Subgroups — #36708113

Order 22784.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	2	yes	maximal, complemented
2	#26022078	C₂	1	4	yes	maximal, complemented
3	D₄×#13083549	C₂	1	2	yes	maximal, complemented
4	#26022055	C₂	1	2	yes	maximal, complemented
5	#26022051	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊#13083549	C₂	1	2	yes	maximal, complemented
8	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	C₂²×#13083549	C₂²	1	4	yes	—
11	C₄⋊#13083549	C₂²	1	4	yes	—
12	C₄⋊#13083549	C₂²	1	2	yes	—
13	C₄⋊#13083549	C₂²	1	1	yes	characteristic
14	C₄⋊#13083549	C₂²	1	2	yes	—
15	C₄⋊#13083549	C₂²	1	1	yes	characteristic
16	C₂²⋊#13083549	C₂²	1	2	yes	—
17	C₂²⋊#13083549	C₂²	1	4	yes	—
18	#19151374	C₂²	1	4	yes	—
19	C₄×#13083549	C₂²	1	2	yes	—
20	C₄×#41×C₈₉	C₄	1	1	yes	characteristic
21	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
22	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₈₉	C₄	1	2	yes	—
24	C₈×D₄×C₈₉	C₄	1	1	yes	characteristic
25	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₈₉	C₂²	1	2	yes	—
27	#431×C₈₉	C₄	1	1	yes	characteristic
28	#431×C₈₉	C₄	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#15667952	—	4	2	no	—
32	C₂×#13083549	—	4	4	no	—
33	C₂×#13083549	C₂³	1	2	yes	—
34	C₂×#13083549	C₂³	1	2	yes	—
35	C₂×#13083549	C₂³	1	4	yes	—
36	C₂×#41×C₈₉	C₈	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₈₉	C₈	1	2	yes	—
44	C₄×D₄×C₈₉	C₈	1	1	yes	characteristic
45	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	#117×C₈₉	C₈	1	1	yes	characteristic
48	#117×C₈₉	C₈	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	2	yes	—
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	#13083549	—	2	2	no	—
66	#13083549	—	2	4	no	—
67	#13083549	—	2	2	no	—
68	#41×C₈₉	—	2	4	no	—
69	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
71	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	4	2	no	—
83	C₂×C₈×C₈₉	—	4	1	no	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
91	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
93	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
94	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
95	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
96	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
97	C₄²×C₈₉	C₂×C₈	1	2	yes	—
98	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×C₈	1	2	yes	—
100	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
105	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
106	Q₈×C₈₉	—	2	2	no	—
107	D₄×C₈₉	—	2	4	no	—
108	D₄×C₈₉	M₃₂	1	4	yes	complemented
109	C₂×C₄×C₈₉	—	2	4	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	4	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₈₉	—	2	2	no	—
127	C₈×C₈₉	—	2	1	no	—
128	C₈×C₈₉	—	2	1	no	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	—	2	2	no	—
131	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
132	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	C₂×M₃₂	1	4	yes	—
139	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
147	C₄×C₈₉	#504	1	2	yes	—
148	C₄×C₈₉	#520	1	1	yes	characteristic
149	C₄×C₈₉	#568	1	1	yes	characteristic
150	#33504	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	—	2	4	no	—
152	C₂×C₈₉	—	4	1	no	—
153	C₂×C₈₉	#2988	1	1	yes	characteristic
154	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
156	C₈×#41	D₈₉	1	1	yes	characteristic
157	#2183	—	89	2	no	—
158	#2168	—	89	4	no	—
159	D₄×C₁₆	—	89	2	no	—
160	#2163	—	89	2	no	—
161	#2162	—	89	1	no	—
162	#2150	—	89	1	no	—
163	C₂×C₄⋊C₁₆	—	89	2	no	—
164	C₈₉	#33504	1	1	yes	characteristic, complemented, Sylow
165	C₄×#41	Dic₈₉	1	1	yes	characteristic
166	C₂×M₃₂	—	89	2	no	—
167	C₂²×C₁₆	—	89	4	no	—
168	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
169	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
170	C₈×D₄	Dic₈₉	1	2	yes	—
171	C₈×D₄	Dic₈₉	1	1	yes	characteristic
172	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
173	C₈×D₄	C₂×D₈₉	1	2	yes	—
174	#431	Dic₈₉	1	1	yes	characteristic
175	#431	Dic₈₉	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
178	C₄⋊C₁₆	—	89	1	no	—
179	C₄⋊C₁₆	—	89	2	no	—
180	C₄⋊C₁₆	—	89	4	no	—
181	C₄⋊C₁₆	—	89	2	no	—
182	C₄⋊C₁₆	—	89	1	no	—
183	C₂²⋊C₁₆	—	89	2	no	—
184	C₂²⋊C₁₆	—	89	4	no	—
185	#346	—	89	4	no	—
186	C₄×C₁₆	—	89	2	no	—
187	C₂×#41	#10623494	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
189	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
