Subgroups — C₄×#14583265 #25104672

Order 11008.

970 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583265	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
3	#19135565	C₂	1	1	yes	maximal, characteristic, complemented
4	#19135418	C₂	1	1	yes	maximal, characteristic
5	#19135291	C₂	1	4	yes	maximal, complemented
6	#19134728	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13063245	C₂	1	4	yes	maximal, complemented
8	C₄×#13063236	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#125×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063322	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₇₂	—	2	2	no	—
12	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
13	#14583265	C₄	1	8	yes	direct product factor
14	C₂×#13063245	C₂²	1	4	yes	—
15	C₂×#13063245	C₂²	1	4	yes	—
16	C₂×#13063236	C₂²	1	2	yes	—
17	C₂×#13063236	C₂²	1	1	yes	characteristic
18	C₂×#13063236	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
20	#14582801	—	2	2	no	—
21	#14582801	C₂²	1	2	yes	—
22	#14582769	C₂²	1	1	yes	characteristic
23	#14582661	C₂²	1	4	yes	—
24	#14582661	C₂²	1	4	yes	—
25	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#389	C₂²	1	2	yes	—
27	C₄₃×#387	—	2	2	no	—
28	C₄₃×#382	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
31	C₄³×C₄₃	C₂²	1	1	yes	characteristic
32	C₂²×D₁₇₂	—	2	2	no	—
33	C₂²×D₁₇₂	C₂³	1	2	yes	—
34	C₂²×C₄×D₄₃	—	2	2	no	—
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
37	#13063322	—	2	2	no	—
38	#13063322	C₂×C₄	1	4	yes	—
39	C₄×D₁₇₂	—	2	8	no	—
40	C₄×D₁₇₂	—	4	8	no	—
41	C₂×C₄⋊Dic₄₃	—	2	2	no	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
44	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
45	#13063245	—	2	8	no	—
46	#13063245	C₂×C₄	1	16	yes	complemented
47	#13063236	—	2	2	no	—
48	#13063236	—	2	4	no	—
49	#13063236	C₂×C₄	1	4	yes	complemented
50	C₄×C₄₃⋊C₈	—	2	4	no	—
51	#125×C₄₃	—	2	2	no	—
52	#125×C₄₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₃	—	2	2	no	—
58	C₂×C₄²×C₄₃	D₄	1	2	yes	—
59	C₂×C₄²×C₄₃	D₄	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	#95×C₄₃	—	2	2	no	—
64	#95×C₄₃	—	2	2	no	—
65	#95×C₄₃	—	2	2	no	—
66	C₂³×D₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	2	2	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₃	—	2	2	no	—
75	C₂×D₁₇₂	—	2	8	no	—
76	C₂×D₁₇₂	—	2	2	no	—
77	C₂×D₁₇₂	—	2	4	no	—
78	C₂×D₁₇₂	—	2	4	no	—
79	C₂×D₁₇₂	—	4	4	no	—
80	C₂×D₁₇₂	—	4	4	no	—
81	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
82	C₂×C₄×D₄₃	—	4	4	no	—
83	C₂×C₄×D₄₃	—	4	8	no	—
84	C₄⋊C₄×C₄₃	—	2	8	no	—
85	C₄⋊C₄×C₄₃	—	2	2	no	—
86	C₄⋊C₄×C₄₃	—	2	4	no	—
87	C₄⋊C₄×C₄₃	—	4	4	no	—
88	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₃	—	2	4	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	4	2	no	—
92	C₄²×C₄₃	—	4	4	no	—
93	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₃	#41	1	2	yes	—
95	C₄²×C₄₃	#41	1	4	yes	—
96	(C₂×D₄₃)⋊C₄	—	4	8	no	—
97	(C₂×D₄₃)⋊C₄	—	4	4	no	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	4	4	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₃⋊C₈	#41	1	4	yes	—
