Subgroups — C₂²×#14583109 #25963399

Order 11008.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10621885	C₂	1	2	yes	maximal
2	C₂²×#13063325	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583109	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13063248	C₂	1	2	yes	maximal
6	C₂²×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₈₆	C₂²	1	2	yes	—
8	C₂²×#10621885	—	2	16	no	—
9	C₂²×#10621885	C₂²	1	24	yes	—
10	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂×#13063325	—	2	6	no	—
12	C₂×#13063325	C₂²	1	12	yes	—
13	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
14	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
15	#14583109	C₂²	1	64	yes	direct product factor
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	12	yes	—
17	C₂×#13063248	C₂²	1	24	yes	—
18	C₂³×Q₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₄₃	—	2	6	no	—
20	C₂×C₄×Q₈×C₄₃	C₂²	1	12	yes	—
21	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₃	—	2	1	no	—
24	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	—	2	8	no	—
26	C₂²×Dic₈₆	—	2	8	no	—
27	C₂²×Dic₈₆	—	2	6	no	—
28	C₂²×Dic₈₆	C₂³	1	12	yes	—
29	C₂³×Dic₄₃	—	2	2	no	—
30	C₂×#10621885	—	2	96	no	—
31	C₂×#10621885	C₂³	1	32	yes	—
32	C₂²×C₄₃⋊C₈	—	2	8	no	—
33	C₂²×C₄₃⋊C₈	C₂³	1	12	yes	—
34	#13063325	—	2	24	no	—
35	#13063325	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₄₃	—	2	12	no	—
37	C₂×C₄⋊Dic₄₃	—	2	12	no	—
38	C₄⋊(C₄₃⋊C₈)	C₂³	1	16	yes	—
39	#13063248	C₂³	1	32	yes	—
40	C₂²×Q₈×C₄₃	—	2	3	no	—
41	C₂²×Q₈×C₄₃	—	2	4	no	—
42	C₂²×Q₈×C₄₃	D₄	1	8	yes	—
43	C₂²×Q₈×C₄₃	C₂³	1	6	yes	—
44	C₂³×C₄×C₄₃	—	2	1	no	—
45	C₂³×C₄×C₄₃	—	2	1	no	—
46	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₄₃	—	2	24	no	—
50	C₄×Q₈×C₄₃	C₂³	1	16	yes	—
51	C₂×C₄⋊C₄×C₄₃	—	2	12	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
54	C₂×C₄²×C₄₃	—	2	12	no	—
55	C₂×C₄²×C₄₃	D₄	1	6	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
57	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₄₃	—	2	24	no	—
59	C₂×Q₈×C₄₃	—	2	6	no	—
60	C₂×Q₈×C₄₃	—	2	12	no	—
61	C₂×Q₈×C₄₃	C₂×D₄	1	24	yes	—
62	C₂×Q₈×C₄₃	C₂⁴	1	4	yes	—
63	C₂²×C₄×C₄₃	—	2	8	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	2	6	no	—
66	C₂²×C₄×C₄₃	—	2	6	no	—
67	C₂²×C₄×C₄₃	—	4	4	no	—
68	C₂²×C₄×C₄₃	Q₁₆	1	8	yes	complemented
69	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
73	C₂²×C₄×C₄₃	#41	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
75	C₂²×Dic₄₃	—	2	12	no	—
76	C₂²×Dic₄₃	—	4	8	no	—
77	C₂×Dic₈₆	—	2	24	no	—
78	C₂×Dic₈₆	—	2	24	no	—
79	C₂×Dic₈₆	—	2	12	no	—
80	C₂×Dic₈₆	—	4	24	no	—
81	C₂×Dic₈₆	C₂⁴	1	8	yes	—
82	C₄⋊C₄×C₄₃	—	2	16	no	—
83	C₄⋊C₄×C₄₃	—	2	6	no	—
84	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
85	C₄²×C₄₃	—	2	16	no	—
86	C₄²×C₄₃	C₂×D₄	1	12	yes	—
87	C₄²×C₄₃	C₂⁴	1	4	yes	—
88	#10621885	—	2	64	no	—
89	C₄⋊Dic₄₃	—	2	16	no	—
90	C₄⋊Dic₄₃	—	2	16	no	—
91	C₂×C₄₃⋊C₈	—	2	24	no	—
92	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
93	C₂³×C₄₃	#121	1	4	yes	—
94	C₂³×C₄₃	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₄₃	#137	1	4	yes	—
96	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
