Subgroups — C₄×#14582926 #25094485

Order 11008.

1174 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582926	C₂	1	2	yes	maximal
2	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19134452	C₂	1	4	yes	maximal, complemented
5	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	4	yes	maximal, complemented
6	#19134292	C₂	1	2	yes	maximal
7	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10621877	C₂²	1	2	yes	—
9	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
10	#14582926	C₄	1	16	yes	direct product factor
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
12	C₄×#10621877	—	2	4	no	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	—	2	4	no	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
18	#14582654	C₂²	1	8	yes	—
19	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₃	C₄	1	4	yes	—
23	C₄×C₂²⋊C₄×C₄₃	C₄	1	4	yes	—
24	C₄₃×#376	C₂²	1	2	yes	—
25	C₄³×C₄₃	C₂²	1	2	yes	—
26	C₂×#10621877	—	2	4	no	—
27	C₂×#10621877	—	2	4	no	—
28	C₂×#10621877	C₂×C₄	1	8	yes	—
29	C₂²×C₄₃⋊C₈	—	2	4	no	—
30	C₂²×C₄₃⋊C₈	—	2	4	no	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
34	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	32	yes	complemented
35	#13063236	—	2	4	no	—
36	C₄×C₄₃⋊C₈	—	2	4	no	—
37	C₄×C₄₃⋊C₈	—	2	4	no	—
38	C₄×C₄₃⋊C₈	—	2	4	no	—
39	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	#117×C₄₃	—	2	4	no	—
42	#117×C₄₃	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₃	D₄	1	4	yes	—
50	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
51	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#95×C₄₃	C₂×C₄	1	8	yes	—
56	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	2	8	no	—
58	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₃	#41	1	4	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₃	—	2	4	no	—
70	C₄⋊C₄×C₄₃	C₄²	1	8	yes	—
71	C₂²⋊C₄×C₄₃	—	2	4	no	—
72	C₂²⋊C₄×C₄₃	C₄²	1	8	yes	—
73	C₄²×C₄₃	—	2	16	no	—
74	C₄²×C₄₃	—	2	4	no	—
75	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₃	C₂×D₄	1	2	yes	—
78	C₄²×C₄₃	C₂×D₄	1	2	yes	—
79	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
81	#10621877	—	2	16	no	—
82	C₂×C₄₃⋊C₈	—	2	16	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
88	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
89	C₂³×C₄₃	—	2	4	no	—
90	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
91	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₄₃	#117	1	2	yes	—
93	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	—	2	8	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	16	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	2	8	no	—
99	C₂×C₄×C₄₃	C₂×C₄²	1	8	yes	—
100	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₃	#117	1	2	yes	—
105	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
106	C₂×C₄×C₄₃	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₄₃	#131	1	8	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
112	C₄₃⋊C₈	—	2	8	no	—
113	C₄₃⋊C₈	—	2	8	no	—
114	C₄×#424	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	4	no	—
116	C₂²×C₄₃	—	2	4	no	—
117	C₂²×C₄₃	—	2	4	no	—
118	C₂²×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
119	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
122	C₂²×C₄₃	C₂×#131	1	4	yes	—
123	C₄×C₄₃	—	2	8	no	—
124	C₄×C₄₃	—	2	4	no	—
125	C₄×C₄₃	—	4	4	no	—
126	C₄×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
127	C₄×C₄₃	#424	1	8	yes	complemented
128	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
129	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
130	C₂×#424	—	43	2	no	—
131	C₂×C₄×M₁₆	—	43	1	no	—
132	C₂²×C₄×C₈	—	43	1	no	—
133	C₄×#117	D₄₃	1	1	yes	characteristic
134	#1768	—	43	4	no	—
135	C₄×C₄⋊C₈	—	43	4	no	—
136	#1729	—	43	2	no	—
137	C₂×C₄₃	—	2	4	no	—
138	C₂×C₄₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄₃	C₄×#131	1	2	yes	—
140	C₂×C₄₃	C₂×#424	1	2	yes	—
141	C₂×C₄₃	#2922	1	2	yes	—
142	C₂²×M₁₆	—	43	2	no	—
143	C₂³×C₈	—	43	2	no	—
144	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
