Subgroups — C₄×#14583304 #25105332

Order 11008.

878 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063325	C₂	1	2	yes	maximal
3	#19135556	C₂	1	2	yes	maximal
4	#19135328	C₂	1	4	yes	maximal
5	#19134755	C₂	1	1	yes	maximal, characteristic
6	C₄×#13063259	C₂	1	4	yes	maximal
7	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	C₂²	1	2	yes	—
9	C₂×C₄×Dic₈₆	—	2	2	no	—
10	#14583304	C₄	1	8	yes	direct product factor
11	C₂×#13063259	C₂²	1	4	yes	—
12	C₂×#13063259	C₂²	1	4	yes	—
13	#14582805	—	2	2	no	—
14	#14582797	—	2	2	no	—
15	#14582797	C₂²	1	4	yes	—
16	#14582769	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₃	—	2	2	no	—
18	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
19	#14582662	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	—	2	2	no	—
26	#13063325	—	2	4	no	—
27	#13063325	C₂×C₄	1	8	yes	—
28	C₄×Dic₈₆	—	4	8	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
34	C₂×Dic₄₃⋊C₄	—	2	4	no	—
35	C₂×C₄×Dic₄₃	—	2	4	no	—
36	#13063259	—	2	8	no	—
37	#13063259	C₂×C₄	1	16	yes	—
38	C₄₃⋊#95	—	2	4	no	—
39	C₄₃⋊#95	—	2	4	no	—
40	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₄₃	—	2	4	no	—
50	C₄×C₈×C₄₃	—	2	2	no	—
51	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₄₃	Q₈	1	8	yes	complemented
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	2	4	no	—
61	C₂²×Dic₄₃	—	2	4	no	—
62	C₂×Dic₈₆	—	2	8	no	—
63	C₂×Dic₈₆	—	4	4	no	—
64	C₂×Dic₈₆	—	4	4	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₄₃	#41	1	4	yes	—
71	C₄²×C₄₃	—	2	2	no	—
72	C₄²×C₄₃	—	2	2	no	—
73	C₄²×C₄₃	—	2	2	no	—
74	C₄²×C₄₃	D₈	1	4	yes	—
75	C₄²×C₄₃	Q₁₆	1	4	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
78	C₄²×C₄₃	#41	1	2	yes	—
79	C₄⋊Dic₄₃	—	2	8	no	—
80	C₄⋊Dic₄₃	—	2	4	no	—
81	C₄⋊Dic₄₃	—	2	8	no	—
82	C₄⋊Dic₄₃	—	4	4	no	—
83	C₄⋊Dic₄₃	—	4	4	no	—
84	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
85	Dic₄₃⋊C₄	—	4	8	no	—
86	C₄×Dic₄₃	—	4	8	no	—
87	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₃	—	2	1	no	—
89	C₂×C₄×C₄₃	—	2	1	no	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	1	no	—
92	C₂×C₄×C₄₃	—	2	1	no	—
93	C₂×C₄×C₄₃	—	2	1	no	—
94	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
97	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₄₃	#135	1	2	yes	—
105	C₂×C₄×C₄₃	#135	1	2	yes	—
106	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₄₃	—	2	8	no	—
111	C₂×Dic₄₃	—	4	4	no	—
112	C₂×Dic₄₃	—	4	4	no	—
113	C₂×Dic₄₃	—	4	4	no	—
114	C₂×Dic₄₃	—	4	4	no	—
115	C₂×Dic₄₃	—	4	4	no	—
116	Dic₈₆	—	4	16	no	—
117	C₈×C₄₃	—	2	4	no	—
118	C₈×C₄₃	C₄×Q₈	1	8	yes	complemented
119	C₄×#500	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₄₃	#532	1	1	yes	characteristic
123	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
124	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
125	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
126	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	—	2	2	no	—
129	C₄×C₄₃	—	2	2	no	—
130	C₄×C₄₃	C₄×D₈	1	4	yes	—
131	C₄×C₄₃	C₄×Q₁₆	1	4	yes	—
132	C₄×C₄₃	#500	1	4	yes	complemented
133	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
134	C₄×C₄₃	#533	1	2	yes	—
135	Dic₄₃	—	4	8	no	—
136	Dic₄₃	—	8	4	no	—
137	C₂×#500	—	43	1	no	—
138	C₄×C₄⋊Q₈	—	43	2	no	—
139	#1990	—	43	2	no	—
140	#1942	—	43	4	no	—
141	#1845	—	43	1	no	—
142	C₄×#107	—	43	4	no	—
143	C₄²×C₈	D₄₃	1	1	yes	characteristic
144	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
146	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
147	C₂×C₄₃	C₂×#500	1	1	yes	characteristic
148	C₂×C₄₃	#3160	1	1	yes	characteristic
149	C₂×C₄₃	#3169	1	1	yes	characteristic
150	C₂×C₄₃	#3170	1	1	yes	characteristic
151	C₂×C₄⋊Q₈	—	43	2	no	—
152	C₂×C₄×Q₈	—	86	2	no	—
153	#500	—	43	8	no	—
154	C₂×#107	—	43	4	no	—
155	C₂×#107	—	43	4	no	—
156	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
160	#391	—	86	2	no	—
161	#389	—	43	4	no	—
162	#389	—	86	2	no	—
163	#384	—	43	2	no	—
164	C₄×C₄⋊C₄	—	43	4	no	—
165	C₄×C₄⋊C₄	—	86	2	no	—
166	C₄³	C₂×D₄₃	1	1	yes	characteristic
167	#340	—	43	8	no	—
168	C₄₃	C₄×#500	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₈	—	86	2	no	—
