Subgroups — #36572858

Order 22784.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151692	C₂	1	2	yes	maximal
3	#26023164	C₂	1	2	yes	maximal
4	#26020983	C₂	1	2	yes	maximal
5	#26020503	C₂	1	4	yes	maximal
6	#26020254	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
11	C₂×#15667977	C₂²	1	4	yes	—
12	C₄×Dic₃₅₆	C₄	1	8	yes	complemented
13	#19151692	—	2	8	no	—
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×#15667937	C₂²	1	2	yes	—
17	C₂×#15667935	C₂²	1	4	yes	—
18	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
19	#19151494	C₂²	1	4	yes	—
20	#19151493	—	2	4	no	—
21	#19151467	—	2	2	no	—
22	C₄⋊C₄×Dic₈₉	—	2	2	no	—
23	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
24	#19151360	C₂²	1	4	yes	—
25	#19151352	C₂²	1	2	yes	—
26	C₂×#107×C₈₉	C₂²	1	2	yes	—
27	C₂×#107×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#384×C₈₉	C₂²	1	2	yes	—
30	C₂²×Dic₁₇₈	C₂³	1	2	yes	—
31	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
32	C₂×Dic₃₅₆	D₄	1	4	yes	—
33	C₂×Dic₃₅₆	Q₈	1	4	yes	—
34	C₂²×#10623494	C₂³	1	2	yes	—
35	C₄×Dic₁₇₈	—	2	4	no	—
36	C₄×Dic₁₇₈	C₂×C₄	1	8	yes	—
37	#15667977	—	2	8	no	—
38	#15667977	—	4	4	no	—
39	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
42	C₂×#13083553	—	2	4	no	—
43	C₂×#13083553	—	2	4	no	—
44	C₂×#13083553	—	2	2	no	—
45	C₂×C₄×Dic₈₉	—	2	4	no	—
46	C₂×C₄×Dic₈₉	—	2	2	no	—
47	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
48	#15667946	C₂×C₄	1	4	yes	—
49	C₄⋊#10623494	—	2	2	no	—
50	#15667937	C₂×C₄	1	8	yes	—
51	#15667935	—	2	8	no	—
52	C₈×Dic₈₉	—	2	4	no	—
53	#15667919	—	2	4	no	—
54	#15667919	D₄	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	#107×C₈₉	—	2	4	no	—
62	#107×C₈₉	—	4	2	no	—
63	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
64	C₂×Dic₁₇₈	—	2	2	no	—
65	C₂×Dic₁₇₈	—	2	2	no	—
66	C₂×Dic₁₇₈	—	2	2	no	—
67	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₇₈	C₂×D₄	1	4	yes	—
69	C₂×Dic₁₇₈	C₂×Q₈	1	4	yes	—
70	C₂²×Dic₈₉	—	2	2	no	—
71	C₂²×Dic₈₉	—	2	2	no	—
72	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
73	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
74	Dic₃₅₆	C₄⋊C₄	1	16	yes	complemented
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	2	no	—
80	C₄⋊Dic₈₉	Q₁₆	1	4	yes	complemented
81	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
82	#13083553	—	2	8	no	—
83	#13083553	—	4	8	no	—
84	#13083553	—	8	4	no	—
85	C₄×Dic₈₉	—	2	8	no	—
86	C₄×Dic₈₉	—	2	2	no	—
87	C₄×Dic₈₉	—	2	2	no	—
88	C₄×Dic₈₉	—	4	8	no	—
89	C₂²×C₄×C₈₉	—	2	4	no	—
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
99	C₄⋊C₄×C₈₉	—	2	4	no	—
100	C₄⋊C₄×C₈₉	—	2	4	no	—
101	C₄⋊C₄×C₈₉	—	4	4	no	—
102	C₄⋊C₄×C₈₉	—	8	2	no	—
103	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
104	C₄²×C₈₉	#41	1	2	yes	—
105	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	4	4	no	—
109	C₂×C₄×C₈₉	—	4	4	no	—
110	C₂×C₄×C₈₉	—	4	4	no	—
111	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₈₉	#135	1	2	yes	—
115	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×Dic₈₉	—	2	4	no	—
122	C₂×Dic₈₉	—	2	2	no	—
123	C₂×Dic₈₉	—	2	4	no	—
124	C₂×Dic₈₉	—	2	4	no	—
