Subgroups — #25001796

Order 11008.

886 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582854	C₂	1	2	yes	maximal
2	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
3	#19134312	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582659	C₂	1	2	yes	maximal
5	C₂²×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	2	yes	—
8	#14582854	—	2	4	no	—
9	#14582854	C₄	1	8	yes	complemented
10	C₂×#13063235	—	2	2	no	—
11	C₂×C₈×Dic₄₃	—	2	2	no	—
12	#14582664	C₄	1	4	yes	complemented
13	#14582662	C₄	1	4	yes	complemented
14	#14582659	C₄	1	8	yes	complemented
15	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₄₃	—	2	4	no	—
19	C₂×#97×C₄₃	C₄	1	4	yes	complemented
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	C₂×C₄	1	4	yes	—
24	C₂×#10621877	C₂×C₄	1	4	yes	—
25	C₂×#10621877	C₂×C₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063284	C₂×C₄	1	4	yes	—
30	#13063284	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄×Dic₄₃	—	2	2	no	—
34	#13063235	—	2	8	no	—
35	C₈×Dic₄₃	—	2	8	no	—
36	C₂³×C₄×C₄₃	—	2	1	no	—
37	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₃	D₄	1	2	yes	—
46	C₂²×C₈×C₄₃	D₄	1	4	yes	—
47	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
48	C₂×C₄²×C₄₃	—	2	2	no	—
49	C₂×C₄²×C₄₃	—	2	1	no	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
52	#97×C₄₃	—	2	8	no	—
53	C₂⁴×C₄₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	2	4	no	—
56	C₂²×C₄×C₄₃	—	4	4	no	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×Dic₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	C₄²	1	4	yes	—
73	C₂×C₈×C₄₃	C₄²	1	4	yes	—
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
76	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
77	C₂×C₈×C₄₃	C₄⋊C₄	1	8	yes	—
78	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	C₄²	1	4	yes	—
82	C₂²⋊Dic₄₃	—	2	4	no	—
83	C₂²⋊Dic₄₃	—	2	4	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	C₄²	1	8	yes	—
86	C₄×Dic₄₃	—	2	4	no	—
87	C₄×Dic₄₃	—	2	4	no	—
88	#10621877	—	2	4	no	—
89	#10621877	—	2	4	no	—
90	#10621877	—	2	4	no	—
91	#10621877	C₄²	1	8	yes	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂³×C₄₃	—	2	4	no	—
96	C₂³×C₄₃	#117	1	2	yes	—
97	C₂³×C₄₃	#117	1	1	yes	characteristic
98	C₂³×C₄₃	C₄×D₄	1	2	yes	—
99	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	4	8	no	—
103	C₂×C₄×C₄₃	—	4	4	no	—
104	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#117	1	2	yes	—
114	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
122	C₂×Dic₄₃	—	2	8	no	—
123	C₂×Dic₄₃	—	4	8	no	—
124	C₈×C₄₃	—	2	4	no	—
125	C₈×C₄₃	#95	1	8	yes	—
126	C₄₃⋊C₈	—	2	8	no	—
127	#10707	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	#376	1	1	yes	characteristic
133	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
135	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
137	C₂²×C₄₃	#444	1	4	yes	—
138	C₄×C₄₃	—	4	4	no	—
139	C₄×C₄₃	C₄³	1	1	yes	characteristic
140	C₄×C₄₃	C₂×#95	1	1	yes	characteristic
141	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
143	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
145	Dic₄₃	—	4	8	no	—
146	C₂×#405	—	43	2	no	—
147	C₂²×#97	D₄₃	1	1	yes	characteristic
148	C₂×#341	—	43	1	no	—
149	#1735	—	43	1	no	—
150	C₂×#339	—	43	2	no	—
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	—	2	1	no	—
153	C₂×C₄₃	—	2	2	no	—
154	C₂×C₄₃	C₄×#95	1	1	yes	characteristic
155	C₂×C₄₃	#1765	1	4	yes	—
156	C₂×C₄₃	#1773	1	2	yes	—
157	C₂²×M₁₆	—	43	2	no	—
158	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
160	C₂×#117	—	43	2	no	—
161	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
162	#405	—	43	8	no	—
163	#405	—	86	4	no	—
164	C₂×#97	—	2	4	no	—
165	C₂×#97	—	86	2	no	—
166	C₂×#97	Dic₄₃	1	4	yes	complemented
167	C₂×C₄×C₈	—	86	2	no	—
168	#341	—	43	4	no	—
169	#340	—	43	4	no	—
170	#339	—	43	8	no	—
171	C₄₃	#10707	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
174	C₂×M₁₆	—	43	4	no	—
175	C₂×M₁₆	—	43	4	no	—
176	C₂×M₁₆	—	43	4	no	—
177	C₂×M₁₆	—	86	2	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	86	2	no	—
183	C₂²×C₈	Dic₈₆	1	2	yes	—
