Subgroups — #36553053

Order 22784.

958 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151725	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667977	C₂	1	1	yes	maximal, characteristic
4	#26020128	C₂	1	2	yes	maximal
5	#26020125	C₂	1	2	yes	maximal
6	#26020121	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
8	#26020100	C₂	1	2	yes	maximal
9	#97×Dic₈₉	C₂	1	2	yes	maximal
10	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1770	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	C₂×#15667977	C₄	1	2	yes	—
14	#19151728	C₄	1	8	yes	complemented
15	#19151725	C₄	1	8	yes	complemented
16	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
17	C₂×#15667941	C₂²	1	2	yes	—
18	C₂×#15667941	C₂²	1	2	yes	—
19	C₂×#15667923	C₂²	1	2	yes	—
20	C₂×#15667922	C₂²	1	2	yes	—
21	C₂×#15667922	C₂²	1	2	yes	—
22	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
25	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×Dic₈₉	C₄	1	2	yes	—
29	C₄×#13083553	C₂²	1	2	yes	—
30	#19151449	C₄	1	4	yes	—
31	C₄²×Dic₈₉	C₂²	1	2	yes	—
32	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
33	#19151353	C₂²	1	2	yes	—
34	#19151352	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#97×C₈₉	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
39	#336×C₈₉	C₂²	1	2	yes	—
40	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×#10623494	C₂³	1	1	yes	characteristic
42	C₂²×#10623494	C₂³	1	2	yes	—
43	C₂²×#10623494	C₂³	1	1	yes	characteristic
44	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
46	#15667977	—	2	8	no	—
47	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×#13083553	C₂×C₄	1	4	yes	—
49	C₂×#13083553	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
52	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
53	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
56	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊#10623494	C₂×C₄	1	4	yes	—
58	#15667941	C₂×C₄	1	8	yes	—
59	#15667941	C₂×C₄	1	8	yes	—
60	#15667923	—	2	4	no	—
61	#15667922	C₂×C₄	1	8	yes	—
62	#15667922	C₂×C₄	1	8	yes	—
63	C₄×#10623494	C₂×C₄	1	4	yes	—
64	C₈×Dic₈₉	—	2	2	no	—
65	C₈×Dic₈₉	—	2	2	no	—
66	C₈×Dic₈₉	—	2	4	no	—
67	C₈×Dic₈₉	—	2	2	no	—
68	C₈×Dic₈₉	Q₈	1	4	yes	complemented
69	#15667919	C₂×C₄	1	4	yes	—
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
74	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
77	#97×C₈₉	—	2	4	no	—
78	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
79	#95×C₈₉	C₂×C₄	1	2	yes	—
80	C₂×Dic₁₇₈	—	2	4	no	—
81	C₂×Dic₁₇₈	C₄²	1	4	yes	—
82	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
86	C₂×#10623494	—	2	1	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	—	2	1	no	—
89	C₂×#10623494	—	2	2	no	—
90	C₂×#10623494	C₂²×C₄	1	2	yes	—
91	C₂×#10623494	C₂²×C₄	1	4	yes	—
92	C₂×#10623494	C₂²×C₄	1	2	yes	—
93	C₂×#10623494	C₂×Q₈	1	2	yes	—
94	C₂×#10623494	#41	1	2	yes	—
95	C₄⋊Dic₈₉	C₄²	1	4	yes	—
96	#13083553	—	2	8	no	—
97	#13083553	C₄²	1	8	yes	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	—	2	1	no	—
100	C₄×Dic₈₉	—	2	8	no	—
101	C₄×Dic₈₉	—	2	4	no	—
102	C₄×Dic₈₉	—	2	8	no	—
103	C₄×Dic₈₉	M₁₆	1	8	yes	—
104	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
105	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
106	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
107	C₄×Dic₈₉	#41	1	2	yes	—
108	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
109	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	1	no	—
114	C₂×C₈×C₈₉	—	2	1	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
