Subgroups — C₂²×#14583186 #25964057

Order 11008.

2790 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583186	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13063256	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13063245	C₂	1	2	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₇₂	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₄₃	—	2	1	no	—
9	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×D₁₇₂	—	2	6	no	—
11	C₂×C₄×D₁₇₂	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂²×(C₂×D₄₃)⋊C₄	—	2	1	no	—
14	#14583186	C₂²	1	64	yes	direct product factor
15	C₂×#13063256	C₂²	1	24	yes	complemented
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	12	yes	complemented
17	C₂×#13063245	C₂²	1	24	yes	—
18	C₂³×Q₈×C₄₃	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₄₃	—	2	6	no	—
20	C₂×C₄⋊Q₈×C₄₃	C₂²	1	12	yes	complemented
21	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₂⁴×D₄₃	—	2	1	no	—
25	C₂²×D₁₇₂	—	2	3	no	—
26	C₂²×D₁₇₂	—	2	4	no	—
27	C₂²×D₁₇₂	Q₈	1	8	yes	complemented
28	C₂²×D₁₇₂	C₂³	1	6	yes	—
29	C₂³×Dic₄₃	—	2	1	no	—
30	C₂²×C₄×D₄₃	—	2	12	no	—
31	C₂²×C₄×D₄₃	—	4	8	no	—
32	C₂²×C₄₃⋊C₈	—	2	8	no	—
33	C₂²×C₄₃⋊C₈	C₂³	1	12	yes	—
34	C₄×D₁₇₂	—	2	24	no	—
35	C₄×D₁₇₂	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₄₃	—	2	3	no	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	6	yes	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	12	no	—
39	#13063256	C₂³	1	32	yes	complemented
40	C₄⋊(C₄₃⋊C₈)	C₂³	1	16	yes	complemented
41	#13063245	C₂³	1	32	yes	—
42	C₂²×Q₈×C₄₃	—	2	12	no	—
43	C₂²×Q₈×C₄₃	—	4	8	no	—
44	C₂³×C₄×C₄₃	—	2	2	no	—
45	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₃	—	2	24	no	—
49	C₄⋊Q₈×C₄₃	C₂³	1	16	yes	complemented
50	C₂×C₄⋊C₄×C₄₃	—	2	12	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	6	no	—
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	12	yes	—
53	C₂×C₄²×C₄₃	D₄	1	6	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
55	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂³×D₄₃	—	2	8	no	—
57	C₂³×D₄₃	—	2	6	no	—
58	C₂³×D₄₃	—	4	4	no	—
59	C₂³×D₄₃	—	4	4	no	—
60	C₂×Q₈×C₄₃	—	2	16	no	—
61	C₂×Q₈×C₄₃	—	4	48	no	—
62	C₂²×C₄×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	—	2	12	no	—
64	C₂²×C₄×C₄₃	—	4	8	no	—
65	C₂²×C₄×C₄₃	SD₁₆	1	8	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₄₃	#41	1	4	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
72	C₂²×Dic₄₃	—	2	6	no	—
73	C₂²×Dic₄₃	—	4	4	no	—
74	C₂×D₁₇₂	—	2	24	no	—
75	C₂×D₁₇₂	—	2	6	no	—
76	C₂×D₁₇₂	—	2	12	no	—
77	C₂×D₁₇₂	C₂×Q₈	1	24	yes	complemented
78	C₂×D₁₇₂	C₂⁴	1	4	yes	—
79	C₂×C₄×D₄₃	—	2	16	no	—
80	C₂×C₄×D₄₃	—	4	48	no	—
81	C₄⋊C₄×C₄₃	—	2	16	no	—
82	C₄⋊C₄×C₄₃	—	2	12	no	—
83	C₄⋊C₄×C₄₃	C₂⁴	1	8	yes	—
84	C₄²×C₄₃	C₂×D₄	1	12	yes	—
85	C₄²×C₄₃	C₂⁴	1	4	yes	—
86	(C₂×D₄₃)⋊C₄	—	2	16	no	—
87	C₄⋊Dic₄₃	—	2	6	no	—
88	C₄⋊Dic₄₃	C₂⁴	1	4	yes	—
89	C₂×C₄₃⋊C₈	—	2	24	no	—
90	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
91	C₂³×C₄₃	C₂²⋊Q₈	1	4	yes	—
92	C₂³×C₄₃	C₂×SD₁₆	1	4	yes	—
93	C₂³×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
94	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
95	C₂²×D₄₃	—	2	24	no	—
96	C₂²×D₄₃	—	2	4	no	—
97	C₂²×D₄₃	—	4	12	no	—
98	C₂²×D₄₃	—	4	12	no	—
99	C₂²×D₄₃	—	4	32	no	—
100	Q₈×C₄₃	—	4	32	no	—
101	C₂×C₄×C₄₃	—	2	12	no	—
102	C₂×C₄×C₄₃	—	2	8	no	—
