
299	D₄	—	178	2	no	—
300	D₄	—	178	2	no	—
301	D₄	—	712	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
318	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
319	C₂×C₄	#15668057	1	1	yes	characteristic
320	C₂×C₄	#15668058	1	1	yes	characteristic
321	C₂×C₄	#15668058	1	1	yes	characteristic
322	C₈	—	4	2	no	—
323	C₈	—	356	2	no	—
324	C₈	—	356	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	356	2	no	—
327	C₂²	—	712	2	no	—
328	C₂²	#19152186	1	1	yes	characteristic
329	C₂²	#19152188	1	1	yes	characteristic
330	C₂²	#19152223	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	8	2	no	—
338	C₄	—	356	2	no	—
339	C₄	—	712	2	no	—
340	C₄	#19152308	1	1	yes	characteristic
341	C₄	#19152309	1	1	yes	characteristic
342	C₄	#19152333	1	1	yes	characteristic
343	C₄	#19152340	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	712	2	no	—
348	C₂	#26028196	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19152340
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1994	C₂
socle	C₂×C₈₉	#2804
prime radical	#19152148	C₂²
hypercenter	#1994	D₈₉
hypocenter	C₈₉	#32704
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1994

Complements

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

N	H ≅ G/N	type
1. #26021830	C₂	semidirect
2. #26021425	C₂	semidirect
3. #26021421	C₂	semidirect
4. #26021229	C₂	semidirect
7. #26020493	C₂	semidirect
8. #26020491	C₂	semidirect
9. C₈₉×#1994	C₂	semidirect
38. #15668012	Q₈	semidirect
39. #15668009	Q₈	semidirect
41. #15668000	Q₈	semidirect
179. #1994	D₈₉	semidirect
184. C₈₉	#32704	semidirect

Maximal subgroups

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

subgroup	index
#26021830	2
#26021425	2
#26021421	2
#26021229	2
#26021080	2
#26020634	2
#26020493	2
#26020491	2
C₈₉×#1994	2
#32704	89