Subgroups — #25115366

Order 11008.

822 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#10621877	C₂	1	2	yes	maximal
2	#19135371	C₂	1	2	yes	maximal, complemented
3	#19134802	C₂	1	4	yes	maximal, complemented
4	#19134733	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134633	C₂	1	2	yes	maximal, complemented
6	C₄×#13063261	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582664	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	4	yes	—
10	#14582977	—	2	2	no	—
11	#14582971	—	2	4	no	—
12	D₄×C₄₃⋊C₈	—	2	2	no	—
13	C₂×#13063261	C₂²	1	2	yes	—
14	C₂×#13063261	C₂²	1	4	yes	—
15	#14582923	C₂²	1	2	yes	—
16	#14582863	C₂²	1	4	yes	—
17	C₄×#10621877	C₂²	1	2	yes	—
18	#14582664	C₂²	1	2	yes	—
19	#14582664	C₂²	1	2	yes	—
20	#14582664	C₂²	1	1	yes	characteristic
21	#14582664	C₂²	1	1	yes	characteristic
22	#14582664	C₂²	1	4	yes	—
23	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₈×D₄×C₄₃	C₄	1	4	yes	—
26	C₈×D₄×C₄₃	C₄	1	2	yes	—
27	C₈×D₄×C₄₃	C₄	1	2	yes	—
28	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×#10621877	—	2	4	no	—
32	C₂×#10621877	—	2	8	no	—
33	C₂×#10621877	—	4	4	no	—
34	C₂×#10621877	C₂³	1	2	yes	—
35	C₂×#10621877	C₂³	1	4	yes	—
36	C₂×#10621877	C₂³	1	2	yes	—
37	C₂²×C₄₃⋊C₈	—	2	4	no	—
38	#13063261	—	2	4	no	—
39	#13063261	—	4	4	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
42	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
43	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
44	#13063236	—	2	2	no	—
45	C₄×C₄₃⋊C₈	—	2	2	no	—
46	C₂²×D₄×C₄₃	Q₈	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₃	—	4	4	no	—
49	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₄×D₄×C₄₃	C₂×C₄	1	2	yes	—
55	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
58	C₄×D₄×C₄₃	D₄	1	2	yes	—
59	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₃	Q₈	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
64	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
66	C₂²⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
69	C₂⁴×C₄₃	C₂×Q₈	1	2	yes	—
70	C₂×D₄×C₄₃	—	2	4	no	—
71	C₂×D₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×D₄×C₄₃	C₄⋊C₄	1	2	yes	—
73	C₂×D₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	—	2	4	no	—
75	C₂²×C₄×C₄₃	—	2	4	no	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	4	no	—
87	C₂×C₈×C₄₃	—	4	4	no	—
88	C₂×C₈×C₄₃	—	4	4	no	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
92	C₄⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
96	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
99	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	#10621877	—	2	2	no	—
104	#10621877	—	2	2	no	—
105	#10621877	—	2	4	no	—
106	#10621877	—	2	2	no	—
107	#10621877	—	2	4	no	—
108	#10621877	—	4	8	no	—
109	#10621877	#41	1	4	yes	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	2	no	—
112	C₂×C₄₃⋊C₈	—	2	4	no	—
113	C₂×C₄₃⋊C₈	—	4	8	no	—
114	C₂³×C₄₃	—	2	4	no	—
115	C₂³×C₄₃	—	2	4	no	—
116	C₂³×C₄₃	—	4	2	no	—
117	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
120	D₄×C₄₃	—	2	4	no	—
121	D₄×C₄₃	—	2	2	no	—
122	D₄×C₄₃	#108	1	4	yes	complemented
123	C₂×C₄×C₄₃	—	2	4	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	4	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
142	C₈×C₄₃	—	2	2	no	—
143	C₈×C₄₃	—	2	2	no	—
144	C₈×C₄₃	—	2	4	no	—
145	C₄₃⋊C₈	—	2	4	no	—
146	C₄₃⋊C₈	—	4	4	no	—
147	C₄₃⋊C₈	—	4	2	no	—
148	#22717	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	2	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	2	1	no	—
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	—	2	2	no	—
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	4	8	no	—
157	C₂²×C₄₃	C₂×#108	1	4	yes	—
158	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₄₃	#554	1	2	yes	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	C₂×#108	1	2	yes	—
166	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
167	C₄×C₄₃	#518	1	1	yes	characteristic
168	C₄×C₄₃	#546	1	2	yes	—
169	D₄×M₁₆	—	43	2	no	—
170	C₂×C₈×D₄	D₄₃	1	1	yes	characteristic
171	#1950	—	43	2	no	—
172	#1855	—	43	4	no	—
173	#1837	—	43	1	no	—
174	#1810	—	43	2	no	—
175	C₄×#108	—	43	1	no	—
176	C₂×#341	—	43	2	no	—
177	C₂×C₄₃	—	2	2	no	—
178	C₂×C₄₃	—	2	4	no	—
179	C₂×C₄₃	—	4	2	no	—
180	C₂×C₄₃	#2349	1	1	yes	characteristic
181	C₂×C₄₃	C₂²×#108	1	1	yes	characteristic
182	C₂×C₄₃	#2916	1	1	yes	characteristic
183	C₂²×M₁₆	—	43	4	no	—
184	C₂³×C₈	C₂×D₄₃	1	2	yes	—
185	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
186	#436	—	86	2	no	—
187	#435	—	86	4	no	—
188	C₈×D₄	—	86	2	no	—
189	C₈×D₄	Dic₄₃	1	4	yes	—
190	C₈×D₄	Dic₄₃	1	2	yes	—
191	C₈×D₄	Dic₄₃	1	2	yes	—
192	C₂×#108	—	43	2	no	—
193	C₂×#108	—	43	4	no	—
194	#423	—	43	2	no	—
195	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
196	#407	—	43	4	no	—
197	C₂×C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
198	C₄×M₁₆	—	43	2	no	—
199	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
200	#341	—	43	1	no	—