103	C₂×C₄×C₄₃	—	4	24	no	—
104	C₂×C₄×C₄₃	C₂²⋊Q₈	1	12	yes	—
105	C₂×C₄×C₄₃	C₂×SD₁₆	1	24	yes	—
106	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
107	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₄₃	C₂²×Q₈	1	6	yes	—
110	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
111	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₄₃	—	2	4	no	—
113	C₂×Dic₄₃	—	4	12	no	—
114	D₁₇₂	—	2	24	no	—
115	D₁₇₂	—	2	4	no	—
116	D₁₇₂	C₂²×Q₈	1	8	yes	complemented
117	C₄×D₄₃	—	4	32	no	—
118	C₄₃⋊C₈	—	2	8	no	—
119	C₂²×#476	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	#476	1	16	yes	complemented
121	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	6	yes	—
122	C₂²×C₄₃	C₂²×SD₁₆	1	6	yes	—
123	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
124	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
125	C₂×D₄₃	—	2	8	no	—
126	C₂×D₄₃	—	4	4	no	—
127	C₂×D₄₃	—	4	4	no	—
128	C₂×D₄₃	—	4	48	no	—
129	C₄×C₄₃	—	2	4	no	—
130	C₄×C₄₃	—	4	8	no	—
131	C₄×C₄₃	C₂×C₂²⋊Q₈	1	6	yes	—
132	C₄×C₄₃	C₂²×SD₁₆	1	8	yes	—
133	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
134	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
135	Dic₄₃	—	4	4	no	—
136	C₂²×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
137	C₂²×C₄×D₄	—	43	1	no	—
138	C₂×#476	—	43	24	no	—
139	C₂²×#106	—	43	2	no	—
140	C₂²×C₄⋊C₈	—	43	1	no	—
141	C₂²×#102	—	43	2	no	—
142	C₂×C₄₃	C₂×#476	1	12	yes	complemented
143	C₂×C₄₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄₃	C₂³×SD₁₆	1	1	yes	characteristic
145	C₂×C₄₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
146	D₄₃	—	4	8	no	—
147	C₂³×Q₈	—	2	1	no	—
148	C₂³×D₄	—	43	1	no	—
149	C₂⁴×C₄	—	86	1	no	—
150	C₂³×C₈	—	43	2	no	—
151	C₂×C₄⋊Q₈	—	2	6	no	—
152	C₂×C₄⋊Q₈	C₂×D₄₃	1	12	yes	complemented
153	C₂×C₄×D₄	—	43	12	no	—
154	C₂×C₄×D₄	—	86	6	no	—
155	C₂²×C₄⋊C₄	—	2	1	no	—
156	C₂²×C₄⋊C₄	—	43	1	no	—
157	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
158	C₂²×C₂²⋊C₄	—	86	1	no	—
159	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
160	#476	—	43	64	no	—
161	C₂×#106	—	43	24	no	—
162	C₂×C₄⋊C₈	—	43	12	no	—
163	C₂×#102	—	43	24	no	—
164	C₄₃	C₂²×#476	1	1	yes	characteristic, complemented, Sylow
165	C₂⁵	—	86	1	no	—
166	C₂²×Q₈	—	2	12	no	—
167	C₂²×Q₈	—	4	8	no	—
168	C₂²×D₄	—	43	8	no	—
169	C₂²×D₄	—	43	6	no	—
170	C₂²×D₄	—	86	4	no	—
171	C₂²×D₄	—	86	3	no	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	86	1	no	—
174	C₂³×C₄	—	86	12	no	—
175	C₂³×C₄	—	172	8	no	—
176	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
179	C₂²×C₈	—	43	12	no	—
180	C₂²×C₈	—	86	8	no	—
181	C₄⋊Q₈	—	2	24	no	—
182	C₄⋊Q₈	C₂²×D₄₃	1	16	yes	complemented
183	C₄×D₄	—	43	16	no	—
184	C₄×D₄	—	86	24	no	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	2	6	no	—
187	C₂×C₄⋊C₄	—	43	6	no	—
188	C₂×C₄⋊C₄	—	86	3	no	—
189	C₂×C₄⋊C₄	C₂²×D₄₃	1	12	yes	—
190	C₂×C₂²⋊C₄	—	86	12	no	—
191	C₂×C₄²	C₂²⋊D₄₃	1	6	yes	—
192	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
193	#106	—	43	32	no	—
194	C₄⋊C₈	—	43	16	no	—
195	#102	—	43	32	no	—
196	C₂⁴	—	86	8	no	—
197	C₂⁴	—	86	6	no	—
198	C₂⁴	—	172	4	no	—
199	C₂⁴	—	172	4	no	—
200	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
201	C₂×Q₈	—	2	16	no	—
202	C₂×Q₈	—	4	48	no	—
203	C₂×D₄	—	43	24	no	—
204	C₂×D₄	—	43	4	no	—
205	C₂×D₄	—	86	24	no	—
206	C₂×D₄	—	86	6	no	—
