Subgroups — #36523917

Order 22784.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029361	C₂	1	2	yes	maximal, complemented
2	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
3	#26019075	C₂	1	2	yes	maximal, complemented
4	#26019070	C₂	1	2	yes	maximal, complemented
5	#26019051	C₂	1	4	yes	maximal, complemented
6	#26018637	C₂	1	2	yes	maximal, complemented
7	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₄	1	1	yes	characteristic
10	C₄×#13083574	C₂²	1	2	yes	—
11	#19151722	C₂²	1	4	yes	—
12	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
13	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
14	C₄×#13083563	C₄	1	2	yes	—
15	C₄×#13083563	C₂²	1	2	yes	—
16	C₄×D₇₁₂	C₄	1	1	yes	characteristic
17	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
18	#19151663	C₂²	1	4	yes	—
19	C₈×C₂²⋊D₈₉	—	2	4	no	—
20	#19151651	C₂²	1	1	yes	characteristic
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151650	—	2	2	no	—
23	#19151636	C₄	1	1	yes	characteristic
24	C₄⋊C₈×D₈₉	—	2	2	no	—
25	#19151595	C₄	1	2	yes	—
26	#19151328	C₂²	1	4	yes	—
27	#19151325	C₂²	1	4	yes	—
28	#19151319	C₂²	1	2	yes	—
29	C₈×#10623494	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#334×C₈₉	C₂²	1	2	yes	—
33	#334×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	C₂×C₄	1	2	yes	—
35	#15668034	C₈	1	8	yes	complemented
36	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083563	C₂×C₄	1	2	yes	—
38	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₈₉	—	2	4	no	—
40	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₇₈	C₂³	1	2	yes	—
42	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
43	C₄×D₃₅₆	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₈₉	—	2	2	no	—
45	C₄×C₂²⋊D₈₉	—	2	2	no	—
46	#15667971	C₂×C₄	1	2	yes	—
47	#15667969	—	2	2	no	—
48	C₄²×D₈₉	—	2	2	no	—
49	#15667946	C₂×C₄	1	1	yes	characteristic
50	#15667946	C₂×C₄	1	1	yes	characteristic
51	#15667944	C₂×C₄	1	2	yes	—
52	C₄⋊#10623494	—	2	1	no	—
53	C₄⋊#10623494	D₄	1	2	yes	—
54	#15667941	—	2	4	no	—
55	#15667937	C₂×C₄	1	2	yes	—
56	#15667937	C₂×C₄	1	2	yes	—
57	#15667934	C₂×C₄	1	2	yes	—
58	#15667934	C₂×C₄	1	2	yes	—
59	C₂²⋊#10623494	—	2	1	no	—
60	C₂²⋊#10623494	D₄	1	2	yes	—
61	#15667924	—	2	4	no	—
62	C₄×#10623494	C₂³	1	2	yes	—
63	C₄×#10623494	C₂³	1	2	yes	—
64	C₈×Dic₈₉	—	2	4	no	—
65	C₂²×C₈×C₈₉	—	2	2	no	—
66	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	—	2	2	no	—
70	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#13083574	—	2	4	no	—
76	#13083574	C₂×C₈	1	8	yes	complemented
77	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
78	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
79	C₂×C₂²⋊D₈₉	—	2	2	no	—
80	C₂×C₄×D₈₉	—	2	2	no	—
81	C₂×C₄×D₈₉	—	2	2	no	—
82	C₂×C₄×D₈₉	—	2	2	no	—
83	Dic₃₅₆	C₂×C₈	1	4	yes	complemented
84	#13083563	C₂×C₈	1	8	yes	complemented
85	D₇₁₂	C₂×C₈	1	4	yes	complemented
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	C₂×D₄	1	4	yes	—
88	C₈×D₈₉	—	8	4	no	—
89	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
91	#13083553	—	2	2	no	—
92	#13083553	—	2	2	no	—
93	C₂²⋊Dic₈₉	—	2	2	no	—
94	#13083551	—	2	2	no	—
95	#13083551	—	2	2	no	—
96	C₄×Dic₈₉	—	2	2	no	—
97	C₄×Dic₈₉	—	2	2	no	—
98	C₄×Dic₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	4	no	—
