Subgroups — C₄×#19151995 #36563154

Order 22784.

878 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151995	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668011	C₂	1	2	yes	maximal
3	#26021211	C₂	1	2	yes	maximal
4	#26020981	C₂	1	1	yes	maximal, characteristic
5	#26020979	C₂	1	2	yes	maximal
6	#26020978	C₂	1	1	yes	maximal, characteristic
7	#26020407	C₂	1	1	yes	maximal, characteristic
8	C₄×#15667943	C₂	1	2	yes	maximal
9	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
10	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
11	C₄×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668011	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₇₈	—	2	2	no	—
14	#19151995	C₄	1	8	yes	direct product factor
15	C₂×#15667944	C₂²	1	1	yes	characteristic
16	C₂×#15667944	C₂²	1	1	yes	characteristic
17	C₂×#15667943	C₂²	1	2	yes	—
18	C₂×#15667943	C₂²	1	2	yes	—
19	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
24	#19151502	—	2	2	no	—
25	#19151497	C₂²	1	2	yes	—
26	#19151495	—	2	2	no	—
27	#19151494	C₂²	1	2	yes	—
28	#19151467	C₂²	1	2	yes	—
29	C₄×#13083553	—	2	2	no	—
30	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
31	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
32	#19151361	C₂²	1	2	yes	—
33	#19151360	C₂²	1	2	yes	—
34	#19151360	C₂²	1	2	yes	—
35	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
39	#340×C₈₉	C₂²	1	2	yes	—
40	C₂²×Dic₁₇₈	—	2	2	no	—
41	C₂²×#10623494	C₂³	1	1	yes	characteristic
42	C₂²×#10623494	C₂³	1	1	yes	characteristic
43	#15668011	—	2	4	no	—
44	#15668011	C₂×C₄	1	8	yes	—
45	C₄×Dic₁₇₈	—	4	8	no	—
46	C₂×C₄⋊Dic₈₉	—	2	2	no	—
47	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
49	C₂×#13083553	—	2	2	no	—
50	C₂×#13083553	—	2	2	no	—
51	C₂×#13083553	—	2	2	no	—
52	C₂×#13083553	—	2	2	no	—
53	C₂×C₄×Dic₈₉	—	2	2	no	—
54	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
55	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
56	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
58	#15667944	—	2	2	no	—
59	#15667944	C₂×C₄	1	4	yes	—
60	#15667943	—	2	4	no	—
61	#15667943	C₂×C₄	1	8	yes	—
62	C₄×#10623494	Q₈	1	4	yes	complemented
63	C₈×Dic₈₉	—	2	2	no	—
64	C₈×Dic₈₉	—	2	2	no	—
65	C₈×Dic₈₉	—	2	2	no	—
66	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
67	#15667919	—	2	2	no	—
68	#15667919	—	2	2	no	—
69	#15667919	—	2	2	no	—
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
74	C₂×C₄²×C₈₉	—	2	2	no	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	#106×C₈₉	—	2	2	no	—
77	#106×C₈₉	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
79	#95×C₈₉	—	2	2	no	—
80	C₂×Dic₁₇₈	—	2	8	no	—
81	C₂×Dic₁₇₈	—	4	4	no	—
82	C₂×Dic₁₇₈	—	4	4	no	—
83	C₂²×Dic₈₉	—	2	2	no	—
84	C₂²×Dic₈₉	—	2	2	no	—
85	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
86	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
87	C₂×#10623494	—	2	1	no	—
88	C₂×#10623494	—	2	1	no	—
89	C₂×#10623494	C₂²×C₄	1	2	yes	—
90	C₂×#10623494	C₂×Q₈	1	2	yes	—
91	C₂×#10623494	C₂×Q₈	1	2	yes	—
92	C₂×#10623494	#41	1	2	yes	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	4	4	no	—
97	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
98	#13083553	—	2	8	no	—
99	#13083553	—	4	4	no	—
100	#13083553	—	4	4	no	—
101	#13083553	—	4	4	no	—
102	C₄×Dic₈₉	—	2	1	no	—
103	C₄×Dic₈₉	—	2	1	no	—
104	C₄×Dic₈₉	—	2	4	no	—
105	C₄×Dic₈₉	—	4	4	no	—
106	C₄×Dic₈₉	SD₁₆	1	8	yes	—
107	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
108	C₄×Dic₈₉	#41	1	2	yes	—
109	C₂²×C₄×C₈₉	—	2	2	no	—
110	C₂²×C₄×C₈₉	—	2	2	no	—
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂×C₈×C₈₉	—	2	1	no	—