191	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
192	C₄×Q₈	#10623494	1	1	yes	characteristic
193	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
194	C₄×D₄	#10623494	1	2	yes	—
195	C₄×D₄	#10623494	1	1	yes	characteristic
196	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
197	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
198	#117	#10623494	1	1	yes	characteristic
199	#117	#10623494	1	2	yes	—
200	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
201	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
202	M₃₂	—	356	2	no	—
203	C₂×C₁₆	—	89	2	no	—
204	C₂×C₁₆	—	89	2	no	—
205	C₂×C₁₆	—	89	4	no	—
206	C₂×C₁₆	—	356	4	no	—
207	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
208	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
209	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
210	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
211	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
212	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
213	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
214	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
215	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
217	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₈₉	1	2	yes	—
220	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	#41	—	2	4	no	—
222	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
223	C₂×D₄	C₂×#10623494	1	2	yes	—
224	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
225	C₂²×C₄	C₂×#10623494	1	2	yes	—
226	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
227	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
228	C₂²×C₄	C₂×#10623494	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
239	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
241	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
242	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
243	C₄⋊C₄	C₂×#10623494	1	2	yes	—
244	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
245	C₄⋊C₄	C₂×#10623494	1	2	yes	—
246	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
247	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
248	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
249	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
250	C₄²	C₂×#10623494	1	2	yes	—
251	C₄²	C₂×#10623494	1	1	yes	characteristic
252	C₄²	C₂×#10623494	1	2	yes	—
253	C₄²	C₂×#10623494	1	1	yes	characteristic
254	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
255	C₄²	C₂²×Dic₈₉	1	2	yes	—
256	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
257	C₁₆	—	178	2	no	—
258	C₁₆	—	178	4	no	—
259	C₁₆	—	178	2	no	—
260	C₂³	C₂²×#10623494	1	1	yes	characteristic
261	C₂³	C₂²×#10623494	1	2	yes	—
262	Q₈	—	2	2	no	—
263	D₄	—	2	4	no	—
264	D₄	#15667952	1	4	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
274	C₂×C₄	C₂²×#10623494	1	2	yes	—
275	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
276	C₂×C₄	C₂²×#10623494	1	2	yes	—
277	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
278	C₂×C₄	C₂²×#10623494	1	2	yes	—
279	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
280	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
281	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	2	no	—
287	C₈	#15668058	1	1	yes	characteristic
288	C₈	#15668059	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₂×#15667952	1	4	yes	—
295	C₂²	C₂³×#10623494	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	C₂×#15667952	1	2	yes	—
303	C₄	#19152015	1	2	yes	—
304	C₄	#19152081	1	1	yes	characteristic
305	C₄	#19152365	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	—	4	1	no	—
308	C₂	#26029363	1	1	yes	characteristic
309	C₂	C₂²×#15667952	1	1	yes	characteristic
310	C₂	C₂×#19152015	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈×#41×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	#33504
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #26022161	C₂	semidirect
2. #26022078	C₂	semidirect
3. D₄×#13083549	C₂	semidirect
4. #26022055	C₂	semidirect
5. #26022051	C₂	semidirect
6. #26022008	C₂	semidirect
7. C₂×C₄⋊#13083549	C₂	semidirect
108. D₄×C₈₉	M₃₂	semidirect
164. C₈₉	#33504	semidirect
264. D₄	#15667952	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
#26022078	2
D₄×#13083549	2
#26022055	2
#26022051	2
#26022008	2
C₂×C₄⋊#13083549	2
C₈×#41×C₈₉	2
#33504	89