97	Q₈×C₄₃	—	2	24	no	—
98	Q₈×C₄₃	—	2	4	no	—
99	Q₈×C₄₃	C₂²×D₄	1	8	yes	—
100	C₂×C₄×C₄₃	—	2	24	no	—
101	C₂×C₄×C₄₃	—	2	12	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	4	12	no	—
105	C₂×C₄×C₄₃	#121	1	12	yes	—
106	C₂×C₄×C₄₃	C₂×Q₁₆	1	24	yes	complemented
107	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
108	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
109	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
110	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
112	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
113	C₂×Dic₄₃	—	2	8	no	—
114	C₂×Dic₄₃	—	4	24	no	—
115	Dic₈₆	—	2	8	no	—
116	Dic₈₆	—	2	8	no	—
117	Dic₈₆	—	4	24	no	—
118	C₄₃⋊C₈	—	2	8	no	—
119	C₂²×#462	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	#462	1	16	yes	complemented
121	C₂²×C₄₃	C₂×#121	1	6	yes	—
122	C₂²×C₄₃	C₂²×Q₁₆	1	6	yes	—
123	C₂²×C₄₃	C₂×#137	1	6	yes	—
124	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₄₃	—	2	8	no	—
126	C₄×C₄₃	—	2	4	no	—
127	C₄×C₄₃	—	4	4	no	—
128	C₄×C₄₃	C₂×#121	1	6	yes	—
129	C₄×C₄₃	C₂²×Q₁₆	1	8	yes	complemented
130	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
131	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
132	Dic₄₃	—	4	8	no	—
133	C₂³×Q₁₆	—	43	2	no	—
134	C₂²×C₄⋊Q₈	—	43	1	no	—
135	C₂²×C₄×Q₈	D₄₃	1	1	yes	characteristic
136	C₂×#462	—	43	24	no	—
137	C₂²×C₄⋊C₈	—	43	1	no	—
138	C₂²×#103	—	43	2	no	—
139	C₂×C₄₃	C₂×#462	1	12	yes	complemented
140	C₂×C₄₃	C₂²×#121	1	1	yes	characteristic
141	C₂×C₄₃	C₂³×Q₁₆	1	1	yes	characteristic
142	C₂×C₄₃	C₂²×#137	1	1	yes	characteristic
143	C₂³×Q₈	—	43	2	no	—
144	C₂³×Q₈	C₂×D₄₃	1	1	yes	characteristic
145	C₂²×Q₁₆	—	43	24	no	—
146	C₂²×Q₁₆	—	86	16	no	—
147	C₂³×C₈	—	43	2	no	—
148	C₂×C₄⋊Q₈	—	43	12	no	—
149	C₂×C₄⋊Q₈	—	86	6	no	—
150	C₂×C₄×Q₈	—	2	6	no	—
151	C₂×C₄×Q₈	C₂×D₄₃	1	12	yes	—
152	C₂²×C₄⋊C₄	—	2	1	no	—
153	C₂²×C₄⋊C₄	—	86	1	no	—
154	C₂²×C₄⋊C₄	—	86	1	no	—
155	C₂²×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
156	C₂²×C₄²	—	2	1	no	—
157	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
158	#462	—	43	64	no	—
159	C₂×C₄⋊C₈	—	43	12	no	—
160	C₂×#103	—	43	24	no	—
161	C₄₃	C₂²×#462	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	—	2	3	no	—
164	C₂²×Q₈	—	43	12	no	—
165	C₂²×Q₈	—	86	6	no	—
166	C₂²×Q₈	—	86	8	no	—
167	C₂²×Q₈	—	86	8	no	—
168	C₂²×Q₈	D₁₇₂	1	8	yes	—
169	C₂²×Q₈	C₂²×D₄₃	1	6	yes	—
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	—	86	2	no	—
173	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
175	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
176	C₂×Q₁₆	—	43	32	no	—
177	C₂×Q₁₆	—	86	96	no	—
178	C₂²×C₈	—	43	12	no	—
179	C₂²×C₈	—	86	8	no	—
180	C₄⋊Q₈	—	43	16	no	—
181	C₄⋊Q₈	—	86	24	no	—
182	C₄×Q₈	—	2	24	no	—
183	C₄×Q₈	C₂²×D₄₃	1	16	yes	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	—	2	3	no	—
186	C₂×C₄⋊C₄	—	86	12	no	—
187	C₂×C₄⋊C₄	—	86	12	no	—
188	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
189	C₂×C₄²	—	2	12	no	—
190	C₂×C₄²	C₂²⋊D₄₃	1	6	yes	—
191	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
192	C₄⋊C₈	—	43	16	no	—
193	#103	—	43	32	no	—
