Subgroups — C₄×#14583278 #25105032

Order 11008.

1086 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583278	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10621885	C₂	1	4	yes	maximal
3	C₄×#13063325	C₂	1	1	yes	maximal, characteristic
4	#19135455	C₂	1	4	yes	maximal
5	#19135398	C₂	1	1	yes	maximal, characteristic
6	#19135396	C₂	1	1	yes	maximal, characteristic
7	#19134754	C₂	1	1	yes	maximal, characteristic
8	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10621885	C₂²	1	4	yes	—
11	C₂×#13063325	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
13	#14583278	C₄	1	8	yes	direct product factor
14	C₄×#10621885	—	2	16	no	—
15	C₂×#13063258	C₂²	1	4	yes	—
16	C₂×#13063249	C₂²	1	4	yes	—
17	C₂×#13063248	C₂²	1	4	yes	—
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
22	#14582805	C₂²	1	2	yes	—
23	#14582797	—	2	2	no	—
24	#14582769	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊Dic₄₃	—	2	2	no	—
26	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
28	C₄₃×#391	C₂²	1	2	yes	—
29	C₄₃×#389	—	2	2	no	—
30	C₄₃×#384	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄³×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₈₆	C₂³	1	2	yes	—
34	C₂×#10621885	—	2	8	no	—
35	C₂×#10621885	—	2	8	no	—
36	C₂×#10621885	C₂×C₄	1	16	yes	—
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
39	#13063325	—	2	2	no	—
40	#13063325	C₂×C₄	1	4	yes	—
41	C₄×Dic₈₆	—	2	8	no	—
42	C₂×C₄⋊Dic₄₃	—	2	2	no	—
43	C₂×C₄⋊Dic₄₃	—	2	2	no	—
44	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
45	C₂×Dic₄₃⋊C₄	—	2	2	no	—
46	C₂×C₄×Dic₄₃	—	2	2	no	—
47	#13063258	—	2	8	no	—
48	#13063249	—	2	8	no	—
49	#13063248	—	2	8	no	—
50	C₄×C₄₃⋊C₈	—	2	2	no	—
51	C₄×C₄₃⋊C₈	—	2	4	no	—
52	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
53	C₄×C₄₃⋊C₈	D₄	1	8	yes	—
54	C₄₃⋊#95	—	2	4	no	—
55	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₄₃	—	2	2	no	—
57	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₄₃	—	2	8	no	—
59	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₃	—	2	2	no	—
64	C₂×C₄²×C₄₃	D₄	1	2	yes	—
65	C₂×C₄²×C₄₃	D₄	1	2	yes	—
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	#95×C₄₃	—	2	4	no	—
70	C₂×Q₈×C₄₃	—	2	2	no	—
71	C₂×Q₈×C₄₃	—	2	4	no	—
72	C₂×Q₈×C₄₃	—	2	4	no	—
73	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₃	—	2	2	no	—
82	C₂²×Dic₄₃	—	2	2	no	—
83	C₂×Dic₈₆	—	2	4	no	—
84	C₂×Dic₈₆	—	2	4	no	—
85	C₂×Dic₈₆	—	2	2	no	—
86	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
87	C₄⋊C₄×C₄₃	—	2	8	no	—
88	C₄⋊C₄×C₄₃	—	2	4	no	—
89	C₄⋊C₄×C₄₃	—	4	4	no	—
90	C₄⋊C₄×C₄₃	—	4	4	no	—
91	C₄²×C₄₃	—	2	4	no	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	4	4	no	—
94	C₄²×C₄₃	Q₁₆	1	8	yes	complemented
95	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₃	C₂×D₄	1	4	yes	—
97	C₄²×C₄₃	#41	1	2	yes	—
98	#10621885	—	2	32	no	—
99	C₄⋊Dic₄₃	—	2	8	no	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	C₄⋊Dic₄₃	—	4	4	no	—
102	Dic₄₃⋊C₄	—	4	4	no	—
103	C₄×Dic₄₃	—	4	4	no	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
108	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
109	C₂×C₄₃⋊C₈	#41	1	4	yes	—
