Subgroups — C₄×#14582993 #25095072

Order 11008.

1110 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582993	C₂	1	2	yes	maximal
3	C₄×#13063290	C₂	1	2	yes	maximal
4	#107×Dic₄₃	C₂	1	2	yes	maximal
5	C₄×#13063248	C₂	1	2	yes	maximal
6	#19134414	C₂	1	4	yes	maximal
7	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
8	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₂×#13063290	C₂²	1	4	yes	—
12	C₄×Dic₁₇₂	C₄	1	8	yes	complemented
13	#14582993	C₄	1	16	yes	direct product factor
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063250	C₂²	1	2	yes	—
16	C₂×#13063248	C₂²	1	4	yes	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
21	C₄×Dic₄₃⋊C₄	—	2	2	no	—
22	#14582750	—	2	4	no	—
23	C₄²×Dic₄₃	—	2	2	no	—
24	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
25	#14582661	C₂²	1	4	yes	—
26	C₂×#107×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	2	yes	—
30	C₂²×Dic₈₆	C₂³	1	2	yes	—
31	C₂×Dic₁₇₂	C₂×C₄	1	8	yes	—
32	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	C₄×Dic₈₆	—	2	4	no	—
35	C₄×Dic₈₆	C₂×C₄	1	8	yes	—
36	#13063290	—	2	8	no	—
37	#13063290	C₂×C₄	1	16	yes	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
39	C₂×Dic₄₃⋊C₄	—	2	2	no	—
40	C₂×Dic₄₃⋊C₄	—	2	4	no	—
41	C₂×C₄×Dic₄₃	—	2	2	no	—
42	C₂×C₄×Dic₄₃	—	2	4	no	—
43	C₂×C₄×Dic₄₃	—	2	4	no	—
44	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
48	#13063259	C₂×C₄	1	4	yes	—
49	#13063250	C₂×C₄	1	8	yes	—
50	#13063248	C₂×C₄	1	16	yes	—
51	C₄×C₄₃⋊C₈	—	2	2	no	—
52	C₈×Dic₄₃	—	2	4	no	—
53	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
54	C₄₃⋊#95	—	2	4	no	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#107×C₄₃	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
62	#95×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
70	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
71	C₂×Dic₈₆	—	2	4	no	—
72	C₂×Dic₈₆	—	2	2	no	—
73	C₂×Dic₈₆	C₂²×C₄	1	8	yes	—
74	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
79	C₄⋊C₄×C₄₃	—	2	4	no	—
80	C₄⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
81	C₄²×C₄₃	—	4	4	no	—
82	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₃	#41	1	2	yes	—
84	C₄⋊Dic₄₃	—	2	2	no	—
85	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
86	Dic₄₃⋊C₄	—	2	8	no	—
87	Dic₄₃⋊C₄	—	2	16	no	—
88	C₄×Dic₄₃	—	2	8	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	16	no	—
91	C₄×Dic₄₃	—	2	4	no	—
92	C₄×Dic₄₃	—	4	8	no	—
93	C₄×Dic₄₃	Q₁₆	1	4	yes	complemented
94	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
95	C₄×Dic₄₃	#41	1	4	yes	—
96	C₂×C₄₃⋊C₈	—	2	2	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
99	Dic₁₇₂	C₄²	1	16	yes	complemented
100	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	8	no	—
103	C₂×C₄×C₄₃	—	4	4	no	—
104	C₂×C₄×C₄₃	—	4	4	no	—
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₄₃	#135	1	2	yes	—
109	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