104	C₂×C₈×C₈₉	—	8	2	no	—
105	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	complemented
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₈₉	#131	1	2	yes	—
123	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
126	C₂²⋊D₈₉	—	2	8	no	—
127	C₂×Dic₈₉	—	2	2	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	D₃₅₆	—	2	2	no	—
131	D₃₅₆	C₂²×C₈	1	4	yes	—
132	C₄×D₈₉	—	2	8	no	—
133	C₄×D₈₉	—	4	2	no	—
134	C₄×D₈₉	—	4	2	no	—
135	C₄×D₈₉	—	8	2	no	—
136	Dic₁₇₈	—	2	2	no	—
137	Dic₁₇₈	C₂²×C₈	1	4	yes	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	4	4	no	—
141	C₈×C₈₉	C₂²×C₈	1	2	yes	—
142	C₈×C₈₉	C₂²×C₈	1	2	yes	—
143	#10623494	—	2	4	no	—
144	#10623494	—	4	2	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	C₈×D₄	1	2	yes	—
148	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×D₈₉	—	2	4	no	—
150	C₂×D₈₉	—	4	2	no	—
151	C₂×D₈₉	—	4	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	C₈×D₄	1	2	yes	—
156	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
157	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
158	C₄×C₈₉	#577	1	2	yes	—
159	Dic₈₉	—	2	4	no	—
160	Dic₈₉	—	4	2	no	—
161	Dic₈₉	—	4	2	no	—
162	#17858	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	2	2	no	—
164	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
165	C₂×C₈₉	#2952	1	1	yes	characteristic
166	C₂×C₈₉	#2957	1	1	yes	characteristic
167	D₈₉	—	4	4	no	—
168	#2988	—	89	2	no	—
169	C₄×#135	—	89	1	no	—
170	#1580	—	89	2	no	—
171	#1579	—	89	2	no	—
172	C₈×SD₁₆	—	89	4	no	—
173	C₂×#334	D₈₉	1	1	yes	characteristic, complemented
174	#1477	—	89	2	no	—
175	#1450	—	89	1	no	—
176	C₈₉	#17858	1	1	yes	characteristic, complemented, Sylow
177	C₂×#135	—	89	1	no	—
178	C₄×#41	—	89	2	no	—
179	C₈×Q₈	—	89	4	no	—
180	C₄×Q₁₆	—	89	1	no	—
181	C₄×Q₁₆	—	89	1	no	—
182	C₄×SD₁₆	—	89	2	no	—
183	C₄×SD₁₆	—	89	2	no	—
184	C₄×D₈	—	89	1	no	—
185	C₄×D₈	—	89	1	no	—
186	C₈×D₄	—	89	4	no	—
187	C₈×D₄	—	178	4	no	—
188	#431	—	89	1	no	—
189	#431	—	89	1	no	—
190	#431	—	178	2	no	—
191	#427	—	89	1	no	—
192	C₂×C₄⋊C₈	—	178	2	no	—
193	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
194	#416	—	89	2	no	—
195	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
196	#334	C₂×D₈₉	1	2	yes	—
197	#334	C₂×D₈₉	1	2	yes	—
198	Q₈⋊C₈	—	89	4	no	—
199	D₄⋊C₈	—	89	4	no	—
200	#322	—	89	2	no	—
201	C₈²	—	89	2	no	—
202	C₂×#41	—	89	2	no	—
203	#135	—	89	8	no	—
204	C₂×Q₁₆	—	89	1	no	—
205	C₂×SD₁₆	—	89	2	no	—
206	C₂×D₈	—	89	1	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	178	4	no	—
209	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
211	C₄×Q₈	—	89	2	no	—
212	C₄×Q₈	—	89	2	no	—
213	C₄×D₄	—	89	2	no	—
214	C₄×D₄	—	89	2	no	—
215	C₄×D₄	—	178	2	no	—
216	C₄×D₄	—	178	2	no	—
217	#117	—	89	2	no	—
218	#117	—	178	2	no	—
219	C₂×C₄²	—	178	2	no	—
220	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
221	#107	—	89	1	no	—
222	#107	—	89	1	no	—
223	#106	—	89	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	89	2	no	—
226	C₄⋊C₈	—	178	4	no	—
227	C₄⋊C₈	—	178	1	no	—
228	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
229	#103	—	89	2	no	—
230	#103	—	89	2	no	—
231	#102	—	89	2	no	—
232	#102	—	89	2	no	—
233	C₂²⋊C₈	—	89	2	no	—
234	C₂²⋊C₈	—	178	4	no	—