106	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	4	no	—
108	C₂²×D₄₃	—	4	2	no	—
109	C₂²×D₄₃	—	4	2	no	—
110	C₂²×D₄₃	—	4	4	no	—
111	C₂²×D₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₄₃	#124	1	2	yes	—
126	C₂×C₄×C₄₃	#124	1	2	yes	—
127	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
128	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
132	C₂×Dic₄₃	—	4	2	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	D₁₇₂	—	2	8	no	—
135	D₁₇₂	—	4	16	no	—
136	D₁₇₂	—	4	4	no	—
137	C₄×D₄₃	—	8	8	no	—
138	C₄₃⋊C₈	—	2	4	no	—
139	C₄₃⋊C₈	—	2	4	no	—
140	C₄×#490	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
143	C₂²×C₄₃	#532	1	1	yes	characteristic
144	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
145	C₂²×C₄₃	#575	1	2	yes	—
146	C₂×D₄₃	—	4	8	no	—
147	C₂×D₄₃	—	4	4	no	—
148	C₂×D₄₃	—	8	2	no	—
149	C₂×D₄₃	—	8	2	no	—
150	C₂×D₄₃	—	8	4	no	—
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	2	4	no	—
153	C₄×C₄₃	—	4	2	no	—
154	C₄×C₄₃	—	4	4	no	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	#490	1	4	yes	complemented
157	C₄×C₄₃	C₄×#41	1	2	yes	—
158	C₄×C₄₃	#529	1	2	yes	—
159	Dic₄₃	—	8	2	no	—
160	C₂×#490	—	43	1	no	—
161	C₄×#127	—	43	1	no	—
162	C₄×#125	D₄₃	1	1	yes	characteristic, complemented
163	#1992	—	43	1	no	—
164	#1961	—	43	1	no	—
165	#1936	—	43	4	no	—
166	#1836	—	43	1	no	—
167	C₄×#102	—	43	4	no	—
168	C₄×#97	—	43	1	no	—
169	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
170	C₂×C₄₃	C₄×AGL(1,ℤ/8)	1	2	yes	—
171	C₂×C₄₃	C₂×#490	1	1	yes	characteristic
172	C₂×C₄₃	#3135	1	1	yes	characteristic
173	C₂×C₄₃	#3171	1	2	yes	—
174	D₄₃	—	8	4	no	—
175	C₂×#127	—	43	1	no	—
176	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
177	C₂×C₄×D₄	—	43	2	no	—
178	C₂×C₄×D₄	—	86	2	no	—
179	#490	—	43	8	no	—
180	C₂×#102	—	43	4	no	—
181	C₂×#102	—	43	4	no	—
182	C₂×#97	—	43	1	no	—
183	C₂×#97	—	43	2	no	—
184	C₂×#97	—	43	1	no	—
185	C₂×C₄×C₈	—	43	2	no	—
186	#390	—	43	2	no	—
187	#390	—	86	2	no	—
188	#389	C₂×D₄₃	1	2	yes	—
189	#387	—	2	2	no	—
190	#384	—	43	1	no	—
191	#382	C₂×D₄₃	1	1	yes	characteristic
192	C₄×C₄⋊C₄	—	2	2	no	—
193	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
194	C₄³	C₂×D₄₃	1	1	yes	characteristic
195	#340	—	43	4	no	—
196	#340	—	43	4	no	—
197	C₄₃	C₄×#490	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	43	2	no	—
199	C₂²×D₄	—	86	2	no	—
200	C₂³×C₄	—	86	2	no	—
201	C₂²×C₈	—	43	2	no	—
202	C₂²×C₈	—	43	2	no	—
203	#127	—	43	4	no	—
204	#127	—	86	2	no	—
205	#125	—	2	2	no	—
206	#125	C₄×D₄₃	1	4	yes	complemented
207	C₄×D₄	—	86	8	no	—
208	C₄×D₄	—	172	8	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	C₂²×D₄₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	86	2	no	—
216	C₂×C₂²⋊C₄	—	86	4	no	—
217	C₂×C₄²	—	2	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	#102	—	43	16	no	—
224	#102	—	86	8	no	—
225	#97	—	43	4	no	—
226	#97	—	86	4	no	—
227	#97	—	86	2	no	—
228	C₄×C₈	—	86	4	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	C₂⁴	—	86	2	no	—
233	C₂×D₄	—	43	4	no	—
234	C₂×D₄	—	86	8	no	—
235	C₂×D₄	—	86	4	no	—
236	C₂×D₄	—	86	2	no	—
