Subgroups — #25259430

Order 11008.

2486 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137769	C₂	1	4	yes	maximal, complemented
2	#19137707	C₂	1	4	yes	maximal, complemented
3	#19137444	C₂	1	8	yes	maximal, complemented
4	#19137410	C₂	1	8	yes	maximal, complemented
5	#19137181	C₂	1	2	yes	maximal, complemented
6	#19137173	C₂	1	2	yes	maximal
7	#19136993	C₂	1	2	yes	maximal, complemented
8	C₂²×#124×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
11	#14582806	C₂²	1	8	yes	complemented
12	#14582806	C₂²	1	8	yes	complemented
13	#14582792	C₂²	1	16	yes	complemented
14	#14582792	C₂²	1	16	yes	complemented
15	#14582779	C₂²	1	8	yes	—
16	#14582779	C₂²	1	8	yes	—
17	#14582774	C₂²	1	16	yes	complemented
18	#14582767	C₂²	1	4	yes	complemented
19	#14582765	C₂²	1	4	yes	complemented
20	#14582762	C₂²	1	4	yes	—
21	#14582762	C₂²	1	4	yes	—
22	#14582759	C₂²	1	4	yes	—
23	#14582755	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	complemented
25	C₂²⋊C₄×Dic₄₃	C₂²	1	16	yes	complemented
26	C₄×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
27	#14582745	C₂²	1	4	yes	complemented
28	C₂³×Q₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂³×D₄×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#124×C₄₃	C₄	1	16	yes	complemented
31	C₂×#124×C₄₃	C₂²	1	4	yes	—
32	C₂×#124×C₄₃	C₂²	1	2	yes	—
33	C₂×#124×C₄₃	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
35	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
36	C₂³×Dic₄₃	C₂³	1	8	yes	—
37	#13063284	—	4	8	no	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
40	C₂×C₂²⋊Dic₄₃	—	4	8	no	—
41	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
42	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
43	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
44	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
45	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
46	C₄₃⋊#95	C₂³	1	16	yes	—
47	C₄₃⋊#95	C₂³	1	8	yes	—
48	C₄₃⋊#95	C₂³	1	8	yes	—
49	C₄₃⋊#95	C₂³	1	8	yes	—
50	C₄₃⋊#95	C₂³	1	8	yes	—
51	C₂⁵×C₄₃	C₂³	1	2	yes	—
52	C₂²×Q₈×C₄₃	—	4	4	no	—
53	C₂²×Q₈×C₄₃	C₂×C₄	1	8	yes	complemented
54	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
55	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₄₃	—	4	4	no	—
57	C₂²×D₄×C₄₃	C₂×C₄	1	8	yes	—
58	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
63	#124×C₄₃	C₂×C₄	1	32	yes	complemented
64	#124×C₄₃	C₂×C₄	1	16	yes	complemented
65	#124×C₄₃	C₂×C₄	1	16	yes	complemented
66	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	32	yes	complemented
67	C₂×C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
68	C₂×C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
69	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	complemented
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
73	C₂⁴×C₄₃	—	2	8	no	—
74	C₂⁴×C₄₃	—	4	8	no	—
75	C₂⁴×C₄₃	C₂²×C₄	1	8	yes	—
76	C₂⁴×C₄₃	C₂⁴	1	4	yes	—
77	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₂×Q₈×C₄₃	—	4	16	no	—
79	C₂×Q₈×C₄₃	—	4	2	no	—
80	C₂×Q₈×C₄₃	—	4	4	no	—
