Subgroups — #25256643

Order 11008.

4110 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137444	C₂	1	16	yes	maximal, complemented
2	#19137209	C₂	1	2	yes	maximal, complemented
3	#19137193	C₂	1	4	yes	maximal, complemented
4	#19136978	C₂	1	4	yes	maximal, complemented
5	C₂×#14582755	C₂	1	2	yes	maximal
6	#19136948	C₂	1	2	yes	maximal, complemented
7	C₂³×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063284	C₂²	1	8	yes	complemented
9	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
11	#14582792	C₂²	1	32	yes	complemented
12	#14582774	C₂²	1	16	yes	complemented
13	#14582762	C₂²	1	8	yes	complemented
14	#14582759	C₂²	1	8	yes	complemented
15	#14582759	C₂²	1	8	yes	complemented
16	#14582755	C₂²	1	8	yes	—
17	#14582755	C₂²	1	4	yes	—
18	#14582755	C₂²	1	4	yes	—
19	C₂²⋊C₄×Dic₄₃	C₂²	1	16	yes	complemented
20	C₄×C₂²⋊Dic₄₃	C₂²	1	8	yes	complemented
21	C₄×C₂²⋊Dic₄₃	C₂²	1	8	yes	complemented
22	C₂⁶×C₄₃	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄×C₄₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₄₃	C₄	1	16	yes	complemented
25	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	8	yes	—
28	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₂³×Dic₄₃	C₂³	1	8	yes	—
30	#13063284	—	4	16	no	—
31	C₂×C₄⋊Dic₄₃	C₂³	1	16	yes	complemented
32	C₂×C₂²⋊Dic₄₃	—	2	16	no	—
33	C₂×C₂²⋊Dic₄₃	—	4	16	no	—
34	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
35	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
36	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
37	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
38	C₂×C₄×Dic₄₃	C₂³	1	16	yes	complemented
39	C₄₃⋊#95	C₂³	1	16	yes	complemented
40	C₄₃⋊#95	C₂³	1	16	yes	complemented
41	C₂⁵×C₄₃	—	2	8	no	—
42	C₂⁵×C₄₃	—	4	8	no	—
43	C₂⁵×C₄₃	C₂×C₄	1	8	yes	—
44	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
45	C₂⁵×C₄₃	C₂³	1	4	yes	—
46	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
47	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	—	4	8	no	—
49	C₂³×C₄×C₄₃	C₂×C₄	1	16	yes	complemented
50	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
51	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
52	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
53	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
55	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
56	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	64	yes	complemented
57	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	16	yes	complemented
58	C₂×C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
59	C₂×C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
61	C₂⁴×C₄₃	—	2	4	no	—
62	C₂⁴×C₄₃	—	2	8	no	—
63	C₂⁴×C₄₃	—	2	32	no	—
64	C₂⁴×C₄₃	—	2	16	no	—
65	C₂⁴×C₄₃	—	2	4	no	—
66	C₂⁴×C₄₃	—	2	16	no	—
67	C₂⁴×C₄₃	—	4	32	no	—
68	C₂⁴×C₄₃	—	4	4	no	—
69	C₂⁴×C₄₃	—	4	8	no	—
70	C₂⁴×C₄₃	—	4	4	no	—
71	C₂⁴×C₄₃	—	4	64	no	—
72	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
73	C₂⁴×C₄₃	C₂²×C₄	1	16	yes	—
74	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
75	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
76	C₂⁴×C₄₃	#41	1	8	yes	—
77	C₂⁴×C₄₃	C₂⁴	1	2	yes	—
78	C₂⁴×C₄₃	C₂⁴	1	2	yes	—
79	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂⁴×C₄₃	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₃	—	2	16	no	—