145	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
146	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
147	#424	—	43	16	no	—
148	C₂×C₄⋊C₈	—	43	8	no	—
149	C₄×M₁₆	—	86	4	no	—
150	C₂×#97	—	43	2	no	—
151	C₂×C₄×C₈	—	43	2	no	—
152	C₂×C₄×C₈	—	43	2	no	—
153	C₂×C₄×C₈	—	43	2	no	—
154	C₂×C₄×C₈	—	86	4	no	—
155	C₄×C₄⋊C₄	Dic₄₃	1	4	yes	—
156	C₄×C₂²⋊C₄	Dic₄₃	1	4	yes	—
157	#376	C₂×D₄₃	1	2	yes	—
158	C₄³	C₂×D₄₃	1	2	yes	—
159	#336	—	43	8	no	—
160	C₄₃	C₄×#424	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
163	C₂×M₁₆	—	43	8	no	—
164	C₂×M₁₆	—	86	4	no	—
165	C₂×M₁₆	—	86	4	no	—
166	C₂²×C₈	—	43	8	no	—
167	C₂²×C₈	—	43	4	no	—
168	C₂²×C₈	—	43	4	no	—
169	C₂²×C₈	—	86	4	no	—
170	C₂²×C₈	—	86	4	no	—
171	#117	—	2	4	no	—
172	#117	C₄×D₄₃	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
177	C₂×C₄²	Dic₈₆	1	4	yes	—
178	C₂×C₄²	D₁₇₂	1	4	yes	—
179	C₂×C₄²	C₂×Dic₄₃	1	8	yes	—
180	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
181	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
182	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	C₄⋊C₈	—	43	32	no	—
186	#97	—	86	4	no	—
187	C₄×C₈	—	86	4	no	—
188	C₄×C₈	—	86	4	no	—
189	C₄×C₈	—	86	4	no	—
190	#95	C₂×Dic₄₃	1	8	yes	—
191	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
194	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
196	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
197	C₂²×C₄	#10621898	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
203	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
204	M₁₆	—	86	16	no	—
205	C₂×C₈	—	43	8	no	—
206	C₂×C₈	—	43	8	no	—
207	C₂×C₈	—	86	16	no	—
208	C₂×C₈	—	86	4	no	—
209	C₂×C₈	—	86	4	no	—
210	C₂×C₈	—	86	4	no	—
211	C₂×C₈	—	86	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	C₄×Dic₄₃	1	8	yes	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	C₄×Dic₄₃	1	8	yes	—
216	C₄²	—	2	16	no	—
217	C₄²	—	2	4	no	—
218	C₄²	C₄⋊Dic₄₃	1	8	yes	—
219	C₄²	C₂×Dic₈₆	1	2	yes	—
220	C₄²	C₂×Dic₈₆	1	2	yes	—
221	C₄²	C₂×C₄×D₄₃	1	8	yes	—
222	C₄²	C₂×D₁₇₂	1	2	yes	—
223	C₄²	C₂×D₁₇₂	1	2	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
226	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
227	C₂³	#13063284	1	2	yes	—
228	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	C₂×C₄×Dic₄₃	1	8	yes	—
235	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
239	C₂×C₄	#13063284	1	2	yes	—
240	C₂×C₄	C₄×D₁₇₂	1	8	yes	—
241	C₂×C₄	C₄×Dic₈₆	1	8	yes	—
242	C₂×C₄	#13063332	1	8	yes	—
243	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
244	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
245	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
246	C₂×C₄	C₂×#10621898	1	2	yes	—
247	C₈	—	86	8	no	—
248	C₈	—	86	8	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₄×C₄⋊Dic₄₃	1	4	yes	—
253	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
254	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
255	C₂²	C₂×#13063284	1	1	yes	characteristic
256	C₂²	C₂×#13063332	1	4	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	2	4	no	—
259	C₄	—	4	4	no	—
260	C₄	C₄×C₄⋊Dic₄₃	1	4	yes	—
261	C₄	#14582926	1	8	yes	direct product factor
262	C₄	C₂×C₄×D₁₇₂	1	2	yes	—
263	C₄	C₂×C₄×Dic₈₆	1	2	yes	—
264	C₂	—	2	4	no	—
265	C₂	C₂×C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
266	C₂	C₄×#13063332	1	2	yes	—
267	C₂	C₂×#14582926	1	2	yes	—
268	C₂	#19143239	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	C₄×#424
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. #19134452	C₂	semidirect
5. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
10. #14582926	C₄	direct (N × H)
34. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
127. C₄×C₄₃	#424	semidirect
160. C₄₃	C₄×#424	semidirect
261. C₄	#14582926	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×#14582926	2
C₂×C₄×#10621877	2
C₂²×C₄×C₄₃⋊C₈	2
#19134452	2
C₄×C₄⋊(C₄₃⋊C₈)	2
#19134292	2
C₄×#117×C₄₃	2
C₄×#424	43

Subgroups — C4×#14582926 #25094485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582926 #25094485