81	C₂×Q₈×C₄₃	—	4	2	no	—
82	C₂×Q₈×C₄₃	C₂²×C₄	1	8	yes	—
83	C₂×Q₈×C₄₃	C₂²×C₄	1	8	yes	—
84	C₂×D₄×C₄₃	—	4	16	no	—
85	C₂×D₄×C₄₃	—	4	2	no	—
86	C₂×D₄×C₄₃	—	4	4	no	—
87	C₂×D₄×C₄₃	—	4	2	no	—
88	C₂×D₄×C₄₃	C₂²×C₄	1	8	yes	—
89	C₂×D₄×C₄₃	C₂²×C₄	1	8	yes	—
90	C₂²×C₄×C₄₃	—	2	4	no	—
91	C₂²×C₄×C₄₃	—	2	8	no	—
92	C₂²×C₄×C₄₃	—	4	4	no	—
93	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
95	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
96	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
97	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
100	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
101	C₂²×Dic₄₃	—	4	8	no	—
102	C₂²×Dic₄₃	—	4	16	no	—
103	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
104	C₂²⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
105	C₂²⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
106	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
107	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
108	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
109	C₂²⋊Dic₄₃	—	2	16	no	—
110	C₂²⋊Dic₄₃	—	4	8	no	—
111	C₂²⋊Dic₄₃	—	4	16	no	—
112	C₄⋊Dic₄₃	—	4	8	no	—
113	C₄⋊Dic₄₃	—	4	8	no	—
114	C₄×Dic₄₃	—	4	8	no	—
115	C₄×Dic₄₃	—	4	8	no	—
116	C₂³×C₄₃	—	2	4	no	—
117	C₂³×C₄₃	—	2	4	no	—
118	C₂³×C₄₃	—	2	4	no	—
119	C₂³×C₄₃	—	2	16	no	—
120	C₂³×C₄₃	—	2	16	no	—
121	C₂³×C₄₃	—	4	16	no	—
122	C₂³×C₄₃	—	8	16	no	—
123	C₂³×C₄₃	C₂³×C₄	1	8	yes	—
124	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
125	C₂³×C₄₃	H(5,𝔽₂)	1	4	yes	—
126	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
127	Q₈×C₄₃	—	4	8	no	—
128	Q₈×C₄₃	—	4	8	no	—
129	D₄×C₄₃	—	4	8	no	—
130	D₄×C₄₃	—	4	8	no	—
131	C₂×C₄×C₄₃	—	2	8	no	—
132	C₂×C₄×C₄₃	—	2	16	no	—
133	C₂×C₄×C₄₃	—	2	4	no	—
134	C₂×C₄×C₄₃	—	2	4	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	2	4	no	—
139	C₂×C₄×C₄₃	—	4	8	no	—
140	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
141	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
142	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
143	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
144	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
145	C₂×Dic₄₃	—	2	16	no	—
146	C₂×Dic₄₃	—	2	16	no	—
147	C₂×Dic₄₃	—	2	16	no	—
148	C₂×Dic₄₃	—	8	16	no	—
149	#34366	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	—	2	4	no	—
151	C₂²×C₄₃	—	2	8	no	—
152	C₂²×C₄₃	—	2	8	no	—
153	C₂²×C₄₃	—	2	16	no	—
154	C₂²×C₄₃	—	4	8	no	—
155	C₂²×C₄₃	—	4	8	no	—
156	C₂²×C₄₃	—	8	16	no	—
157	C₂²×C₄₃	#518	1	4	yes	—
158	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
160	C₂²×C₄₃	#585	1	4	yes	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	8	no	—
164	C₄×C₄₃	—	2	4	no	—
165	C₄×C₄₃	—	2	4	no	—
166	Dic₄₃	—	4	16	no	—
167	C₂²×#124	D₄₃	1	1	yes	characteristic
168	#2379	—	43	4	no	—
169	#2372	—	43	4	no	—
170	#2342	—	43	8	no	—
171	#2337	—	43	8	no	—
172	#2310	—	43	2	no	—
173	#2308	—	43	2	no	—
174	#2283	—	43	2	no	—
175	C₂×C₄₃	—	2	4	no	—
176	C₂×C₄₃	—	4	8	no	—
177	C₂×C₄₃	C₂×#518	1	1	yes	characteristic
178	C₂×C₄₃	#3428	1	2	yes	—