82	C₂²×C₄×C₄₃	—	4	32	no	—
83	C₂²×C₄×C₄₃	—	4	4	no	—
84	C₂²×C₄×C₄₃	—	4	8	no	—
85	C₂²×C₄×C₄₃	—	4	4	no	—
86	C₂²×C₄×C₄₃	C₂²×C₄	1	16	yes	—
87	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
88	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
89	C₂²×C₄×C₄₃	C₂²×C₄	1	16	yes	—
90	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
92	C₂²×Dic₄₃	—	2	16	no	—
93	C₂²×Dic₄₃	—	4	16	no	—
94	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
95	C₂²⋊C₄×C₄₃	C₂²×C₄	1	16	yes	—
96	C₂²⋊C₄×C₄₃	C₂²×C₄	1	16	yes	—
97	C₂²⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
98	C₂²⋊Dic₄₃	—	2	16	no	—
99	C₂²⋊Dic₄₃	—	2	16	no	—
100	C₂²⋊Dic₄₃	—	2	32	no	—
101	C₂²⋊Dic₄₃	—	4	16	no	—
102	C₄⋊Dic₄₃	—	4	16	no	—
103	C₄×Dic₄₃	—	4	16	no	—
104	C₂³×C₄₃	—	2	32	no	—
105	C₂³×C₄₃	—	2	8	no	—
106	C₂³×C₄₃	—	2	16	no	—
107	C₂³×C₄₃	—	2	4	no	—
108	C₂³×C₄₃	—	2	8	no	—
109	C₂³×C₄₃	—	2	4	no	—
110	C₂³×C₄₃	—	2	4	no	—
111	C₂³×C₄₃	—	2	32	no	—
112	C₂³×C₄₃	—	2	8	no	—
113	C₂³×C₄₃	—	2	32	no	—
114	C₂³×C₄₃	—	2	4	no	—
115	C₂³×C₄₃	—	2	8	no	—
116	C₂³×C₄₃	—	4	8	no	—
117	C₂³×C₄₃	—	4	16	no	—
118	C₂³×C₄₃	—	4	8	no	—
119	C₂³×C₄₃	—	4	16	no	—
120	C₂³×C₄₃	—	4	64	no	—
121	C₂³×C₄₃	—	4	128	no	—
122	C₂³×C₄₃	—	4	16	no	—
123	C₂³×C₄₃	#117	1	16	yes	complemented
124	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
125	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
126	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
127	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
128	C₂³×C₄₃	C₂×#41	1	4	yes	—
129	C₂³×C₄₃	C₂×#41	1	8	yes	—
130	C₂³×C₄₃	H(5,𝔽₂)	1	8	yes	—
131	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
132	C₂×C₄×C₄₃	—	2	32	no	—
133	C₂×C₄×C₄₃	—	2	8	no	—
134	C₂×C₄×C₄₃	—	2	8	no	—
135	C₂×C₄×C₄₃	—	4	8	no	—
136	C₂×C₄×C₄₃	—	4	8	no	—
137	C₂×C₄×C₄₃	—	4	16	no	—
138	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
139	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
140	C₂×Dic₄₃	—	2	16	no	—
141	C₂×Dic₄₃	—	2	32	no	—
142	C₂×Dic₄₃	—	8	16	no	—
143	#34176	—	43	1	no	maximal, Sylow
144	C₂²×C₄₃	—	2	32	no	—
145	C₂²×C₄₃	—	2	4	no	—
146	C₂²×C₄₃	—	2	8	no	—
147	C₂²×C₄₃	—	2	4	no	—
148	C₂²×C₄₃	—	2	4	no	—
149	C₂²×C₄₃	—	2	8	no	—
150	C₂²×C₄₃	—	2	16	no	—
151	C₂²×C₄₃	—	2	16	no	—
152	C₂²×C₄₃	—	2	4	no	—
153	C₂²×C₄₃	—	2	8	no	—
154	C₂²×C₄₃	—	4	16	no	—
155	C₂²×C₄₃	—	4	64	no	—
156	C₂²×C₄₃	—	4	16	no	—
157	C₂²×C₄₃	—	4	8	no	—
158	C₂²×C₄₃	C₂×#117	1	4	yes	—
159	C₂²×C₄₃	C₂×#117	1	8	yes	—
160	C₂²×C₄₃	#518	1	8	yes	—
161	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
162	C₂²×C₄₃	C₂²×#41	1	2	yes	—
163	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	4	yes	—
164	C₄×C₄₃	—	2	8	no	—
165	C₄×C₄₃	—	2	8	no	—
166	Dic₄₃	—	4	16	no	—
167	C₂³×C₂²⋊C₄	D₄₃	1	1	yes	characteristic
168	#2342	—	43	16	no	—
169	#2315	—	43	2	no	—
170	#2312	—	43	4	no	—
171	#2280	—	43	4	no	—
172	C₂×#379	—	43	2	no	—
173	#2273	—	43	2	no	—
174	C₂×C₄₃	—	2	4	no	—
175	C₂×C₄₃	—	2	8	no	—
176	C₂×C₄₃	—	2	8	no	—
177	C₂×C₄₃	—	4	4	no	—
178	C₂×C₄₃	C₂²×#117	1	1	yes	characteristic
179	C₂×C₄₃	C₂×#518	1	2	yes	—
180	C₂×C₄₃	#3490	1	4	yes	—
181	C₂⁶	C₂×D₄₃	1	1	yes	characteristic
182	C₂⁴×C₄	C₂×D₄₃	1	2	yes	—
183	C₂×#117	—	43	8	no	—