194	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	2	12	no	—
198	C₂×Q₈	—	43	8	no	—
199	C₂×Q₈	—	86	12	no	—
200	C₂×Q₈	—	86	24	no	—
201	C₂×Q₈	—	86	24	no	—
202	C₂×Q₈	—	172	24	no	—
203	C₂×Q₈	C₂×D₁₇₂	1	24	yes	—
204	C₂×Q₈	C₂³×D₄₃	1	4	yes	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	86	12	no	—
211	C₂²×C₄	—	172	8	no	—
212	C₂²×C₄	#10621885	1	8	yes	complemented
213	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
214	C₂²×C₄	#10621898	1	4	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
218	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
219	Q₁₆	—	86	64	no	—
220	C₂×C₈	—	43	8	no	—
221	C₂×C₈	—	86	24	no	—
222	C₄⋊C₄	—	2	16	no	—
223	C₄⋊C₄	—	2	6	no	—
224	C₄⋊C₄	—	86	16	no	—
225	C₄⋊C₄	—	86	16	no	—
226	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
227	C₄²	—	2	16	no	—
228	C₄²	C₂×C₂²⋊D₄₃	1	12	yes	—
229	C₄²	C₂³×D₄₃	1	4	yes	—
230	C₂³	#13063299	1	4	yes	—
231	C₂³	C₂×#10621885	1	4	yes	—
232	C₂³	#13063355	1	4	yes	—
233	C₂³	C₂²×C₂²⋊D₄₃	1	3	yes	—
234	Q₈	—	2	24	no	—
235	Q₈	—	2	4	no	—
236	Q₈	—	86	8	no	—
237	Q₈	—	86	8	no	—
238	Q₈	—	172	24	no	—
239	Q₈	C₂²×D₁₇₂	1	8	yes	—
240	C₂×C₄	—	2	24	no	—
241	C₂×C₄	—	2	12	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	86	8	no	—
246	C₂×C₄	—	172	24	no	—
247	C₂×C₄	#13063299	1	12	yes	—
248	C₂×C₄	C₂×#10621885	1	24	yes	complemented
249	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
250	C₂×C₄	C₂²×C₂²⋊D₄₃	1	3	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
252	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
253	C₂×C₄	C₂×#10621898	1	6	yes	—
254	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
255	C₈	—	86	8	no	—
256	C₂²	#14583109	1	16	yes	direct product factor
257	C₂²	C₂×#13063299	1	6	yes	—
258	C₂²	C₂²×#10621885	1	6	yes	—
259	C₂²	C₂×#13063355	1	6	yes	—
260	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	4	4	no	—
264	C₄	—	172	8	no	—
265	C₄	C₂×#13063299	1	6	yes	—
266	C₄	C₂²×#10621885	1	8	yes	complemented
267	C₄	C₂³×D₁₇₂	1	1	yes	characteristic
268	C₄	C₂²×#10621898	1	1	yes	characteristic
269	C₂	C₂×#14583109	1	12	yes	direct product factor
270	C₂	C₂²×#13063299	1	1	yes	characteristic
271	C₂	C₂³×#10621885	1	1	yes	characteristic
272	C₂	C₂²×#13063355	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, prime radical	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂²×C₄×Q₈	D₄₃
hypocenter	C₄₃	C₂²×#462
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
3. C₂×#14583109	C₂	direct (N × H)
15. #14583109	C₂²	direct (N × H)
68. C₂²×C₄×C₄₃	Q₁₆	semidirect
106. C₂×C₄×C₄₃	C₂×Q₁₆	semidirect
120. C₂²×C₄₃	#462	semidirect
129. C₄×C₄₃	C₂²×Q₁₆	semidirect
139. C₂×C₄₃	C₂×#462	semidirect
161. C₄₃	C₂²×#462	semidirect
212. C₂²×C₄	#10621885	semidirect
248. C₂×C₄	C₂×#10621885	semidirect
256. C₂²	#14583109	direct (N × H)
266. C₄	C₂²×#10621885	semidirect
269. C₂	C₂×#14583109	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#10621885	2
C₂²×#13063325	2
C₂×#14583109	2
C₂²×C₄⋊(C₄₃⋊C₈)	2
C₂²×#13063248	2
C₂²×C₄×Q₈×C₄₃	2
C₂²×#462	43

Subgroups — C22×#14583109 #25963399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583109 #25963399