110	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₄₃	—	2	8	no	—
112	Q₈×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	4	no	—
121	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
122	C₂×C₄×C₄₃	#127	1	2	yes	—
123	C₂×C₄×C₄₃	#127	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
125	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
126	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
127	C₂×C₄×C₄₃	#135	1	4	yes	—
128	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₄₃	—	2	4	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	—	4	4	no	—
137	Dic₈₆	—	2	8	no	—
138	Dic₈₆	—	4	4	no	—
139	C₄₃⋊C₈	—	2	4	no	—
140	C₄₃⋊C₈	C₄×D₄	1	8	yes	—
141	C₄×#494	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
144	C₂²×C₄₃	#532	1	1	yes	characteristic
145	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
146	C₂²×C₄₃	C₂×#135	1	2	yes	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	—	2	4	no	—
149	C₄×C₄₃	—	4	4	no	—
150	C₄×C₄₃	—	8	2	no	—
151	C₄×C₄₃	C₄×Q₁₆	1	8	yes	complemented
152	C₄×C₄₃	#494	1	4	yes	complemented
153	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
154	C₄×C₄₃	#532	1	2	yes	—
155	Dic₄₃	—	4	4	no	—
156	Dic₄₃	—	8	2	no	—
157	C₂×#494	—	43	1	no	—
158	C₂×C₄×Q₁₆	—	43	4	no	—
159	C₄×C₄⋊Q₈	—	43	1	no	—
160	C₄×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
161	#1969	—	43	4	no	—
162	#1956	—	43	1	no	—
163	#1956	—	43	1	no	—
164	#1845	—	43	1	no	—
165	C₄²×C₈	—	43	1	no	—
166	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
167	C₂×C₄₃	C₂×C₄×Q₁₆	1	2	yes	—
168	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
169	C₂×C₄₃	#3147	1	1	yes	characteristic
170	C₂×C₄₃	#3170	1	2	yes	—
171	C₂²×Q₁₆	—	43	4	no	—
172	C₂×C₄⋊Q₈	—	43	1	no	—
173	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
174	C₂×C₄×Q₈	—	43	2	no	—
175	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
176	#494	—	43	8	no	—
177	C₄×Q₁₆	—	86	16	no	—
178	C₂×#107	—	43	4	no	—
179	C₂×#103	—	43	4	no	—
180	C₂×#103	—	43	4	no	—
181	C₂×C₄×C₈	—	43	1	no	—
182	C₂×C₄×C₈	—	43	2	no	—
183	C₂×C₄×C₈	—	43	1	no	—
184	C₂×C₄×C₈	—	43	2	no	—
185	#391	—	43	2	no	—
186	#391	C₂×D₄₃	1	2	yes	—
187	#389	—	2	2	no	—
188	#389	—	86	2	no	—
189	#384	—	43	1	no	—
190	#384	C₂×D₄₃	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	2	2	no	—
192	C₄×C₄⋊C₄	—	86	2	no	—
193	C₄³	C₂×D₄₃	1	1	yes	characteristic
194	C₄₃	C₄×#494	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	43	2	no	—
196	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
197	C₂×Q₁₆	—	43	16	no	—
198	C₂×Q₁₆	—	86	8	no	—
199	C₂×Q₁₆	—	86	8	no	—
200	C₂²×C₈	—	43	2	no	—
201	C₂²×C₈	—	43	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	43	4	no	—
204	C₄⋊Q₈	—	86	2	no	—
205	C₄⋊Q₈	C₄×D₄₃	1	4	yes	—
206	C₄×Q₈	—	2	8	no	—
207	C₄×Q₈	—	86	8	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	43	2	no	—
212	C₂×C₄⋊C₄	—	86	2	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₄⋊C₄	—	86	2	no	—
215	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	86	2	no	—
218	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
219	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