112	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₄₃	—	2	8	no	—
114	C₂×Dic₄₃	—	2	4	no	—
115	C₂×Dic₄₃	—	2	2	no	—
116	C₂×Dic₄₃	—	2	2	no	—
117	C₂×Dic₄₃	—	2	4	no	—
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	C₄×D₄	1	4	yes	—
120	C₂×Dic₄₃	C₄×D₄	1	4	yes	—
121	C₂×Dic₄₃	C₂×Q₁₆	1	4	yes	—
122	C₂×Dic₄₃	#135	1	4	yes	—
123	Dic₈₆	—	2	4	no	—
124	Dic₈₆	—	2	8	no	—
125	Dic₈₆	C₂×C₄²	1	8	yes	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	C₂×C₄²	1	4	yes	—
128	C₄₃⋊C₈	—	2	4	no	—
129	C₄²×Q₁₆	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	C₂×C₄×D₄	1	2	yes	—
131	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂²×C₄₃	C₂×#135	1	2	yes	—
134	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	4	8	no	—
137	C₄×C₄₃	C₄×Q₁₆	1	4	yes	complemented
138	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
140	C₄×C₄₃	C₄×#41	1	2	yes	—
141	Dic₄₃	—	2	4	no	—
142	Dic₄₃	—	2	8	no	—
143	Dic₄₃	—	4	4	no	—
144	Dic₄₃	C₄×Q₁₆	1	8	yes	complemented
145	C₂×C₄×Q₁₆	—	43	2	no	—
146	C₂×C₄×Q₁₆	—	43	1	no	—
147	C₄²×Q₈	—	43	2	no	—
148	C₄×#107	—	43	2	no	—
149	C₄×#107	D₄₃	1	1	yes	characteristic
150	C₄×#103	—	43	2	no	—
151	C₄×#103	—	43	4	no	—
152	C₄²×C₈	—	43	1	no	—
153	C₂×C₄₃	C₄²×D₄	1	1	yes	characteristic
154	C₂×C₄₃	C₂×C₄×Q₁₆	1	2	yes	—
155	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
156	C₂×C₄₃	C₄×#135	1	2	yes	—
157	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
158	C₂²×Q₁₆	—	43	1	no	—
159	C₂×C₄×Q₈	—	43	4	no	—
160	C₂×C₄×Q₈	—	43	2	no	—
161	C₄×Q₁₆	—	43	16	no	—
162	C₄×Q₁₆	—	43	8	no	—
163	C₂×#107	—	43	1	no	—
164	C₂×#107	C₂×D₄₃	1	2	yes	—
165	C₂×#103	—	43	2	no	—
166	C₂×#103	—	43	4	no	—
167	C₂×C₄×C₈	—	43	2	no	—
168	C₂×C₄×C₈	—	43	2	no	—
169	C₂×C₄×C₈	—	43	1	no	—
170	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
171	C₄×C₄⋊C₄	—	43	4	no	—
172	C₄×C₄⋊C₄	—	86	4	no	—
173	C₄×C₄⋊C₄	—	86	2	no	—
174	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
175	C₄³	—	43	1	no	—
176	C₄³	—	86	2	no	—
177	#340	—	43	4	no	—
178	#340	C₂×D₄₃	1	2	yes	—
179	C₄₃	C₄²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	43	2	no	—
181	C₂×Q₁₆	—	43	8	no	—
182	C₂×Q₁₆	—	43	4	no	—
183	C₂²×C₈	—	43	2	no	—
184	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
186	C₄×Q₈	—	43	16	no	—
187	C₄×Q₈	—	43	8	no	—
188	C₄×Q₈	—	86	8	no	—
189	C₄×Q₈	—	86	4	no	—
190	C₂×C₄⋊C₄	—	43	2	no	—
191	C₂×C₄⋊C₄	—	86	4	no	—
192	C₂×C₄⋊C₄	—	86	2	no	—
193	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	43	1	no	—
196	C₂×C₄²	—	43	2	no	—
197	C₂×C₄²	—	43	2	no	—
198	C₂×C₄²	—	43	1	no	—
199	C₂×C₄²	—	86	4	no	—
200	C₂×C₄²	—	86	2	no	—
201	C₂×C₄²	—	86	4	no	—
202	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
203	#107	—	43	4	no	—
204	#107	C₄×D₄₃	1	8	yes	—
205	#103	—	43	8	no	—
206	#103	—	43	16	no	—
207	C₄×C₈	—	43	8	no	—
208	C₄×C₈	—	86	4	no	—
209	C₄×C₈	—	86	2	no	—
210	C₄×C₈	C₄×D₄₃	1	4	yes	—
211	#95	—	2	2	no	—
212	#95	—	86	4	no	—