179	C₂×C₄₃	#3545	1	4	yes	—
180	C₂³×Q₈	C₂×D₄₃	1	1	yes	characteristic
181	C₂³×D₄	C₂×D₄₃	1	1	yes	characteristic
182	C₂×#124	Dic₄₃	1	16	yes	complemented
183	C₂×#124	C₂×D₄₃	1	4	yes	—
184	C₂×#124	C₂×D₄₃	1	2	yes	—
185	C₂×#124	C₂×D₄₃	1	2	yes	—
186	C₂×#117	—	43	4	no	—
187	C₂²×C₂²⋊C₄	—	43	4	no	—
188	C₂²×C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
189	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
190	#391	—	43	8	no	—
191	#391	—	43	8	no	—
192	#388	—	43	16	no	—
193	#388	—	43	16	no	—
194	#386	—	43	8	no	—
195	#386	—	43	8	no	—
196	#385	—	43	16	no	—
197	#383	—	43	4	no	—
198	#382	—	43	4	no	—
199	#381	—	43	4	no	—
200	#381	—	43	4	no	—
201	#380	—	43	4	no	—
202	#379	—	43	4	no	—
203	C₄×C₄⋊C₄	—	43	4	no	—
204	C₄×C₂²⋊C₄	—	43	4	no	—
205	C₄×C₂²⋊C₄	—	43	16	no	—
206	#376	—	43	4	no	—
207	C₄₃	#34366	1	1	yes	characteristic, complemented, Sylow
208	C₂⁵	C₂²×D₄₃	1	2	yes	—
209	C₂²×Q₈	—	4	4	no	—
210	C₂²×Q₈	C₂×Dic₄₃	1	8	yes	complemented
211	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
212	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
213	C₂²×D₄	—	4	4	no	—
214	C₂²×D₄	C₂×Dic₄₃	1	8	yes	—
215	C₂²×D₄	C₂²×D₄₃	1	2	yes	—
216	C₂²×D₄	C₂²×D₄₃	1	2	yes	—
217	C₂³×C₄	—	43	8	no	—
218	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
219	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
220	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
221	#124	C₂×Dic₄₃	1	32	yes	complemented
222	#124	C₂×Dic₄₃	1	16	yes	complemented
223	#124	C₂×Dic₄₃	1	16	yes	complemented
224	#117	—	172	8	no	—
225	C₂×C₄⋊C₄	—	43	8	no	—
226	C₂×C₄⋊C₄	—	43	8	no	—
227	C₂×C₂²⋊C₄	—	43	8	no	—
228	C₂×C₂²⋊C₄	—	43	8	no	—
229	C₂×C₂²⋊C₄	—	43	16	no	—
230	C₂×C₂²⋊C₄	—	172	8	no	—
231	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	32	yes	complemented
232	C₂×C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
233	C₂×C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
234	C₂×C₄²	—	43	8	no	—
235	C₂×C₄²	—	43	8	no	—
236	C₂×C₄²	C₂×Dic₄₃	1	8	yes	complemented
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	#95	—	43	8	no	—
241	#95	—	43	8	no	—
242	#95	—	43	16	no	—
243	#95	—	43	8	no	—
244	#95	—	43	8	no	—
245	C₂⁴	—	2	8	no	—
246	C₂⁴	—	4	8	no	—
247	C₂⁴	C₂²×Dic₄₃	1	8	yes	—
248	C₂⁴	C₂³×D₄₃	1	4	yes	—
249	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	16	no	—
254	C₂×Q₈	C₂²×Dic₄₃	1	8	yes	—
255	C₂×Q₈	C₂²×Dic₄₃	1	8	yes	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	16	no	—
260	C₂×D₄	C₂²×Dic₄₃	1	8	yes	—
261	C₂×D₄	C₂²×Dic₄₃	1	8	yes	—
262	C₂²×C₄	—	2	8	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	4	4	no	—
265	C₂²×C₄	—	43	16	no	—
266	C₂²×C₄	—	172	8	no	—
267	C₂²×C₄	—	172	16	no	—
268	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
269	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
270	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
271	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
274	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
275	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
276	C₄⋊C₄	—	172	8	no	—
