Subgroups — C₄×#14582708 #25011848

Order 11008.

466 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063347	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582708	C₂	1	2	yes	maximal
3	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132702	C₂	1	2	yes	maximal, complemented
5	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063347	C₂²	1	1	yes	characteristic
7	C₄×#10621898	—	2	1	no	—
8	C₂×#13063265	C₂²	1	2	yes	—
9	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic, complemented
10	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic, complemented
11	C₄×#10621873	—	2	1	no	—
12	C₄×#10621873	—	2	1	no	—
13	C₄×D₃₄₄	C₄	1	1	yes	characteristic, complemented
14	C₄×D₃₄₄	C₄	1	1	yes	characteristic, complemented
15	#14582937	C₂²	1	1	yes	characteristic
16	#14582896	—	2	1	no	—
17	#14582708	C₄	1	8	yes	direct product factor
18	#14582675	—	2	1	no	—
19	C₄×C₄₃⋊C₁₆	—	2	1	no	—
20	C₂×#106×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
23	C₄₃×#340	—	2	1	no	—
24	C₂×#10621898	—	2	1	no	—
25	#13063347	—	2	2	no	—
26	#13063347	C₂×C₄	1	4	yes	—
27	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
28	C₂×#10621873	—	2	1	no	—
29	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₈₆	—	2	1	no	—
31	C₄×Dic₈₆	—	2	1	no	—
32	C₄×D₁₇₂	—	2	1	no	—
33	C₄×D₁₇₂	—	2	1	no	—
34	C₄×C₂²⋊D₄₃	—	4	1	no	—
35	C₄×C₂²⋊D₄₃	—	4	1	no	—
36	#13063284	—	2	1	no	—
37	#13063282	—	4	1	no	—
38	C₄²×D₄₃	—	4	1	no	—
39	#13063265	—	2	2	no	—
40	#13063265	C₂×C₄	1	4	yes	complemented
41	C₂×C₄₃⋊C₁₆	—	2	2	no	—
42	#13063259	C₂×C₄	1	1	yes	characteristic
43	#13063259	C₂×C₄	1	1	yes	characteristic
44	#13063257	—	2	1	no	—
45	#13063250	—	2	1	no	—
46	#13063250	—	2	1	no	—
47	#13063247	—	2	1	no	—
48	#13063247	—	2	1	no	—
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	—	4	1	no	—
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	#106×C₄₃	—	2	2	no	—
55	#106×C₄₃	C₂×C₄	1	4	yes	complemented
56	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
60	#95×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₃	—	4	1	no	—
66	#10621898	—	2	4	no	—
67	#10621898	—	4	2	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×C₄×D₄₃	—	4	1	no	—
70	C₂×C₄×D₄₃	—	4	1	no	—
71	C₂×C₄×D₄₃	—	4	1	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	#41	1	2	yes	—
80	C₄⋊C₄×C₄₃	—	2	4	no	—
81	C₄⋊C₄×C₄₃	—	4	2	no	—
82	C₄²×C₄₃	—	8	2	no	—
83	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
86	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
87	C₂²⋊Dic₄₃	—	4	1	no	—
88	(C₂×D₄₃)⋊C₄	—	4	1	no	—
89	(C₂×D₄₃)⋊C₄	—	4	1	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	Dic₄₃⋊C₄	—	4	1	no	—
93	Dic₄₃⋊C₄	—	4	1	no	—
94	C₄×Dic₄₃	—	4	1	no	—
95	C₄×Dic₄₃	—	4	1	no	—
96	C₄×Dic₄₃	—	4	1	no	—
97	Dic₁₇₂	—	2	1	no	—
98	Dic₁₇₂	C₄²	1	2	yes	complemented
99	D₃₄₄	—	2	1	no	—
100	D₃₄₄	C₄²	1	2	yes	complemented
101	#10621873	—	2	4	no	—
102	C₄₃⋊C₁₆	—	2	4	no	—
103	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₄₃	—	4	1	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	4	1	no	—
107	C₂×C₄×C₄₃	—	4	4	no	—
108	C₂×C₄×C₄₃	—	8	2	no	—
109	C₂×C₄×C₄₃	—	8	2	no	—
110	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#135	1	2	yes	—