118	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₈₉	#41	1	2	yes	—
121	C₄⋊C₄×C₈₉	—	2	4	no	—
122	C₄²×C₈₉	—	2	2	no	—
123	C₄²×C₈₉	—	2	2	no	—
124	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
125	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
126	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	4	no	—
129	C₂×C₄×C₈₉	—	2	4	no	—
130	C₂×C₄×C₈₉	—	2	4	no	—
131	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₈₉	—	2	2	no	—
141	C₂×Dic₈₉	—	2	4	no	—
142	C₂×Dic₈₉	—	2	4	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	C₂×C₄²	1	4	yes	—
145	C₂×Dic₈₉	C₂×C₄²	1	4	yes	—
146	C₂×Dic₈₉	C₄×Q₈	1	4	yes	—
147	C₂×Dic₈₉	C₂×M₁₆	1	4	yes	—
148	C₂×Dic₈₉	C₂×M₁₆	1	4	yes	—
149	C₂×Dic₈₉	#131	1	4	yes	—
150	Dic₁₇₈	—	2	8	no	—
151	C₈×C₈₉	—	2	4	no	—
152	C₈×C₈₉	—	2	2	no	—
153	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
154	#10623494	—	2	2	no	—
155	#10623494	—	2	4	no	—
156	#10623494	C₄×Q₈	1	4	yes	complemented
157	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
158	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
160	C₂²×C₈₉	C₂²×M₁₆	1	2	yes	—
161	C₂²×C₈₉	C₂×#131	1	2	yes	—
162	C₄×C₈₉	—	2	2	no	—
163	C₄×C₈₉	—	4	4	no	—
164	C₄×C₈₉	C₄×M₁₆	1	4	yes	—
165	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
168	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
169	Dic₈₉	—	2	4	no	—
170	Dic₈₉	—	2	4	no	—
171	Dic₈₉	—	2	4	no	—
172	Dic₈₉	#446	1	8	yes	complemented
173	C₄×#446	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	C₄²×Q₈	1	1	yes	characteristic
175	C₂×C₈₉	C₂×C₄×M₁₆	1	1	yes	characteristic
176	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
177	C₂×C₈₉	C₂×#446	1	2	yes	—
178	C₂×C₈₉	#2965	1	2	yes	—
179	C₂×#446	—	89	1	no	—
180	C₂×#446	—	89	1	no	—
181	C₄²×Q₈	—	89	1	no	—
182	#1771	—	89	2	no	—
183	#1771	—	89	2	no	—
184	#1770	—	89	1	no	—
185	#1770	D₈₉	1	1	yes	characteristic
186	C₄×C₄⋊C₈	—	89	1	no	—
187	C₄×C₄⋊C₈	—	89	2	no	—
188	C₄×#97	—	89	2	no	—
189	C₄²×C₈	—	89	1	no	—
190	C₈₉	C₄×#446	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₄×Q₈	—	89	2	no	—
192	C₂×C₄×Q₈	—	89	1	no	—
193	#446	—	89	8	no	—
194	#446	—	89	8	no	—
195	C₂×C₄⋊C₈	—	89	2	no	—
196	C₂×C₄⋊C₈	—	89	2	no	—
197	C₂×C₄⋊C₈	—	89	1	no	—
198	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
199	C₂×#97	—	89	2	no	—
200	C₂×#97	—	89	2	no	—
201	C₂×#97	—	89	2	no	—
202	C₂×#97	C₂×D₈₉	1	2	yes	—
203	C₂×C₄×C₈	—	89	1	no	—
204	C₂×C₄×C₈	—	89	1	no	—
205	C₂×C₄×C₈	—	89	1	no	—
206	C₂×C₄×C₈	—	89	1	no	—
207	C₂×C₄×C₈	—	89	1	no	—
208	C₂×C₄×C₈	—	89	2	no	—
209	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
210	C₄×C₄⋊C₄	—	89	2	no	—
211	C₄×C₄⋊C₄	—	89	2	no	—
212	C₄×C₄⋊C₄	—	89	4	no	—
213	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
214	C₄³	—	89	1	no	—
215	C₄³	—	89	2	no	—
216	#336	—	89	2	no	—
217	#336	—	89	2	no	—
218	#336	C₂×D₈₉	1	2	yes	—
219	C₂²×Q₈	—	89	1	no	—
220	C₂²×C₈	—	89	1	no	—
221	C₂²×C₈	—	89	1	no	—
222	C₂²×C₈	—	89	2	no	—
223	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
224	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
226	C₄×Q₈	—	89	4	no	—
227	C₄×Q₈	—	89	4	no	—
228	C₄×Q₈	—	178	8	no	—
229	C₂×C₄⋊C₄	—	89	2	no	—
230	C₂×C₄⋊C₄	—	89	1	no	—
231	C₂×C₄⋊C₄	—	89	4	no	—
232	C₂×C₄⋊C₄	C₄×D₈₉	1	2	yes	—
233	C₂×C₄²	—	89	2	no	—
234	C₂×C₄²	—	89	2	no	—
235	C₂×C₄²	—	89	1	no	—
236	C₂×C₄²	—	89	4	no	—
237	C₂×C₄²	—	89	4	no	—
238	C₂×C₄²	—	89	2	no	—
239	C₂×C₄²	—	89	1	no	—
240	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
241	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
242	C₄⋊C₈	—	89	8	no	—
243	C₄⋊C₈	—	89	8	no	—
244	C₄⋊C₈	—	89	4	no	—
245	C₄⋊C₈	C₄×D₈₉	1	4	yes	—