115	C₂×C₈×C₈₉	—	2	1	no	—
116	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₈₉	#41	1	2	yes	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
124	C₄²×C₈₉	—	4	4	no	—
125	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
126	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
127	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	4	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
138	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	4	no	—
147	C₂×Dic₈₉	—	4	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	C₄×D₄	1	4	yes	—
153	C₂×Dic₈₉	C₂×SD₁₆	1	4	yes	—
154	C₂×Dic₈₉	C₂×SD₁₆	1	4	yes	—
155	C₂×Dic₈₉	#135	1	4	yes	—
156	Dic₁₇₈	—	4	16	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
159	#10623494	—	2	2	no	—
160	#10623494	C₄×Q₈	1	4	yes	complemented
161	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₈₉	#532	1	1	yes	characteristic
164	C₂²×C₈₉	C₂²×SD₁₆	1	2	yes	—
165	C₂²×C₈₉	C₂×#135	1	2	yes	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	4	4	no	—
168	C₄×C₈₉	—	8	2	no	—
169	C₄×C₈₉	#498	1	4	yes	complemented
170	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
172	C₄×C₈₉	#533	1	1	yes	characteristic
173	C₄×C₈₉	#533	1	1	yes	characteristic
174	Dic₈₉	—	2	4	no	—
175	Dic₈₉	—	4	4	no	—
176	Dic₈₉	—	8	2	no	—
177	Dic₈₉	C₄×SD₁₆	1	8	yes	—
178	C₄×#498	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
180	C₂×C₈₉	C₂×C₄×SD₁₆	1	2	yes	—
181	C₂×C₈₉	C₂×#498	1	1	yes	characteristic
182	C₂×C₈₉	#3160	1	1	yes	characteristic
183	C₂×C₈₉	#3167	1	2	yes	—
184	C₂×#498	—	89	1	no	—
185	C₄×C₄⋊Q₈	—	89	2	no	—
186	#1990	—	89	2	no	—
187	#1941	—	89	1	no	—
188	#1941	—	89	1	no	—
189	#1941	—	89	2	no	—
190	#1844	—	89	1	no	—
191	C₄×#106	—	89	1	no	—
192	C₄×#106	—	89	2	no	—
193	C₄×#106	D₈₉	1	1	yes	characteristic
194	C₄²×C₈	—	89	1	no	—
195	C₈₉	C₄×#498	1	1	yes	characteristic, complemented, Sylow
196	C₂×C₄⋊Q₈	—	89	2	no	—
197	C₂×C₄×Q₈	—	178	2	no	—
198	#498	—	89	8	no	—
199	C₂×#106	—	89	1	no	—
200	C₂×#106	—	89	2	no	—
201	C₂×#106	—	89	1	no	—
202	C₂×#106	—	89	2	no	—
203	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
204	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
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	1	no	—
209	C₂×C₄×C₈	—	89	1	no	—
210	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
211	#391	—	178	2	no	—
212	#389	—	89	2	no	—
213	#389	—	89	2	no	—
214	#389	—	178	2	no	—
215	#384	—	89	2	no	—
216	C₄×C₄⋊C₄	—	89	2	no	—
217	C₄×C₄⋊C₄	—	178	2	no	—
218	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
219	C₄³	—	89	1	no	—
220	#340	—	89	2	no	—
221	#340	—	89	2	no	—
222	#340	—	89	2	no	—
223	#340	C₂×D₈₉	1	2	yes	—
224	C₂²×Q₈	—	178	2	no	—
225	C₂²×C₈	—	89	1	no	—
226	C₂²×C₈	—	89	1	no	—
227	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	89	8	no	—
230	C₄⋊Q₈	—	178	4	no	—
231	C₄×Q₈	—	356	8	no	—
232	C₂×C₄⋊C₄	—	89	2	no	—
233	C₂×C₄⋊C₄	—	89	2	no	—
234	C₂×C₄⋊C₄	—	178	2	no	—
235	C₂×C₄⋊C₄	—	178	2	no	—
236	C₂×C₄⋊C₄	—	178	2	no	—
237	C₂×C₄⋊C₄	—	178	2	no	—
238	C₂×C₄⋊C₄	—	178	2	no	—
239	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
240	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	—	89	2	no	—
243	C₂×C₄²	—	89	1	no	—
244	C₂×C₄²	—	89	2	no	—
245	C₂×C₄²	—	89	1	no	—
246	C₂×C₄²	—	178	2	no	—
247	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
248	#106	—	2	2	no	—
249	#106	—	89	4	no	—
250	#106	—	89	8	no	—
251	#106	—	178	4	no	—
252	#106	—	178	2	no	—
253	#106	C₄×D₈₉	1	4	yes	—
254	C₄×C₈	—	89	4	no	—
255	C₄×C₈	—	89	4	no	—
256	C₄×C₈	—	178	2	no	—
257	C₄×C₈	—	178	2	no	—
258	C₄×C₈	—	178	2	no	—