237	C₂×D₄	—	86	4	no	—
238	C₂×D₄	—	172	4	no	—
239	C₂×D₄	—	172	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	86	2	no	—
243	C₂²×C₄	—	172	4	no	—
244	C₂²×C₄	—	172	8	no	—
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	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
250	C₂×C₈	—	43	4	no	—
251	C₂×C₈	—	43	4	no	—
252	C₂×C₈	—	86	2	no	—
253	C₂×C₈	—	86	2	no	—
254	C₂×C₈	—	86	2	no	—
255	C₂×C₈	—	86	2	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	86	4	no	—
261	C₄⋊C₄	—	172	4	no	—
262	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
263	C₂²⋊C₄	—	172	8	no	—
264	C₂²⋊C₄	—	172	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₂×C₄×D₄₃	1	2	yes	—
270	C₄²	#10621898	1	2	yes	—
271	C₄²	#10621902	1	4	yes	—
272	C₂³	—	86	4	no	—
273	C₂³	—	172	2	no	—
274	C₂³	—	172	4	no	—
275	C₂³	—	172	4	no	—
276	C₂³	—	172	2	no	—
277	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
278	D₄	—	86	8	no	—
279	D₄	—	172	16	no	—
280	D₄	—	172	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	344	8	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	4	no	—
296	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
297	C₂×C₄	#13063314	1	2	yes	—
298	C₂×C₄	#13063314	1	2	yes	—
299	C₂×C₄	#13063350	1	4	yes	—
300	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	#41×D₄₃	1	2	yes	—
302	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621902	1	2	yes	—
304	C₈	—	86	4	no	—
305	C₈	—	86	4	no	—
306	C₂²	—	344	2	no	—
307	C₂²	—	344	4	no	—
308	C₂²	—	344	2	no	—
309	C₂²	—	172	4	no	—
310	C₂²	—	172	8	no	—
311	C₂²	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
312	C₂²	C₂×#13063314	1	1	yes	characteristic
313	C₂²	#14583445	1	1	yes	characteristic
314	C₂²	C₂×#13063350	1	2	yes	—
315	C₂²	#14583697	1	2	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	344	2	no	—
322	C₄	#14583265	1	4	yes	direct product factor
323	C₄	C₄×#10621902	1	2	yes	—
324	C₄	#14583436	1	2	yes	—
325	C₂	—	344	4	no	—
326	C₂	C₄×#13063314	1	1	yes	characteristic
327	C₂	C₄×#13063350	1	2	yes	—
328	C₂	C₂×#14583265	1	1	yes	characteristic
329	C₂	#19144943	1	1	yes	characteristic
330	C₂	#19145191	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×#125×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#125	D₄₃
hypocenter	C₄₃	C₄×#490
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
3. #19135565	C₂	semidirect
5. #19135291	C₂	semidirect
6. #19134728	C₂	semidirect
7. C₄×#13063245	C₂	semidirect
8. C₄×#13063236	C₂	semidirect
9. C₄×#125×C₄₃	C₂	semidirect
13. #14583265	C₄	direct (N × H)
46. #13063245	C₂×C₄	semidirect
49. #13063236	C₂×C₄	semidirect
52. #125×C₄₃	C₂×C₄	semidirect
156. C₄×C₄₃	#490	semidirect
162. C₄×#125	D₄₃	semidirect
197. C₄₃	C₄×#490	semidirect
206. #125	C₄×D₄₃	semidirect
322. C₄	#14583265	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×#14583265	2
C₄×#13063322	2
#19135565	2
#19135418	2
#19135291	2
#19134728	2
C₄×#13063245	2
C₄×#13063236	2
C₄×#125×C₄₃	2
C₄×#490	43

Subgroups — C4×#14583265 #25104672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14583265 #25104672