125	C₂×Dic₈₉	—	2	2	no	—
126	C₂×Dic₈₉	—	2	4	no	—
127	C₂×Dic₈₉	—	2	2	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	C₂×Dic₈₉	C₂×Q₁₆	1	4	yes	—
133	Dic₁₇₈	—	2	4	no	—
134	Dic₁₇₈	—	2	4	no	—
135	Dic₁₇₈	—	2	4	no	—
136	Dic₁₇₈	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₈₉	—	2	2	no	—
138	C₈×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
139	#10623494	—	4	2	no	—
140	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₈₉	D₄²	1	1	yes	characteristic
142	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
143	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
145	C₂²×C₈₉	#576	1	1	yes	characteristic
146	C₂²×C₈₉	#578	1	1	yes	characteristic
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	8	4	no	—
149	C₄×C₈₉	C₄×Q₁₆	1	4	yes	complemented
150	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈₉	#520	1	1	yes	characteristic
152	C₄×C₈₉	#547	1	1	yes	characteristic
153	C₄×C₈₉	#554	1	1	yes	characteristic
154	Dic₈₉	—	2	8	no	—
155	Dic₈₉	—	2	4	no	—
156	Dic₈₉	—	4	4	no	—
157	Dic₈₉	—	4	2	no	—
158	C₄⋊C₄×Q₁₆	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
161	C₂×C₈₉	#2951	1	1	yes	characteristic
162	C₂×C₈₉	D₄×Q₁₆	1	1	yes	characteristic
163	C₂×C₈₉	#3287	1	1	yes	characteristic
164	C₂×C₈₉	Q₈×Q₁₆	1	1	yes	characteristic
165	C₂×C₈₉	#3383	1	1	yes	characteristic
166	C₂×C₄×Q₁₆	—	89	2	no	—
167	C₂×C₄×Q₁₆	—	89	1	no	—
168	Q₈×C₄⋊C₄	—	89	2	no	—
169	#1942	—	89	2	no	—
170	#1863	—	89	4	no	—
171	#1845	D₈₉	1	1	yes	characteristic
172	#1803	—	89	2	no	—
173	C₈×C₄⋊C₄	—	89	1	no	—
174	C₈₉	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	89	1	no	—
176	C₂×C₄×Q₈	—	89	4	no	—
177	C₂×C₄×Q₈	—	89	2	no	—
178	C₄×Q₁₆	—	89	8	no	—
179	C₄×Q₁₆	—	178	8	no	—
180	C₂×#107	—	89	1	no	—
181	C₂×#107	C₂×D₈₉	1	2	yes	—
182	C₂×#107	C₂×D₈₉	1	2	yes	—
183	C₂×C₄⋊C₈	—	89	1	no	—
184	C₂×#103	—	89	2	no	—
185	C₂×#103	—	89	4	no	—
186	C₂×C₄×C₈	—	89	2	no	—
187	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
188	#389	—	89	4	no	—
189	#389	—	178	4	no	—
190	#384	—	178	2	no	—
191	#384	C₂×D₈₉	1	2	yes	—
192	C₄×C₄⋊C₄	—	89	1	no	—
193	C₄×C₄⋊C₄	—	178	2	no	—
194	#340	—	89	4	no	—
195	#336	—	89	2	no	—
196	C₂²×Q₈	—	89	2	no	—
197	C₂×Q₁₆	—	89	4	no	—
198	C₂×Q₁₆	—	89	4	no	—
199	C₂×Q₁₆	—	89	4	no	—
200	C₂²×C₈	—	89	2	no	—
201	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
203	C₄×Q₈	—	89	8	no	—
204	C₄×Q₈	—	356	4	no	—
205	C₄×Q₈	—	178	8	no	—
206	C₄×Q₈	—	178	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	89	1	no	—
210	C₂×C₄⋊C₄	—	89	2	no	—
211	C₂×C₄⋊C₄	—	89	1	no	—
212	C₂×C₄⋊C₄	—	178	4	no	—
213	C₂×C₄⋊C₄	—	178	2	no	—
214	C₂×C₄⋊C₄	—	178	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
216	C₂×C₄²	—	89	2	no	—
217	C₂×C₄²	—	178	4	no	—
218	C₂×C₄²	—	178	2	no	—
219	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	#107	—	4	2	no	—
222	#107	—	89	4	no	—
223	C₄⋊C₈	—	178	2	no	—
224	#103	—	89	8	no	—
225	#103	—	178	8	no	—
226	C₄×C₈	—	178	4	no	—
227	C₄×C₈	C₄×D₈₉	1	4	yes	—
228	#95	—	89	2	no	—
229	#95	—	178	4	no	—
230	C₂×Q₈	—	89	4	no	—
231	C₂×Q₈	—	89	4	no	—
232	C₂×Q₈	—	89	4	no	—
233	C₂×Q₈	—	178	2	no	—