184	C₂²×C₂²⋊C₄	—	43	4	no	—
185	C₂²×C₂²⋊C₄	—	43	8	no	—
186	C₂²×C₂²⋊C₄	Dic₄₃	1	16	yes	complemented
187	C₂²×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
188	C₂²×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
190	C₂²×C₂²⋊C₄	C₂×D₄₃	1	8	yes	—
191	#388	—	43	32	no	—
192	#385	—	43	16	no	—
193	#381	—	43	8	no	—
194	#380	—	43	8	no	—
195	#380	—	43	8	no	—
196	#379	—	43	4	no	—
197	#379	—	43	8	no	—
198	#379	—	43	4	no	—
199	C₄×C₂²⋊C₄	—	43	8	no	—
200	C₄×C₂²⋊C₄	—	43	8	no	—
201	C₄×C₂²⋊C₄	—	43	16	no	—
202	C₄₃	#34176	1	1	yes	characteristic, complemented, Sylow
203	C₂⁵	—	2	8	no	—
204	C₂⁵	—	4	8	no	—
205	C₂⁵	C₂×Dic₄₃	1	8	yes	—
206	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
207	C₂⁵	C₂²×D₄₃	1	4	yes	—
208	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
209	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
210	C₂³×C₄	—	4	8	no	—
211	C₂³×C₄	—	43	8	no	—
212	C₂³×C₄	C₂×Dic₄₃	1	16	yes	complemented
213	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
214	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
215	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
216	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
217	#117	—	172	16	no	—
218	C₂×C₄⋊C₄	—	43	16	no	—
219	C₂×C₂²⋊C₄	—	43	16	no	—
220	C₂×C₂²⋊C₄	—	43	8	no	—
221	C₂×C₂²⋊C₄	—	43	8	no	—
222	C₂×C₂²⋊C₄	—	43	16	no	—
223	C₂×C₂²⋊C₄	—	86	16	no	—
224	C₂×C₂²⋊C₄	—	172	16	no	—
225	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
226	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
227	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	64	yes	complemented
228	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	16	yes	complemented
229	C₂×C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
230	C₂×C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
231	C₂×C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
232	C₂×C₄²	—	43	16	no	—
233	#95	—	43	16	no	—
234	#95	—	43	16	no	—
235	C₂⁴	—	2	4	no	—
236	C₂⁴	—	2	8	no	—
237	C₂⁴	—	2	32	no	—
238	C₂⁴	—	2	16	no	—
239	C₂⁴	—	2	4	no	—
240	C₂⁴	—	2	16	no	—
241	C₂⁴	—	4	4	no	—
242	C₂⁴	—	4	8	no	—
243	C₂⁴	—	4	4	no	—
244	C₂⁴	—	4	32	no	—
245	C₂⁴	—	4	64	no	—
246	C₂⁴	#10621898	1	8	yes	—
247	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
248	C₂⁴	C₂²×Dic₄₃	1	16	yes	—
249	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
250	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
251	C₂⁴	C₂³×D₄₃	1	2	yes	—
252	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
253	C₂⁴	C₂³×D₄₃	1	2	yes	—
254	C₂⁴	C₂³×D₄₃	1	2	yes	—
255	C₂²×C₄	—	2	16	no	—
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	—	4	8	no	—
258	C₂²×C₄	—	4	4	no	—
259	C₂²×C₄	—	4	32	no	—
260	C₂²×C₄	—	43	16	no	—
261	C₂²×C₄	—	86	16	no	—
262	C₂²×C₄	—	172	16	no	—
263	C₂²×C₄	C₂²×Dic₄₃	1	16	yes	—
264	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
265	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
266	C₂²×C₄	C₂²×Dic₄₃	1	16	yes	—
267	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
268	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
269	C₄⋊C₄	—	172	16	no	—
270	C₂²⋊C₄	—	86	16	no	—
271	C₂²⋊C₄	—	86	16	no	—
272	C₂²⋊C₄	—	86	32	no	—
273	C₂²⋊C₄	—	172	16	no	—
274	C₂²⋊C₄	C₂²×Dic₄₃	1	16	yes	—