184	C₂²×C₈	C₄×D₄₃	1	2	yes	—
185	C₂²×C₈	C₄×D₄₃	1	2	yes	—
186	C₂²×C₈	C₄×D₄₃	1	4	yes	—
187	C₂²×C₈	D₁₇₂	1	2	yes	—
188	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
189	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
191	#117	—	43	4	no	—
192	#117	—	43	4	no	—
193	#117	—	86	2	no	—
194	#117	—	86	2	no	—
195	C₂×C₄⋊C₄	—	43	4	no	—
196	C₂×C₂²⋊C₄	—	86	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	86	2	no	—
200	C₂×C₄²	C₄×D₄₃	1	4	yes	—
201	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
202	#97	—	2	8	no	—
203	#97	—	86	8	no	—
204	C₄×C₈	—	86	8	no	—
205	C₂⁴	#10621898	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	86	4	no	—
210	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
215	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
216	C₂²×C₄	#10621898	1	1	yes	characteristic
217	C₂²×C₄	#10621898	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
220	M₁₆	—	43	8	no	—
221	M₁₆	—	86	4	no	—
222	M₁₆	—	86	4	no	—
223	M₁₆	—	86	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	86	4	no	—
229	C₂×C₈	—	86	4	no	—
230	C₂×C₈	—	86	4	no	—
231	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
232	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
233	C₂×C₈	Dic₄₃⋊C₄	1	8	yes	—
234	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
235	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
236	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
237	C₂×C₈	C₂²⋊Dic₄₃	1	4	yes	—
238	C₄⋊C₄	—	43	8	no	—
239	C₄⋊C₄	—	86	4	no	—
240	C₂²⋊C₄	—	86	4	no	—
241	C₂²⋊C₄	—	86	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	86	4	no	—
245	C₄²	—	86	4	no	—
246	C₄²	C₄×Dic₄₃	1	4	yes	—
247	C₂³	—	2	4	no	—
248	C₂³	#13063282	1	2	yes	—
249	C₂³	#13063284	1	1	yes	characteristic
250	C₂³	C₄×C₂²⋊D₄₃	1	2	yes	—
251	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
252	C₂³	C₄×Dic₈₆	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	86	8	no	—
257	C₂×C₄	—	172	8	no	—
258	C₂×C₄	C₄²×D₄₃	1	2	yes	—
259	C₂×C₄	C₄²×D₄₃	1	2	yes	—
260	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
263	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
265	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
267	C₂×C₄	#13063282	1	2	yes	—
268	C₂×C₄	#13063284	1	1	yes	characteristic
269	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
270	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
271	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
272	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
273	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
274	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
275	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	86	8	no	—
278	C₈	C₄₃⋊#95	1	8	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	#14582745	1	1	yes	characteristic
284	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
285	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	2	yes	—
286	C₂²	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
287	C₂²	C₄×Dic₄₃⋊C₄	1	2	yes	—
288	C₂²	#14583022	1	4	yes	—
289	C₄	—	4	4	no	—
290	C₄	—	172	8	no	—
291	C₄	C₄²×Dic₄₃	1	1	yes	characteristic
292	C₄	C₂×C₄₃⋊#95	1	1	yes	characteristic
293	C₄	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
294	C₄	C₄×(C₂×D₄₃)⋊C₄	1	2	yes	—
295	C₄	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
296	C₄	C₄×Dic₄₃⋊C₄	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₄×C₄₃⋊#95	1	1	yes	characteristic
301	C₂	#19134439	1	4	yes	—
302	C₂	#19134479	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₄×C₄₃	C₄³
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂²×#97×C₄₃	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂²×#97	D₄₃
hypocenter	C₄₃	#10707
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
9. #14582854	C₄	semidirect
12. #14582664	C₄	semidirect
13. #14582662	C₄	semidirect
14. #14582659	C₄	semidirect
19. C₂×#97×C₄₃	C₄	semidirect
166. C₂×#97	Dic₄₃	semidirect
171. C₄₃	#10707	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582854	2
C₂×#14582664	2
#19134312	2
C₂×#14582659	2
C₂²×#97×C₄₃	2
#10707	43