234	C₂×Q₈	—	178	2	no	—
235	C₂×Q₈	—	178	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	89	2	no	—
238	C₂²×C₄	—	89	2	no	—
239	C₂²×C₄	—	178	2	no	—
240	C₂²×C₄	—	178	2	no	—
241	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
244	Q₁₆	—	89	16	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	178	2	no	—
248	C₂×C₈	—	178	2	no	—
249	C₂×C₈	—	178	2	no	—
250	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
252	C₂×C₈	D₄×D₈₉	1	2	yes	—
253	C₂×C₈	Q₈×D₈₉	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	89	4	no	—
259	C₄⋊C₄	—	89	4	no	—
260	C₄⋊C₄	—	356	8	no	—
261	C₄⋊C₄	—	178	8	no	—
262	C₄⋊C₄	—	178	2	no	—
263	C₄⋊C₄	—	178	2	no	—
264	C₄⋊C₄	—	712	4	no	—
265	C₄²	—	356	8	no	—
266	C₄²	—	178	8	no	—
267	C₄²	—	178	2	no	—
268	C₄²	—	178	2	no	—
269	C₄²	C₂×C₄×D₈₉	1	2	yes	—
270	C₄²	#13083575	1	2	yes	—
271	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
272	Q₈	—	89	8	no	—
273	Q₈	—	178	4	no	—
274	Q₈	—	178	4	no	—
275	Q₈	—	178	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	89	4	no	—
282	C₂×C₄	—	356	2	no	—
283	C₂×C₄	—	356	2	no	—
284	C₂×C₄	—	178	2	no	—
285	C₂×C₄	—	178	2	no	—
286	C₂×C₄	—	178	4	no	—
287	C₂×C₄	—	178	2	no	—
288	C₂×C₄	—	178	2	no	—
289	C₂×C₄	—	178	2	no	—
290	C₂×C₄	—	178	4	no	—
291	C₂×C₄	—	178	4	no	—
292	C₂×C₄	—	178	4	no	—
293	C₂×C₄	#15667974	1	2	yes	—
294	C₂×C₄	#15667974	1	2	yes	—
295	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
296	C₂×C₄	#15668031	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
301	C₂×C₄	#15668057	1	1	yes	characteristic
302	C₂×C₄	#15668060	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	356	2	no	—
305	C₈	C₄⋊C₄×D₈₉	1	4	yes	—
306	C₂²	C₂×#15667974	1	1	yes	characteristic
307	C₂²	#19152233	1	1	yes	characteristic
308	C₂²	#19152263	1	1	yes	characteristic
309	C₂²	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
310	C₂²	C₂×#15668031	1	1	yes	characteristic
311	C₂²	#19152399	1	1	yes	characteristic
312	C₂²	#19152411	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	8	4	no	—
315	C₄	—	356	4	no	—
316	C₄	—	356	2	no	—
317	C₄	—	178	4	no	—
318	C₄	—	178	8	no	—
319	C₄	#19151692	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
321	C₄	#19152079	1	1	yes	characteristic
322	C₄	#19152251	1	1	yes	characteristic
323	C₄	#19152309	1	1	yes	characteristic
324	C₂	#26023152	1	1	yes	characteristic
325	C₂	C₂×#19151692	1	1	yes	characteristic
326	C₂	#26029126	1	1	yes	characteristic
327	C₂	#26032002	1	1	yes	characteristic
328	C₂	#26032010	1	1	yes	characteristic
329	C₂	#26033135	1	1	yes	characteristic
330	C₂	#26033145	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1845	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1845	D₈₉
hypocenter	C₈₉	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₃₅₆	C₄	semidirect
74. Dic₃₅₆	C₄⋊C₄	semidirect
80. C₄⋊Dic₈₉	Q₁₆	semidirect
149. C₄×C₈₉	C₄×Q₁₆	semidirect
174. C₈₉	C₄⋊C₄×Q₁₆	semidirect
319. C₄	#19151692	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₅₆	2
C₂×#19151692	2
#26023164	2
#26020983	2
#26020503	2
#26020254	2
C₈×C₄⋊Dic₈₉	2
C₈₉×#1845	2
C₄⋊C₄×Q₁₆	89