118	C₂²⋊D₄₃	—	4	4	no	—
119	C₂×Dic₄₃	—	4	1	no	—
120	C₂×Dic₄₃	—	4	1	no	—
121	C₂×Dic₄₃	—	4	1	no	—
122	D₁₇₂	—	2	2	no	—
123	D₁₇₂	—	4	1	no	—
124	C₄×D₄₃	—	4	4	no	—
125	C₄×D₄₃	—	8	2	no	—
126	C₄×D₄₃	—	8	2	no	—
127	Dic₈₆	—	2	2	no	—
128	Dic₈₆	—	4	1	no	—
129	C₈×C₄₃	—	2	1	no	—
130	C₈×C₄₃	—	2	1	no	—
131	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
132	C₈×C₄₃	#117	1	1	yes	characteristic
133	C₈×C₄₃	C₄×D₄	1	2	yes	—
134	C₄×#360	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	—	2	1	no	—
137	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₄₃	C₄×D₈	1	2	yes	—
139	C₂×D₄₃	—	4	2	no	—
140	C₂×D₄₃	—	8	1	no	—
141	C₂×D₄₃	—	8	1	no	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	8	4	no	—
146	C₄×C₄₃	#360	1	2	yes	complemented
147	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₄₃	#416	1	1	yes	characteristic
149	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
150	Dic₄₃	—	4	2	no	—
151	Dic₄₃	—	8	1	no	—
152	Dic₄₃	—	8	1	no	—
153	C₄×#135	—	43	1	no	—
154	C₂×#360	—	43	2	no	—
155	C₄×M₃₂	—	43	1	no	—
156	C₄×#106	D₄₃	1	1	yes	characteristic, complemented
157	#1386	—	43	2	no	—
158	C₂×C₄₃	—	2	2	no	—
159	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
160	C₂×C₄₃	C₂×#360	1	1	yes	characteristic
161	C₂×C₄₃	#2145	1	1	yes	characteristic
162	D₄₃	—	8	2	no	—
163	C₂×#135	—	43	1	no	—
164	C₄×#41	—	86	1	no	—
165	C₂×M₃₂	—	43	2	no	—
166	C₄×Q₁₆	—	43	1	no	—
167	C₄×Q₁₆	—	43	1	no	—
168	C₄×SD₁₆	—	86	1	no	—
169	C₄×SD₁₆	—	86	1	no	—
170	C₄×D₈	—	43	1	no	—
171	C₄×D₈	—	43	1	no	—
172	#427	—	43	1	no	—
173	C₂×#106	C₂×D₄₃	1	2	yes	—
174	#416	—	86	1	no	—
175	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	2	1	no	—
177	#360	—	43	8	no	—
178	#346	—	86	1	no	—
179	C₄×C₁₆	—	86	1	no	—
180	#340	—	2	1	no	—
181	C₄₃	C₄×#360	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	86	1	no	—
183	#135	—	43	4	no	—
184	#135	—	86	2	no	—
185	C₂×Q₁₆	—	43	1	no	—
186	C₂×SD₁₆	—	86	1	no	—
187	C₂×D₈	—	43	1	no	—
188	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	C₄×Q₈	—	86	1	no	—
191	C₄×Q₈	—	86	1	no	—
192	C₄×D₄	—	86	1	no	—
193	C₄×D₄	—	86	1	no	—
194	C₄×D₄	—	172	1	no	—
195	C₄×D₄	—	172	1	no	—
196	#117	—	86	1	no	—
197	#117	—	172	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄²	—	4	1	no	—
200	C₂×C₄²	—	172	1	no	—
201	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
202	M₃₂	—	43	4	no	—
203	M₃₂	—	86	2	no	—
204	C₂×C₁₆	—	86	2	no	—
205	#107	—	43	1	no	—
206	#107	—	43	1	no	—
207	#106	—	2	2	no	—
208	#106	—	86	1	no	—
209	#106	C₄×D₄₃	1	4	yes	complemented
210	#103	—	86	1	no	—
211	#103	—	86	1	no	—
212	#102	—	86	1	no	—
213	#102	—	86	1	no	—
214	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
215	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
216	C₄×C₈	D₁₇₂	1	1	yes	characteristic
217	C₄×C₈	D₁₇₂	1	1	yes	characteristic
218	#95	—	4	1	no	—
219	#41	—	86	4	no	—
220	#41	—	172	2	no	—
221	C₂×Q₈	—	86	1	no	—
222	C₂×D₄	—	86	1	no	—
223	C₂×D₄	—	172	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	172	1	no	—
226	C₂²×C₄	—	172	1	no	—
227	C₂²×C₄	—	172	1	no	—
228	C₂²×C₄	#10621882	1	1	yes	characteristic
229	C₂²×C₄	#10621882	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