170	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
171	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
172	C₄⋊Q₈	—	43	8	no	—
173	C₄⋊Q₈	—	86	4	no	—
174	C₄×Q₈	—	172	8	no	—
175	C₂×C₄⋊C₄	—	43	4	no	—
176	C₂×C₄⋊C₄	—	43	4	no	—
177	C₂×C₄⋊C₄	—	86	2	no	—
178	C₂×C₄⋊C₄	—	86	4	no	—
179	C₂×C₄⋊C₄	—	86	2	no	—
180	C₂×C₄⋊C₄	—	86	2	no	—
181	C₂×C₄²	—	86	4	no	—
182	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
183	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
184	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
185	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	#107	—	43	16	no	—
190	#107	—	86	8	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	Dic₈₆	1	8	yes	complemented
194	C₄×C₈	C₄×D₄₃	1	4	yes	—
195	#95	—	86	4	no	—
196	#95	—	86	4	no	—
197	C₂×Q₈	—	86	8	no	—
198	C₂×Q₈	—	172	4	no	—
199	C₂×Q₈	—	172	4	no	—
200	C₂²×C₄	—	86	4	no	—
201	C₂²×C₄	—	86	4	no	—
202	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
208	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
212	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
213	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
214	C₂×C₈	#10621898	1	4	yes	—
215	C₄⋊C₄	—	43	8	no	—
216	C₄⋊C₄	—	86	8	no	—
217	C₄⋊C₄	—	86	4	no	—
218	C₄⋊C₄	—	86	8	no	—
219	C₄⋊C₄	—	172	8	no	—
220	C₄⋊C₄	—	172	4	no	—
221	C₄⋊C₄	—	172	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	172	8	no	—
226	C₄²	D₃₄₄	1	4	yes	—
227	C₄²	Dic₁₇₂	1	4	yes	—
228	C₄²	C₂×Dic₈₆	1	4	yes	—
229	C₄²	C₂×C₄×D₄₃	1	2	yes	—
230	C₄²	#10621898	1	2	yes	—
231	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
232	Q₈	—	172	16	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	86	8	no	—
240	C₂×C₄	—	172	4	no	—
241	C₂×C₄	—	172	4	no	—
242	C₂×C₄	—	172	4	no	—
243	C₂×C₄	—	172	4	no	—
244	C₂×C₄	—	172	4	no	—
245	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
246	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
247	C₂×C₄	#13063325	1	2	yes	—
248	C₂×C₄	#13063325	1	2	yes	—
249	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
250	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
251	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
252	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
253	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
254	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
255	C₂×C₄	#13063347	1	2	yes	—
256	C₂×C₄	#13063347	1	2	yes	—
257	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
259	C₂×C₄	C₂×#10621898	1	2	yes	—
260	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	C₄×Dic₈₆	1	8	yes	complemented
263	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
264	C₂²	C₂×#13063325	1	1	yes	characteristic
265	C₂²	#14583449	1	1	yes	characteristic
266	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
267	C₂²	C₂²×Dic₁₇₂	1	1	yes	characteristic
268	C₂²	C₂×#13063347	1	1	yes	characteristic
269	C₂²	C₂×#13063347	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	344	4	no	—
274	C₄	—	172	8	no	—
275	C₄	C₄×D₃₄₄	1	4	yes	—
276	C₄	C₄×Dic₁₇₂	1	4	yes	—
277	C₄	#14583304	1	4	yes	direct product factor
278	C₄	C₂×C₄×Dic₈₆	1	2	yes	—
279	C₄	#14583456	1	2	yes	—
280	C₂	C₄×#13063325	1	1	yes	characteristic
281	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic
282	C₂	C₂×C₄×Dic₁₇₂	1	1	yes	characteristic
283	C₂	C₂×#14583304	1	1	yes	characteristic
284	C₂	#19145109	1	1	yes	characteristic
285	C₂	#19145174	1	1	yes	characteristic
286	C₂	#19145181	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	C₄×#500
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
10. #14583304	C₄	direct (N × H)
52. C₄×C₈×C₄₃	Q₈	semidirect
118. C₈×C₄₃	C₄×Q₈	semidirect
132. C₄×C₄₃	#500	semidirect
168. C₄₃	C₄×#500	semidirect
193. C₄×C₈	Dic₈₆	semidirect
262. C₈	C₄×Dic₈₆	semidirect
277. C₄	#14583304	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×#14583304	2
C₄×#13063325	2
#19135556	2
#19135328	2
#19134755	2
C₄×#13063259	2
C₄²×C₈×C₄₃	2
C₄×#500	43

Subgroups — C4×#14583304 #25105332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14583304 #25105332