213	C₂×Q₈	—	43	8	no	—
214	C₂×Q₈	—	43	4	no	—
215	C₂×Q₈	—	86	4	no	—
216	C₂×Q₈	—	86	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	43	2	no	—
219	C₂²×C₄	—	43	2	no	—
220	C₂²×C₄	—	86	2	no	—
221	C₂²×C₄	—	86	2	no	—
222	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
225	Q₁₆	—	43	16	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	43	4	no	—
228	C₂×C₈	—	86	2	no	—
229	C₂×C₈	—	86	2	no	—
230	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
232	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	43	4	no	—
235	C₄⋊C₄	—	86	16	no	—
236	C₄⋊C₄	—	86	8	no	—
237	C₄⋊C₄	—	86	2	no	—
238	C₄⋊C₄	C₂×C₄×D₄₃	1	8	yes	—
239	C₄²	—	4	4	no	—
240	C₄²	—	43	4	no	—
241	C₄²	—	43	4	no	—
242	C₄²	—	43	4	no	—
243	C₄²	—	86	4	no	—
244	C₄²	—	86	16	no	—
245	C₄²	—	86	8	no	—
246	C₄²	—	86	2	no	—
247	C₄²	—	172	8	no	—
248	C₄²	C₂×C₄×D₄₃	1	2	yes	—
249	C₄²	#10621902	1	2	yes	—
250	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
251	Q₈	—	43	8	no	—
252	Q₈	—	86	4	no	—
253	Q₈	—	86	8	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	43	4	no	—
259	C₂×C₄	—	43	4	no	—
260	C₂×C₄	—	43	4	no	—
261	C₂×C₄	—	43	4	no	—
262	C₂×C₄	—	86	8	no	—
263	C₂×C₄	—	86	4	no	—
264	C₂×C₄	—	86	2	no	—
265	C₂×C₄	—	86	2	no	—
266	C₂×C₄	—	86	4	no	—
267	C₂×C₄	—	172	4	no	—
268	C₂×C₄	#13063287	1	2	yes	—
269	C₂×C₄	#13063287	1	2	yes	—
270	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
271	C₂×C₄	#13063344	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
274	C₂×C₄	#41×D₄₃	1	2	yes	—
275	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	86	4	no	—
278	C₈	C₄²×D₄₃	1	4	yes	—
279	C₂²	C₄×D₄×D₄₃	1	2	yes	—
280	C₂²	C₂×#13063287	1	1	yes	characteristic
281	C₂²	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
282	C₂²	#135×D₄₃	1	2	yes	—
283	C₂²	C₂×#13063344	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	4	8	no	—
286	C₄	—	43	8	no	—
287	C₄	—	86	8	no	—
288	C₄	—	86	4	no	—
289	C₄	—	172	4	no	—
290	C₄	#14582993	1	4	yes	direct product factor
291	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
292	C₄	#14583367	1	1	yes	characteristic
293	C₄	C₄×#10621902	1	2	yes	—
294	C₂	C₄×#13063287	1	1	yes	characteristic
295	C₂	C₄×Q₁₆×D₄₃	1	2	yes	—
296	C₂	C₂×#14582993	1	1	yes	characteristic
297	C₂	C₄×#13063344	1	2	yes	—
298	C₂	#19143366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	C₄²×Q₁₆
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₁₇₂	C₄	semidirect
13. #14582993	C₄	direct (N × H)
93. C₄×Dic₄₃	Q₁₆	semidirect
99. Dic₁₇₂	C₄²	semidirect
137. C₄×C₄₃	C₄×Q₁₆	semidirect
144. Dic₄₃	C₄×Q₁₆	semidirect
179. C₄₃	C₄²×Q₁₆	semidirect
290. C₄	#14582993	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×C₄×Dic₁₇₂	2
C₂×#14582993	2
C₄×#13063290	2
#107×Dic₄₃	2
C₄×#13063248	2
#19134414	2
C₄×C₈×Dic₄₃	2
C₄×#107×C₄₃	2
C₄²×Q₁₆	43

Subgroups — C4×#14582993 #25095072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582993 #25095072