Subgroups — #36460296

Order 22784.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151368	C₂	1	4	yes	maximal
4	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	2	yes	—
6	C₂²×#13083558	C₂²	1	2	yes	—
7	#19151622	—	2	1	no	—
8	#19151622	—	2	1	no	—
9	#19151622	—	2	2	no	—
10	C₂×C₄⋊#10623494	C₄	1	4	yes	complemented
11	C₄×#13083558	—	2	1	no	—
12	C₄×#13083558	—	2	1	no	—
13	C₄×#13083558	C₄	1	2	yes	complemented
14	C₄×#13083558	C₄	1	2	yes	complemented
15	C₂×#15667923	—	2	1	no	—
16	C₂×C₄×#10623494	—	2	1	no	—
17	#19151368	—	2	8	no	—
18	C₂×#117×C₈₉	C₂²	1	2	yes	—
19	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#384×C₈₉	—	2	1	no	—
21	#382×C₈₉	—	2	1	no	—
22	#380×C₈₉	C₄	1	4	yes	—
23	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	—	2	2	no	—
27	C₂×#13083558	—	2	4	no	—
28	C₂×#13083558	—	2	2	no	—
29	C₂×#13083558	—	2	2	no	—
30	C₂×#13083558	—	2	4	no	—
31	C₂×#13083558	C₂×C₄	1	4	yes	—
32	C₂×#13083558	D₄	1	4	yes	—
33	C₂×#13083558	Q₈	1	4	yes	—
34	C₂²×#10623494	—	2	2	no	—
35	C₂²×#10623494	C₂×C₄	1	4	yes	—
36	C₄⋊#10623494	—	2	4	no	—
37	C₄⋊#10623494	—	2	4	no	—
38	#15667923	—	2	2	no	—
39	#15667923	—	2	2	no	—
40	C₄×#10623494	—	2	2	no	—
41	C₄×#10623494	—	2	2	no	—
42	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	#117×C₈₉	—	2	4	no	—
45	#117×C₈₉	—	2	4	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	2	yes	—
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
59	#95×C₈₉	—	2	4	no	—
60	#13083558	—	2	4	no	—
61	#13083558	—	2	4	no	—
62	#13083558	—	2	4	no	—
63	#13083558	—	4	4	no	—
64	#13083558	—	4	2	no	—
65	#13083558	—	4	2	no	—
66	#13083558	C₄⋊C₄	1	8	yes	complemented
67	C₂×#10623494	—	2	4	no	—
68	C₂×#10623494	—	2	4	no	—
69	C₂×#10623494	—	2	4	no	—
70	C₂×#10623494	—	2	4	no	—
71	C₂×#10623494	—	2	8	no	—
72	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	—	2	4	no	—
74	C₂²×C₄×C₈₉	—	4	4	no	—
75	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
76	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₈₉	#41	1	2	yes	—
90	C₂²×C₄×C₈₉	#41	1	2	yes	—
91	C₄⋊C₄×C₈₉	—	4	8	no	—
92	C₄⋊C₄×C₈₉	—	4	4	no	—
93	C₂²⋊C₄×C₈₉	—	2	8	no	—
94	C₄²×C₈₉	—	4	2	no	—
95	C₄²×C₈₉	—	4	2	no	—
96	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
103	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
104	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
105	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
106	C₂³×C₈₉	—	4	2	no	—
107	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₈₉	C₄×D₄	1	2	yes	—
111	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	2	4	no	—
114	C₂×C₄×C₈₉	—	2	4	no	—
115	C₂×C₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	8	no	—
119	C₂×C₄×C₈₉	—	4	4	no	—
120	C₂×C₄×C₈₉	#95	1	2	yes	—
121	C₂×C₄×C₈₉	#95	1	2	yes	—
122	C₂×C₄×C₈₉	#95	1	2	yes	—
123	C₂×C₄×C₈₉	#95	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#117	1	2	yes	—
130	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₈₉	#121	1	2	yes	—
133	C₂×C₄×C₈₉	#121	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#125	1	2	yes	—
139	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
142	#10623494	—	2	8	no	—
143	#10623494	—	4	4	no	—
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	4	2	no	—
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
150	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
151	C₂²×C₈₉	#389	1	2	yes	—
152	C₂²×C₈₉	#390	1	1	yes	characteristic
153	C₂²×C₈₉	#391	1	1	yes	characteristic
154	C₄×C₈₉	—	2	2	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	8	4	no	—
158	C₄×C₈₉	#344	1	4	yes	complemented
159	C₄×C₈₉	#380	1	2	yes	—
160	C₄×C₈₉	#382	1	1	yes	characteristic
161	C₄×C₈₉	#384	1	1	yes	characteristic
162	C₄×C₈₉	#389	1	2	yes	—
163	C₄×C₈₉	#390	1	1	yes	characteristic
164	C₄×C₈₉	#391	1	1	yes	characteristic
165	#10855	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	2	1	no	—
167	C₂×C₈₉	—	2	2	no	—
168	C₂×C₈₉	—	2	2	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
171	C₂×C₈₉	C₂×#344	1	2	yes	—
172	C₂×#423	—	89	1	no	—
173	C₂×C₄×M₁₆	—	89	1	no	—
174	#2285	D₈₉	1	1	yes	characteristic
175	C₂×#344	—	89	4	no	—
176	C₈₉	#10855	1	1	yes	characteristic, complemented, Sylow
177	C₂²×M₁₆	—	89	2	no	—
178	C₂²×M₁₆	—	89	2	no	—
179	C₂×#117	C₂×D₈₉	1	2	yes	—
180	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
181	#423	—	178	1	no	—
182	#423	—	178	1	no	—
183	#423	—	178	2	no	—
184	C₂×C₄⋊C₈	—	89	4	no	—
185	C₄×M₁₆	—	89	2	no	—
186	C₄×M₁₆	—	89	2	no	—
187	C₄×M₁₆	—	178	1	no	—
188	C₄×M₁₆	—	178	1	no	—
189	C₂×#97	—	178	1	no	—
190	C₂×C₄×C₈	—	178	1	no	—