277	C₄⋊C₄	—	172	8	no	—
278	C₂²⋊C₄	—	86	16	no	—
279	C₂²⋊C₄	—	172	8	no	—
280	C₂²⋊C₄	—	172	16	no	—
281	C₂²⋊C₄	C₂²×Dic₄₃	1	32	yes	—
282	C₂²⋊C₄	C₂²×Dic₄₃	1	32	yes	—
283	C₄²	—	172	8	no	—
284	C₄²	—	172	8	no	—
285	C₄²	C₂²×Dic₄₃	1	4	yes	—
286	C₄²	C₂²×Dic₄₃	1	8	yes	—
287	C₄²	C₂²×Dic₄₃	1	4	yes	—
288	C₂³	—	2	4	no	—
289	C₂³	—	2	4	no	—
290	C₂³	—	2	4	no	—
291	C₂³	—	2	16	no	—
292	C₂³	—	2	16	no	—
293	C₂³	—	4	16	no	—
294	C₂³	—	8	16	no	—
295	C₂³	C₂³×Dic₄₃	1	8	yes	—
296	C₂³	C₂³×Dic₄₃	1	2	yes	—
297	C₂³	#13063370	1	4	yes	—
298	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
299	Q₈	—	4	8	no	—
300	Q₈	—	4	8	no	—
301	D₄	—	4	8	no	—
302	D₄	—	4	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	16	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	4	8	no	—
312	C₂×C₄	—	86	16	no	—
313	C₂×C₄	—	86	16	no	—
314	C₂×C₄	—	86	16	no	—
315	C₂×C₄	—	344	16	no	—
316	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
317	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
318	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
319	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
320	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	8	no	—
323	C₂²	—	2	8	no	—
324	C₂²	—	2	16	no	—
325	C₂²	—	4	8	no	—
326	C₂²	—	4	8	no	—
327	C₂²	—	8	16	no	—
328	C₂²	#14583374	1	4	yes	—
329	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
330	C₂²	C₂×#13063370	1	2	yes	—
331	C₂²	#14583735	1	4	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	8	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	4	no	—
337	C₄	—	172	16	no	—
338	C₂	—	2	4	no	—
339	C₂	—	4	8	no	—
340	C₂	C₂×#14583374	1	1	yes	characteristic
341	C₂	#19147798	1	2	yes	—
342	C₂	#19148952	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Fitting	C₂²×#124×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂³×D₄×C₄₃	C₂²
hypercenter	C₂²×#124	D₄₃
hypocenter	C₄₃	#34366
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. #19137769	C₂	semidirect
2. #19137707	C₂	semidirect
3. #19137444	C₂	semidirect
4. #19137410	C₂	semidirect
5. #19137181	C₂	semidirect
7. #19136993	C₂	semidirect
11. #14582806	C₂²	semidirect
12. #14582806	C₂²	semidirect
13. #14582792	C₂²	semidirect
14. #14582792	C₂²	semidirect
17. #14582774	C₂²	semidirect
18. #14582767	C₂²	semidirect
19. #14582765	C₂²	semidirect
24. C₄×C₄⋊Dic₄₃	C₂²	semidirect
25. C₂²⋊C₄×Dic₄₃	C₂²	semidirect
27. #14582745	C₂²	semidirect
30. C₂×#124×C₄₃	C₄	semidirect
53. C₂²×Q₈×C₄₃	C₂×C₄	semidirect
63. #124×C₄₃	C₂×C₄	semidirect
64. #124×C₄₃	C₂×C₄	semidirect
65. #124×C₄₃	C₂×C₄	semidirect
66. C₂×C₂²⋊C₄×C₄₃	C₂×C₄	semidirect
69. C₂×C₄²×C₄₃	C₂×C₄	semidirect
182. C₂×#124	Dic₄₃	semidirect
207. C₄₃	#34366	semidirect
210. C₂²×Q₈	C₂×Dic₄₃	semidirect
221. #124	C₂×Dic₄₃	semidirect
222. #124	C₂×Dic₄₃	semidirect
223. #124	C₂×Dic₄₃	semidirect
231. C₂×C₂²⋊C₄	C₂×Dic₄₃	semidirect
236. C₂×C₄²	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19137769	2
#19137707	2
#19137444	2
#19137410	2
#19137181	2
#19137173	2
#19136993	2
C₂²×#124×C₄₃	2
#34366	43