231	Q₁₆	—	43	2	no	—
232	Q₁₆	—	86	1	no	—
233	SD₁₆	—	86	4	no	—
234	D₈	—	43	2	no	—
235	D₈	—	86	1	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
238	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
241	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂×C₈	#10621898	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	86	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	172	1	no	—
248	C₄⋊C₄	—	172	1	no	—
249	C₂²⋊C₄	—	172	1	no	—
250	C₂²⋊C₄	—	172	1	no	—
251	C₂²⋊C₄	—	172	1	no	—
252	C₄²	—	8	2	no	—
253	C₄²	—	172	1	no	—
254	C₄²	—	172	1	no	—
255	C₄²	—	172	1	no	—
256	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
257	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
258	C₄²	#10621884	1	1	yes	characteristic
259	C₄²	#10621884	1	1	yes	characteristic
260	C₁₆	—	86	4	no	—
261	C₂³	—	172	1	no	—
262	C₂³	C₂×#10621882	1	1	yes	characteristic
263	Q₈	—	86	2	no	—
264	Q₈	—	172	1	no	—
265	D₄	—	86	2	no	—
266	D₄	—	172	4	no	—
267	D₄	—	172	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	344	2	no	—
274	C₂×C₄	—	344	2	no	—
275	C₂×C₄	—	172	1	no	—
276	C₂×C₄	—	172	1	no	—
277	C₂×C₄	—	172	1	no	—
278	C₂×C₄	—	172	4	no	—
279	C₂×C₄	#13063245	1	1	yes	characteristic
280	C₂×C₄	#13063245	1	1	yes	characteristic
281	C₂×C₄	#13063245	1	1	yes	characteristic
282	C₂×C₄	#13063245	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
285	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
286	C₂×C₄	#13063345	1	2	yes	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₄²×D₄₃	1	1	yes	characteristic
290	C₈	#13063282	1	1	yes	characteristic
291	C₈	C₄×D₁₇₂	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	344	1	no	—
295	C₂²	—	344	1	no	—
296	C₂²	—	172	2	no	—
297	C₂²	C₂×#13063245	1	1	yes	characteristic
298	C₂²	C₄×#10621882	1	2	yes	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	4	no	—
303	C₄	—	344	1	no	—
304	C₄	—	344	1	no	—
305	C₄	—	172	2	no	—
306	C₄	#14582708	1	2	yes	direct product factor
307	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
308	C₄	#14582893	1	1	yes	characteristic
309	C₄	C₄×#10621884	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	344	2	no	—
312	C₂	C₄×#13063245	1	1	yes	characteristic
313	C₂	C₂×#14582708	1	1	yes	characteristic
314	C₂	#19136330	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063245
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	C₄×#360
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
3. C₄×#13063265	C₂	semidirect
4. #19132702	C₂	semidirect
5. C₄×#106×C₄₃	C₂	semidirect
9. C₄×Dic₁₇₂	C₄	semidirect
10. C₄×Dic₁₇₂	C₄	semidirect
13. C₄×D₃₄₄	C₄	semidirect
14. C₄×D₃₄₄	C₄	semidirect
17. #14582708	C₄	direct (N × H)
40. #13063265	C₂×C₄	semidirect
55. #106×C₄₃	C₂×C₄	semidirect
98. Dic₁₇₂	C₄²	semidirect
100. D₃₄₄	C₄²	semidirect
146. C₄×C₄₃	#360	semidirect
156. C₄×#106	D₄₃	semidirect
181. C₄₃	C₄×#360	semidirect
209. #106	C₄×D₄₃	semidirect
306. C₄	#14582708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13063347	2
C₂×#14582708	2
C₄×#13063265	2
#19132702	2
C₄×#106×C₄₃	2
C₄×#360	43

Subgroups — C4×#14582708 #25011848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582708 #25011848