191	#384	—	2	1	no	—
192	#382	—	2	1	no	—
193	#380	Dic₈₉	1	4	yes	—
194	C₄×C₄⋊C₄	—	2	2	no	—
195	#344	—	178	8	no	—
196	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
197	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
198	C₂×M₁₆	—	89	4	no	—
199	C₂×M₁₆	—	89	4	no	—
200	C₂×M₁₆	—	89	4	no	—
201	C₂×M₁₆	—	178	4	no	—
202	C₂×M₁₆	—	178	2	no	—
203	C₂×M₁₆	—	178	2	no	—
204	C₂×M₁₆	—	178	4	no	—
205	C₂×M₁₆	—	178	2	no	—
206	C₂×M₁₆	—	178	2	no	—
207	C₂×M₁₆	—	178	2	no	—
208	C₂²×C₈	—	89	4	no	—
209	C₂²×C₈	—	178	2	no	—
210	#117	—	2	4	no	—
211	#117	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	Dic₁₇₈	1	2	yes	—
217	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₈₉	1	2	yes	—
220	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
221	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
223	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
225	C₄⋊C₈	—	178	4	no	—
226	C₄⋊C₈	—	178	4	no	—
227	#97	—	178	2	no	—
228	#97	—	178	2	no	—
229	C₄×C₈	—	178	2	no	—
230	C₄×C₈	—	178	2	no	—
231	#95	—	2	4	no	—
232	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
236	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
237	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
238	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
239	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
240	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₈₉	1	2	yes	—
244	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
248	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
249	C₂²×C₄	#13083573	1	2	yes	—
250	C₂²×C₄	#13083575	1	2	yes	—
251	M₁₆	—	89	8	no	—
252	M₁₆	—	356	4	no	—
253	M₁₆	—	356	2	no	—
254	M₁₆	—	356	2	no	—
255	M₁₆	—	178	4	no	—
256	M₁₆	—	178	4	no	—
257	M₁₆	—	178	4	no	—
258	C₂×C₈	—	178	8	no	—
259	C₂×C₈	—	178	4	no	—
260	C₂×C₈	—	178	4	no	—
261	C₂×C₈	—	178	4	no	—
262	C₂×C₈	—	178	4	no	—
263	C₄⋊C₄	—	4	8	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	#13083551	1	1	yes	characteristic
269	C₄²	#13083551	1	1	yes	characteristic
270	C₄²	#13083551	1	1	yes	characteristic
271	C₄²	#13083551	1	1	yes	characteristic
272	C₄²	#13083551	1	2	yes	—
273	C₄²	#13083551	1	2	yes	—
274	C₄²	#13083553	1	2	yes	—
275	C₄²	#13083553	1	2	yes	—
276	C₄²	#13083553	1	2	yes	—
277	C₄²	#13083553	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
280	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
281	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
282	C₂³	D₄×Dic₈₉	1	2	yes	—
283	C₂³	Q₈×Dic₈₉	1	2	yes	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	#15667919	1	2	yes	—
293	C₂×C₄	#15667919	1	2	yes	—
294	C₂×C₄	#15667919	1	2	yes	—
295	C₂×C₄	#15667919	1	2	yes	—
296	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
297	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
299	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
300	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
301	C₂×C₄	#15667970	1	2	yes	—
302	C₂×C₄	#15667974	1	2	yes	—
303	C₂×C₄	#15667977	1	2	yes	—
304	C₂×C₄	#15667983	1	2	yes	—
305	C₂×C₄	#15667987	1	2	yes	—
306	C₂×C₄	#15667988	1	2	yes	—
307	C₂×C₄	#15667992	1	2	yes	—
308	C₂×C₄	#15667999	1	1	yes	characteristic
309	C₂×C₄	#15668001	1	1	yes	characteristic
310	C₂×C₄	#15668003	1	2	yes	—
311	C₂×C₄	#15668008	1	1	yes	characteristic
312	C₂×C₄	#15668010	1	1	yes	characteristic
313	C₂×C₄	#15668011	1	2	yes	—
314	C₈	—	356	4	no	—
315	C₈	—	178	8	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×#15667919	1	1	yes	characteristic
322	C₂²	C₄⋊C₄×Dic₈₉	1	2	yes	—
323	C₂²	#19151497	1	2	yes	—
324	C₂²	#19151501	1	1	yes	characteristic
325	C₂²	#19151505	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	8	4	no	—
330	C₄	#19151368	1	4	yes	complemented
331	C₄	#19151455	1	2	yes	—
332	C₄	#19151463	1	1	yes	characteristic
333	C₄	#19151467	1	1	yes	characteristic
334	C₄	#19151494	1	2	yes	—
335	C₄	#19151499	1	1	yes	characteristic
336	C₄	#19151503	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	—	2	1	no	—
341	C₂	#26018481	1	1	yes	characteristic
342	C₂	C₂×#19151368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#2285	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#2285	D₈₉
hypocenter	C₈₉	#10855
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
10. C₂×C₄⋊#10623494	C₄	semidirect
13. C₄×#13083558	C₄	semidirect
14. C₄×#13083558	C₄	semidirect
66. #13083558	C₄⋊C₄	semidirect
158. C₄×C₈₉	#344	semidirect
176. C₈₉	#10855	semidirect
330. C₄	#19151368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
C₂×C₄×#13083558	2
C₂×#19151368	2
C₈₉×#2285	2
#10855	89