220	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	#107	—	86	8	no	—
224	#103	—	86	8	no	—
225	#103	—	86	8	no	—
226	C₄×C₈	—	43	4	no	—
227	C₄×C₈	—	43	8	no	—
228	C₄×C₈	—	86	4	no	—
229	C₄×C₈	—	86	2	no	—
230	#95	—	2	4	no	—
231	#95	—	86	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	43	4	no	—
236	C₂×Q₈	—	86	2	no	—
237	C₂×Q₈	—	86	4	no	—
238	C₂×Q₈	—	86	4	no	—
239	C₂×Q₈	C₂×C₄×D₄₃	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	86	2	no	—
243	C₂²×C₄	—	86	2	no	—
244	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
249	Q₁₆	—	86	32	no	—
250	C₂×C₈	—	43	4	no	—
251	C₂×C₈	—	43	4	no	—
252	C₂×C₈	—	43	4	no	—
253	C₂×C₈	—	43	4	no	—
254	C₂×C₈	—	86	2	no	—
255	C₂×C₈	—	86	2	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	86	8	no	—
261	C₄⋊C₄	—	86	4	no	—
262	C₄⋊C₄	—	172	4	no	—
263	C₄⋊C₄	—	172	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	172	4	no	—
268	C₄²	#10621885	1	8	yes	complemented
269	C₄²	C₂×C₄×D₄₃	1	2	yes	—
270	C₄²	#10621898	1	2	yes	—
271	C₄²	D₄×D₄₃	1	4	yes	—
272	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
273	Q₈	—	2	8	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	86	8	no	—
276	Q₈	—	172	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	86	4	no	—
286	C₂×C₄	—	172	4	no	—
287	C₂×C₄	—	172	2	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	—	172	2	no	—
290	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
291	C₂×C₄	#13063321	1	2	yes	—
292	C₂×C₄	#13063321	1	2	yes	—
293	C₂×C₄	C₂×#10621885	1	4	yes	—
294	C₂×C₄	C₂×#10621885	1	4	yes	—
295	C₂×C₄	C₂×#10621885	1	4	yes	—
296	C₂×C₄	#13063345	1	4	yes	—
297	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
299	C₂×C₄	#41×D₄₃	1	2	yes	—
300	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
301	C₈	—	43	8	no	—
302	C₈	—	86	4	no	—
303	C₂²	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
304	C₂²	C₂×#13063321	1	1	yes	characteristic
305	C₂²	#14583445	1	1	yes	characteristic
306	C₂²	C₂²×#10621885	1	2	yes	—
307	C₂²	C₂×#13063345	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	344	2	no	—
313	C₄	—	172	4	no	—
314	C₄	C₄×#10621885	1	8	yes	complemented
315	C₄	#14583278	1	4	yes	direct product factor
316	C₄	C₄×D₄×D₄₃	1	2	yes	—
317	C₄	#14583446	1	2	yes	—
318	C₂	C₄×#13063321	1	1	yes	characteristic
319	C₂	C₂×C₄×#10621885	1	2	yes	—
320	C₂	C₂×#14583278	1	1	yes	characteristic
321	C₂	#19145021	1	1	yes	characteristic
322	C₂	#19145177	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊Q₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	C₄×#494
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
13. #14583278	C₄	direct (N × H)
94. C₄²×C₄₃	Q₁₆	semidirect
151. C₄×C₄₃	C₄×Q₁₆	semidirect
152. C₄×C₄₃	#494	semidirect
194. C₄₃	C₄×#494	semidirect
268. C₄²	#10621885	semidirect
314. C₄	C₄×#10621885	semidirect
315. C₄	#14583278	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×#14583278	2
C₂×C₄×#10621885	2
C₄×#13063325	2
#19135455	2
#19135398	2
#19135396	2
#19134754	2
C₄²×C₄₃⋊C₈	2
C₄×C₄⋊Q₈×C₄₃	2
C₄×#494	43

Subgroups — C4×#14583278 #25105032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14583278 #25105032