207	C₂×D₄	—	86	12	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	86	6	no	—
212	C₂²×C₄	—	86	16	no	—
213	C₂²×C₄	—	172	4	no	—
214	C₂²×C₄	—	172	48	no	—
215	C₂²×C₄	#10621884	1	8	yes	—
216	C₂²×C₄	Q₈×D₄₃	1	4	yes	—
217	C₂²×C₄	#10621902	1	4	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
221	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
222	C₂×C₈	—	43	8	no	—
223	C₂×C₈	—	86	24	no	—
224	C₄⋊C₄	—	2	16	no	—
225	C₄⋊C₄	—	2	12	no	—
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	86	6	no	—
228	C₄⋊C₄	C₂³×D₄₃	1	8	yes	—
229	C₂²⋊C₄	—	86	16	no	—
230	C₄²	C₂×C₂²⋊D₄₃	1	12	yes	—
231	C₄²	C₂³×D₄₃	1	4	yes	—
232	C₂³	—	86	24	no	—
233	C₂³	—	86	4	no	—
234	C₂³	—	172	12	no	—
235	C₂³	—	172	12	no	—
236	C₂³	—	172	32	no	—
237	C₂³	#13063301	1	4	yes	—
238	C₂³	C₂×#10621884	1	4	yes	—
239	C₂³	#13063351	1	4	yes	—
240	C₂³	C₂²×C₂²⋊D₄₃	1	3	yes	—
241	Q₈	—	4	32	no	—
242	D₄	—	43	8	no	—
243	D₄	—	86	24	no	—
244	D₄	—	86	4	no	—
245	C₂×C₄	—	2	12	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	24	no	—
248	C₂×C₄	—	86	4	no	—
249	C₂×C₄	—	172	12	no	—
250	C₂×C₄	—	172	32	no	—
251	C₂×C₄	#13063301	1	12	yes	—
252	C₂×C₄	C₂×#10621884	1	24	yes	—
253	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₄₃	1	3	yes	—
255	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
256	C₂×C₄	C₂×Q₈×D₄₃	1	6	yes	—
257	C₂×C₄	C₂×#10621902	1	6	yes	—
258	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
259	C₈	—	86	8	no	—
260	C₂²	—	86	8	no	—
261	C₂²	—	172	4	no	—
262	C₂²	—	172	4	no	—
263	C₂²	—	172	48	no	—
264	C₂²	#14583186	1	16	yes	direct product factor
265	C₂²	C₂×#13063301	1	6	yes	—
266	C₂²	C₂²×#10621884	1	6	yes	—
267	C₂²	C₂×#13063351	1	6	yes	—
268	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	4	8	no	—
271	C₄	—	172	4	no	—
272	C₄	C₂×#13063301	1	6	yes	—
273	C₄	C₂²×#10621884	1	8	yes	—
274	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
275	C₄	C₂²×#10621902	1	1	yes	characteristic
276	C₂	—	172	8	no	—
277	C₂	C₂×#14583186	1	12	yes	direct product factor
278	C₂	C₂²×#13063301	1	1	yes	characteristic
279	C₂	C₂³×#10621884	1	1	yes	characteristic
280	C₂	C₂²×#13063351	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₄⋊Q₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂³×D₁₇₂	C₂²
hypercenter	C₂²×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	C₂²×#476
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14583186	C₂	direct (N × H)
3. C₂²×#13063256	C₂	semidirect
4. C₂²×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
6. C₂²×C₄⋊Q₈×C₄₃	C₂	semidirect
14. #14583186	C₂²	direct (N × H)
15. C₂×#13063256	C₂²	semidirect
16. C₂×C₄⋊(C₄₃⋊C₈)	C₂²	semidirect
20. C₂×C₄⋊Q₈×C₄₃	C₂²	semidirect
27. C₂²×D₁₇₂	Q₈	semidirect
39. #13063256	C₂³	semidirect
40. C₄⋊(C₄₃⋊C₈)	C₂³	semidirect
49. C₄⋊Q₈×C₄₃	C₂³	semidirect
77. C₂×D₁₇₂	C₂×Q₈	semidirect
116. D₁₇₂	C₂²×Q₈	semidirect
120. C₂²×C₄₃	#476	semidirect
136. C₂²×C₄⋊Q₈	D₄₃	semidirect
142. C₂×C₄₃	C₂×#476	semidirect
152. C₂×C₄⋊Q₈	C₂×D₄₃	semidirect
164. C₄₃	C₂²×#476	semidirect
182. C₄⋊Q₈	C₂²×D₄₃	semidirect
264. C₂²	#14583186	direct (N × H)
277. C₂	C₂×#14583186	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₁₇₂	2
C₂×#14583186	2
C₂²×#13063256	2
C₂²×C₄⋊(C₄₃⋊C₈)	2
C₂²×#13063245	2
C₂²×C₄⋊Q₈×C₄₃	2
C₂²×#476	43

Subgroups — C22×#14583186 #25964057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583186 #25964057