235	C₂²⋊C₈	—	178	1	no	—
236	C₄×C₈	—	89	2	no	—
237	C₄×C₈	—	89	2	no	—
238	C₄×C₈	—	178	4	no	—
239	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
240	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	#41	—	89	8	no	—
244	#41	—	178	4	no	—
245	C₂×Q₈	—	89	2	no	—
246	C₂×D₄	—	89	2	no	—
247	C₂×D₄	—	178	2	no	—
248	C₂²×C₄	—	178	2	no	—
249	C₂²×C₄	—	178	2	no	—
250	C₂²×C₄	—	178	2	no	—
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	#13083575	1	1	yes	characteristic
254	Q₁₆	—	89	4	no	—
255	SD₁₆	—	89	8	no	—
256	D₈	—	89	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	8	2	no	—
260	C₂×C₈	—	89	4	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	712	4	no	—
263	C₂×C₈	C₈×D₈₉	1	4	yes	complemented
264	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	89	2	no	—
269	C₄⋊C₄	—	89	2	no	—
270	C₄⋊C₄	—	178	2	no	—
271	C₄⋊C₄	—	178	2	no	—
272	C₂²⋊C₄	—	178	2	no	—
273	C₂²⋊C₄	—	178	2	no	—
274	C₂²⋊C₄	—	178	2	no	—
275	C₄²	—	178	2	no	—
276	C₄²	—	178	2	no	—
277	C₄²	—	178	2	no	—
278	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₄²	D₄×D₈₉	1	1	yes	characteristic
280	C₄²	#13083575	1	1	yes	characteristic
281	C₄²	C₂³×D₈₉	1	1	yes	characteristic
282	C₂³	—	178	2	no	—
283	C₂³	#15667974	1	1	yes	characteristic
284	Q₈	—	89	4	no	—
285	Q₈	—	178	2	no	—
286	D₄	—	89	4	no	—
287	D₄	—	178	8	no	—
288	D₄	—	178	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	356	2	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	178	2	no	—
294	C₂×C₄	—	178	2	no	—
295	C₂×C₄	—	178	2	no	—
296	C₂×C₄	—	178	8	no	—
297	C₂×C₄	—	712	2	no	—
298	C₂×C₄	#15667974	1	1	yes	characteristic
299	C₂×C₄	#15667974	1	1	yes	characteristic
300	C₂×C₄	#15667974	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
302	C₂×C₄	#15668020	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	4	no	—
309	C₈	—	356	2	no	—
310	C₈	—	178	4	no	—
311	C₈	C₂×C₈×D₈₉	1	2	yes	—
312	C₈	C₂×C₈×D₈₉	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	356	2	no	—
316	C₂²	—	356	2	no	—
317	C₂²	—	178	4	no	—
318	C₂²	#19151663	1	2	yes	—
319	C₂²	C₂×#15667974	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	356	2	no	—
324	C₄	—	356	2	no	—
325	C₄	—	178	4	no	—
326	C₄	#19151663	1	2	yes	—
327	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
328	C₄	C₂×#15668020	1	1	yes	characteristic
329	C₄	#19152404	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	356	4	no	—
332	C₂	C₂×#19151663	1	1	yes	characteristic
333	C₂	#26029135	1	1	yes	characteristic
334	C₂	#26029168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#17858
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
1. #26029361	C₂	semidirect
3. #26019075	C₂	semidirect
4. #26019070	C₂	semidirect
5. #26019051	C₂	semidirect
6. #26018637	C₂	semidirect
7. #26018535	C₂	semidirect
8. C₂×#334×C₈₉	C₂	semidirect
35. #15668034	C₈	semidirect
76. #13083574	C₂×C₈	semidirect
83. Dic₃₅₆	C₂×C₈	semidirect
84. #13083563	C₂×C₈	semidirect
85. D₇₁₂	C₂×C₈	semidirect
105. C₂×C₈×C₈₉	C₂×C₈	semidirect
173. C₂×#334	D₈₉	semidirect
176. C₈₉	#17858	semidirect
263. C₂×C₈	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029361	2
C₄×#15668034	2
#26019075	2
#26019070	2
#26019051	2
#26018637	2
#26018535	2
C₂×#334×C₈₉	2
#17858	89