275	C₂²⋊C₄	C₂²×Dic₄₃	1	16	yes	—
276	C₂²⋊C₄	C₂²×Dic₄₃	1	32	yes	—
277	C₄²	—	172	16	no	—
278	C₂³	—	2	32	no	—
279	C₂³	—	2	8	no	—
280	C₂³	—	2	16	no	—
281	C₂³	—	2	4	no	—
282	C₂³	—	2	4	no	—
283	C₂³	—	2	4	no	—
284	C₂³	—	2	8	no	—
285	C₂³	—	2	32	no	—
286	C₂³	—	2	8	no	—
287	C₂³	—	2	32	no	—
288	C₂³	—	2	4	no	—
289	C₂³	—	2	8	no	—
290	C₂³	—	4	16	no	—
291	C₂³	—	4	8	no	—
292	C₂³	—	4	16	no	—
293	C₂³	—	4	8	no	—
294	C₂³	—	4	64	no	—
295	C₂³	—	4	128	no	—
296	C₂³	—	4	16	no	—
297	C₂³	#13063284	1	16	yes	complemented
298	C₂³	C₂³×Dic₄₃	1	4	yes	—
299	C₂³	C₂³×Dic₄₃	1	4	yes	—
300	C₂³	C₂³×Dic₄₃	1	2	yes	—
301	C₂³	C₂³×Dic₄₃	1	4	yes	—
302	C₂³	C₂×#10621898	1	4	yes	—
303	C₂³	C₂×#10621898	1	8	yes	—
304	C₂³	#13063370	1	8	yes	—
305	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
306	C₂×C₄	—	2	32	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	4	8	no	—
310	C₂×C₄	—	4	8	no	—
311	C₂×C₄	—	4	16	no	—
312	C₂×C₄	—	86	16	no	—
313	C₂×C₄	—	86	32	no	—
314	C₂×C₄	—	344	16	no	—
315	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
316	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
317	C₂²	—	2	32	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	8	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	16	no	—
323	C₂²	—	2	8	no	—
324	C₂²	—	2	16	no	—
325	C₂²	—	2	8	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	4	16	no	—
328	C₂²	—	4	64	no	—
329	C₂²	—	4	16	no	—
330	C₂²	—	4	8	no	—
331	C₂²	C₂×#13063284	1	4	yes	—
332	C₂²	C₂×#13063284	1	8	yes	—
333	C₂²	#14583374	1	8	yes	—
334	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
335	C₂²	C₂²×#10621898	1	2	yes	—
336	C₂²	C₂×#13063370	1	4	yes	—
337	C₄	—	2	8	no	—
338	C₄	—	2	8	no	—
339	C₄	—	172	16	no	—
340	C₂	—	2	4	no	—
341	C₂	—	2	8	no	—
342	C₂	—	2	8	no	—
343	C₂	—	4	4	no	—
344	C₂	C₂²×#13063284	1	1	yes	characteristic
345	C₂	C₂×#14583374	1	2	yes	—
346	C₂	#19148284	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Fitting	C₂³×C₂²⋊C₄×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂⁶×C₄₃	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₄₃
hypocenter	C₄₃	#34176
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #19137444	C₂	semidirect
2. #19137209	C₂	semidirect
3. #19137193	C₂	semidirect
4. #19136978	C₂	semidirect
6. #19136948	C₂	semidirect
8. C₂×#13063284	C₂²	semidirect
11. #14582792	C₂²	semidirect
12. #14582774	C₂²	semidirect
13. #14582762	C₂²	semidirect
14. #14582759	C₂²	semidirect
15. #14582759	C₂²	semidirect
19. C₂²⋊C₄×Dic₄₃	C₂²	semidirect
20. C₄×C₂²⋊Dic₄₃	C₂²	semidirect
21. C₄×C₂²⋊Dic₄₃	C₂²	semidirect
24. C₂²×C₂²⋊C₄×C₄₃	C₄	semidirect
31. C₂×C₄⋊Dic₄₃	C₂³	semidirect
38. C₂×C₄×Dic₄₃	C₂³	semidirect
39. C₄₃⋊#95	C₂³	semidirect
40. C₄₃⋊#95	C₂³	semidirect
49. C₂³×C₄×C₄₃	C₂×C₄	semidirect
56. C₂×C₂²⋊C₄×C₄₃	C₂×C₄	semidirect
57. C₂×C₂²⋊C₄×C₄₃	C₂×C₄	semidirect
123. C₂³×C₄₃	#117	semidirect
186. C₂²×C₂²⋊C₄	Dic₄₃	semidirect
202. C₄₃	#34176	semidirect
212. C₂³×C₄	C₂×Dic₄₃	semidirect
227. C₂×C₂²⋊C₄	C₂×Dic₄₃	semidirect
228. C₂×C₂²⋊C₄	C₂×Dic₄₃	semidirect
297. C₂³	#13063284	semidirect

Maximal subgroups

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

subgroup	index
#19137444	2
#19137209	2
#19137193	2
#19136978	2
C₂×#14582755	2
#19136948	2
C₂³×C₂²⋊C₄×C₄₃	2
#34176	43