201	#341	—	43	2	no	—
202	#341	—	43	2	no	—
203	#341	—	43	4	no	—
204	#341	—	43	1	no	—
205	C₄₃	#22717	1	1	yes	characteristic, complemented, Sylow
206	C₂²×D₄	Dic₈₆	1	1	yes	characteristic
207	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
208	C₂×M₁₆	—	43	2	no	—
209	C₂×M₁₆	—	43	4	no	—
210	C₂×M₁₆	—	43	2	no	—
211	C₂×M₁₆	—	86	4	no	—
212	C₂×M₁₆	—	86	8	no	—
213	C₂×M₁₆	—	172	4	no	—
214	C₂²×C₈	—	4	4	no	—
215	C₂²×C₈	—	86	4	no	—
216	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
217	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
218	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
220	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
221	C₄×D₄	D₁₇₂	1	1	yes	characteristic
222	C₄×D₄	D₁₇₂	1	2	yes	—
223	C₄×D₄	D₁₇₂	1	1	yes	characteristic
224	C₄×D₄	C₂×Dic₄₃	1	2	yes	—
225	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
226	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
227	C₂×C₄⋊C₄	Dic₈₆	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
229	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
230	#108	—	86	4	no	—
231	#108	—	172	4	no	—
232	C₄⋊C₈	—	86	2	no	—
233	C₄⋊C₈	—	86	2	no	—
234	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
235	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
236	C₂²⋊C₈	—	86	4	no	—
237	C₂²⋊C₈	—	86	4	no	—
238	C₂²⋊C₈	C₂×Dic₄₃	1	4	yes	—
239	C₂²⋊C₈	C₂×Dic₄₃	1	4	yes	—
240	#97	—	86	2	no	—
241	C₄×C₈	—	86	2	no	—
242	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
243	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
244	C₂⁴	C₂×Dic₈₆	1	2	yes	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	C₄⋊Dic₄₃	1	2	yes	—
247	C₂×D₄	C₄⋊Dic₄₃	1	1	yes	characteristic
248	C₂×D₄	C₄⋊Dic₄₃	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
252	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
253	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
255	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
258	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
259	M₁₆	—	43	4	no	—
260	M₁₆	—	86	2	no	—
261	M₁₆	—	86	2	no	—
262	M₁₆	—	86	4	no	—
263	M₁₆	—	86	2	no	—
264	M₁₆	—	86	4	no	—
265	M₁₆	—	172	8	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	4	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	—	86	2	no	—
272	C₂×C₈	—	86	2	no	—
273	C₂×C₈	—	86	4	no	—
274	C₂×C₈	—	172	8	no	—
275	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
276	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
277	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
278	C₄⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
279	C₄⋊C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
280	C₄⋊C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
281	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
282	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
283	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
284	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
285	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
286	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
287	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
288	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
289	C₂³	—	2	4	no	—
290	C₂³	—	2	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
293	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
294	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	D₄	#13063261	1	4	yes	complemented
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
315	C₂×C₄	#41×D₄₃	1	2	yes	—
316	C₂×C₄	#13063370	1	2	yes	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	4	no	—
320	C₈	—	86	4	no	—
321	C₈	—	172	4	no	—
322	C₈	—	172	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	8	no	—
331	C₂²	C₂×#13063261	1	4	yes	—
332	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
333	C₂²	#14583608	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	C₂×#13063261	1	2	yes	—
340	C₄	#41×Dic₄₃	1	1	yes	characteristic
341	C₄	#14583374	1	1	yes	characteristic
342	C₄	#14583544	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	4	no	—
345	C₂	—	4	2	no	—
346	C₂	#19137505	1	1	yes	characteristic
347	C₂	C₂²×#13063261	1	1	yes	characteristic
348	C₂	#19143206	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₈×D₄×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₄₃	Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₄₃
hypocenter	C₄₃	#22717
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #19135371	C₂	semidirect
3. #19134802	C₂	semidirect
4. #19134733	C₂	semidirect
5. #19134633	C₂	semidirect
6. C₄×#13063261	C₂	semidirect
7. C₂×#14582664	C₂	semidirect
122. D₄×C₄₃	#108	semidirect
205. C₄₃	#22717	semidirect
297. D₄	#13063261	semidirect

Maximal subgroups

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

subgroup	index
D₄×#10621877	2
#19135371	2
#19134802	2
#19134733	2
#19134633	2
C₄×#13063261	2
C₂×#14582664	2
C₂×C₈×D₄×C₄₃	2
#22717	43