259	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
260	#95	—	2	2	no	—
261	#95	—	178	2	no	—
262	#95	—	178	2	no	—
263	#95	—	178	2	no	—
264	C₂×Q₈	—	356	4	no	—
265	C₂×Q₈	—	356	4	no	—
266	C₂×Q₈	—	178	8	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	89	2	no	—
270	C₂²×C₄	—	89	2	no	—
271	C₂²×C₄	—	178	2	no	—
272	C₂²×C₄	—	178	2	no	—
273	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	2	no	—
281	C₂×C₈	—	89	2	no	—
282	C₂×C₈	—	178	1	no	—
283	C₂×C₈	—	178	1	no	—
284	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
285	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
286	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
287	C₂×C₈	#13083574	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	89	4	no	—
292	C₄⋊C₄	—	356	4	no	—
293	C₄⋊C₄	—	356	4	no	—
294	C₄⋊C₄	—	356	4	no	—
295	C₄⋊C₄	—	356	4	no	—
296	C₄⋊C₄	—	178	8	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₄⋊C₄	C₂×C₄×D₈₉	1	4	yes	—
301	C₄²	—	4	4	no	—
302	C₄²	—	89	8	no	—
303	C₄²	—	89	2	no	—
304	C₄²	—	89	2	no	—
305	C₄²	—	356	4	no	—
306	C₄²	—	178	4	no	—
307	C₄²	—	178	1	no	—
308	C₄²	—	178	1	no	—
309	C₄²	Q₈×D₈₉	1	2	yes	—
310	C₄²	C₂×Dic₁₇₈	1	2	yes	—
311	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
312	Q₈	—	356	16	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	89	4	no	—
321	C₂×C₄	—	89	4	no	—
322	C₂×C₄	—	89	4	no	—
323	C₂×C₄	—	89	4	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	178	4	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	—	178	2	no	—
332	C₂×C₄	#15668011	1	2	yes	—
333	C₂×C₄	#15668011	1	2	yes	—
334	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
335	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
337	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
338	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
339	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
340	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	89	4	no	—
343	C₈	—	178	2	no	—
344	C₈	C₄×Dic₁₇₈	1	4	yes	complemented
345	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
346	C₂²	C₂×#15668011	1	1	yes	characteristic
347	C₂²	#19152145	1	1	yes	characteristic
348	C₂²	C₂×SD₁₆×D₈₉	1	2	yes	—
349	C₂²	#135×D₈₉	1	2	yes	—
350	C₄	—	2	2	no	—
351	C₄	—	4	4	no	—
352	C₄	—	8	2	no	—
353	C₄	—	89	8	no	—
354	C₄	—	356	4	no	—
355	C₄	—	178	4	no	—
356	C₄	—	712	2	no	—
357	C₄	#19151995	1	4	yes	direct product factor
358	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
359	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
360	C₄	#19152151	1	1	yes	characteristic
361	C₄	#19152152	1	1	yes	characteristic
362	C₂	C₄×#15668011	1	1	yes	characteristic
363	C₂	C₄×SD₁₆×D₈₉	1	2	yes	—
364	C₂	C₂×#19151995	1	1	yes	characteristic
365	C₂	#26030764	1	1	yes	characteristic
366	C₂	#26030808	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#106×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#106	D₈₉
hypocenter	C₈₉	C₄×#498
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
14. #19151995	C₄	direct (N × H)
62. C₄×#10623494	Q₈	semidirect
78. C₄×C₈×C₈₉	Q₈	semidirect
158. C₈×C₈₉	C₄×Q₈	semidirect
160. #10623494	C₄×Q₈	semidirect
169. C₄×C₈₉	#498	semidirect
195. C₈₉	C₄×#498	semidirect
259. C₄×C₈	Dic₁₇₈	semidirect
344. C₈	C₄×Dic₁₇₈	semidirect
357. C₄	#19151995	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19151995	2
C₄×#15668011	2
#26021211	2
#26020981	2
#26020979	2
#26020978	2
#26020407	2
C₄×#15667943	2
C₄×#15667944	2
C₄×C₈×Dic₈₉	2
C₄×#106×C₈₉	2
C₄×#498	89

Subgroups — C4×#19151995 #36563154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151995 #36563154