246	#97	—	2	4	no	—
247	#97	—	89	8	no	—
248	#97	—	89	8	no	—
249	#97	—	178	4	no	—
250	C₄×C₈	—	89	4	no	—
251	C₄×C₈	—	89	4	no	—
252	C₄×C₈	—	178	2	no	—
253	C₄×C₈	—	178	4	no	—
254	C₄×C₈	—	178	2	no	—
255	C₄×C₈	—	178	2	no	—
256	C₄×C₈	C₄×D₈₉	1	4	yes	—
257	#95	—	89	4	no	—
258	#95	C₄×D₈₉	1	2	yes	—
259	C₂×Q₈	—	89	4	no	—
260	C₂×Q₈	—	178	4	no	—
261	C₂²×C₄	—	89	2	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	89	2	no	—
264	C₂²×C₄	—	89	2	no	—
265	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
266	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	89	2	no	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	—	89	2	no	—
276	C₂×C₈	—	89	2	no	—
277	C₂×C₈	—	89	4	no	—
278	C₂×C₈	—	178	1	no	—
279	C₂×C₈	—	178	2	no	—
280	C₂×C₈	—	178	2	no	—
281	C₂×C₈	—	178	1	no	—
282	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
283	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
284	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
285	C₂×C₈	Q₈×D₈₉	1	2	yes	—
286	C₂×C₈	#13083573	1	2	yes	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	89	8	no	—
289	C₄⋊C₄	—	89	4	no	—
290	C₄⋊C₄	—	178	8	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	89	8	no	—
294	C₄²	—	89	4	no	—
295	C₄²	—	89	4	no	—
296	C₄²	—	89	2	no	—
297	C₄²	—	89	2	no	—
298	C₄²	—	178	8	no	—
299	C₄²	—	178	4	no	—
300	C₄²	—	178	8	no	—
301	C₄²	—	178	1	no	—
302	C₄²	—	178	1	no	—
303	C₄²	C₂×C₄×D₈₉	1	2	yes	—
304	C₄²	C₂×C₄×D₈₉	1	2	yes	—
305	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
306	Q₈	—	178	8	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	89	4	no	—
312	C₂×C₄	—	89	4	no	—
313	C₂×C₄	—	89	4	no	—
314	C₂×C₄	—	89	4	no	—
315	C₂×C₄	—	89	4	no	—
316	C₂×C₄	—	89	4	no	—
317	C₂×C₄	—	178	2	no	—
318	C₂×C₄	—	178	4	no	—
319	C₂×C₄	—	178	4	no	—
320	C₂×C₄	—	178	2	no	—
321	C₂×C₄	C₄²×D₈₉	1	2	yes	—
322	C₂×C₄	C₄²×D₈₉	1	2	yes	—
323	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
324	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
327	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
328	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	—	89	4	no	—
333	C₈	—	178	2	no	—
334	C₈	—	178	4	no	—
335	C₈	#15667977	1	4	yes	complemented
336	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
337	C₂²	C₄×Q₈×D₈₉	1	1	yes	characteristic
338	C₂²	C₄×#13083573	1	1	yes	characteristic
339	C₂²	C₂×M₁₆×D₈₉	1	2	yes	—
340	C₂²	#131×D₈₉	1	2	yes	—
341	C₄	—	2	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	89	8	no	—
344	C₄	—	178	4	no	—
345	C₄	—	178	4	no	—
346	C₄	—	178	4	no	—
347	C₄	#19151550	1	4	yes	—
348	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
349	C₄	C₂×#15667977	1	1	yes	characteristic
350	C₄	C₄×#13083573	1	1	yes	characteristic
351	C₄	#19152066	1	1	yes	characteristic
352	C₂	C₄×#15667977	1	1	yes	characteristic
353	C₂	C₂×#19151550	1	1	yes	characteristic
354	C₂	#26028674	1	1	yes	characteristic
355	C₂	#446×D₈₉	1	2	yes	—
356	C₂	#26029215	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈₉×#1770	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1770	D₈₉
hypocenter	C₈₉	C₄×#446
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
14. #19151728	C₄	semidirect
15. #19151725	C₄	semidirect
68. C₈×Dic₈₉	Q₈	semidirect
153. C₈×C₈₉	C₄×Q₈	semidirect
156. #10623494	C₄×Q₈	semidirect
172. Dic₈₉	#446	semidirect
190. C₈₉	C₄×#446	semidirect
335. C₈	#15667977	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151728	2
C₂×#19151725	2
C₄×#15667977	2
#26020128	2
#26020125	2
#26020121	2
C₄⋊C₈×Dic₈₉	2
#26020100	2
#97×Dic₈₉	2
C₄×C₈×Dic₈₉	2
C₈₉×#1770	2
C₄×#446	89