Subgroups — #25114742

Order 11008.

982 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582937	C₂	1	2	yes	maximal, complemented
2	#19137544	C₂	1	2	yes	maximal
3	#19135359	C₂	1	2	yes	maximal, complemented
4	#19134796	C₂	1	4	yes	maximal, complemented
5	#19134755	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134624	C₂	1	2	yes	maximal, complemented
7	C₄×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	4	yes	—
10	#14582937	—	2	8	no	—
11	C₂×#13063259	C₂²	1	2	yes	—
12	C₂×#13063259	C₂²	1	1	yes	characteristic
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063257	C₂²	1	4	yes	—
16	#14582802	—	2	2	no	—
17	#14582792	—	2	4	no	—
18	#14582769	C₂²	1	2	yes	—
19	#14582759	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₄₃	—	2	2	no	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
22	#14582662	C₂²	1	2	yes	—
23	#14582662	C₂²	1	4	yes	—
24	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₄₃	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	#13063284	—	2	8	no	—
31	#13063284	—	4	4	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
37	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
38	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
39	C₂×C₄×Dic₄₃	—	2	4	no	—
40	C₂×C₄×Dic₄₃	—	2	2	no	—
41	#13063259	—	2	4	no	—
42	#13063259	—	2	2	no	—
43	#13063259	—	4	2	no	—
44	#13063259	D₄	1	4	yes	complemented
45	#13063257	—	2	8	no	—
46	C₄₃⋊#95	—	2	2	no	—
47	C₄₃⋊#95	—	2	4	no	—
48	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₃	—	2	4	no	—
51	C₂²×C₈×C₄₃	—	2	4	no	—
52	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	—
53	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₄₃	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₄₃	Q₈	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
62	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
64	C₂⁴×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₄₃	—	2	2	no	—
66	C₂×D₄×C₄₃	—	2	2	no	—
67	C₂×D₄×C₄₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₃	—	2	2	no	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	2	2	no	—
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₃	—	2	2	no	—
79	C₂²×Dic₄₃	—	2	2	no	—
80	C₂²×Dic₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	16	no	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₄₃	#41	1	2	yes	—
91	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	8	yes	—
93	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
95	C₂²⋊Dic₄₃	—	4	8	no	—
96	C₂²⋊Dic₄₃	—	8	4	no	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	2	2	no	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	C₄⋊Dic₄₃	—	2	2	no	—
102	C₄⋊Dic₄₃	—	4	4	no	—
103	C₄⋊Dic₄₃	—	8	2	no	—
104	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
105	C₄×Dic₄₃	—	4	8	no	—
106	C₄×Dic₄₃	—	4	4	no	—
107	C₂³×C₄₃	—	2	2	no	—
108	C₂³×C₄₃	—	2	2	no	—
109	C₂³×C₄₃	—	2	2	no	—
110	C₂³×C₄₃	—	2	2	no	—
111	C₂³×C₄₃	—	2	2	no	—
112	C₂³×C₄₃	—	2	2	no	—
113	C₂³×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₄₃	—	2	8	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	4	no	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	4	no	—
123	C₂×C₄×C₄₃	—	2	4	no	—
124	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₃	#135	1	2	yes	—
128	C₂×C₄×C₄₃	#135	1	2	yes	—
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	4	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	4	4	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	4	no	—
142	C₂×Dic₄₃	—	4	4	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	C₈×C₄₃	—	2	4	no	—
145	C₈×C₄₃	—	2	2	no	—
146	C₈×C₄₃	C₄×D₄	1	4	yes	—
147	#22658	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	4	no	—
149	C₂²×C₄₃	—	2	4	no	—
150	C₂²×C₄₃	—	2	4	no	—
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	—	4	2	no	—
153	C₂²×C₄₃	—	4	2	no	—
154	C₂²×C₄₃	—	4	2	no	—
155	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄₃	D₄²	1	1	yes	characteristic
157	C₂²×C₄₃	#547	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
160	C₂²×C₄₃	#576	1	1	yes	characteristic
161	C₂²×C₄₃	#578	1	1	yes	characteristic
162	C₄×C₄₃	—	2	4	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	#427	1	4	yes	complemented
165	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₄₃	#520	1	1	yes	characteristic
167	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
168	C₄×C₄₃	#554	1	1	yes	characteristic
169	Dic₄₃	—	4	4	no	—
170	Dic₄₃	—	8	2	no	—
171	Dic₄₃	—	8	4	no	—
172	C₂×C₈×D₄	D₄₃	1	1	yes	characteristic
173	C₂×#427	—	43	2	no	—
174	#2354	—	43	2	no	—
175	#1948	—	43	2	no	—
176	#1854	—	43	4	no	—
177	#1845	—	43	1	no	—
178	#1808	—	43	2	no	—
179	C₄×#107	—	43	1	no	—
180	C₂×C₄₃	—	4	4	no	—
181	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
183	C₂×C₄₃	#2913	1	1	yes	characteristic
184	C₂×C₄₃	#3283	1	1	yes	characteristic
185	C₂×C₄₃	#3287	1	1	yes	characteristic
186	C₂×C₄₃	#3341	1	1	yes	characteristic
187	C₂×C₄₃	#3342	1	1	yes	characteristic
188	C₂³×C₈	C₂×D₄₃	1	2	yes	—
189	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#117	—	43	4	no	—
191	C₈×D₄	Dic₄₃	1	8	yes	complemented
192	#427	—	86	8	no	—
193	C₂×#107	—	43	1	no	—
194	C₂×#107	—	43	1	no	—
195	C₂×#107	—	43	2	no	—
196	C₂×#107	—	43	2	no	—
197	C₂×#106	—	43	4	no	—
198	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
200	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
201	#390	—	86	2	no	—
202	#388	—	86	4	no	—
203	#384	—	43	2	no	—
204	#380	—	43	4	no	—
205	C₄×C₄⋊C₄	—	43	2	no	—
206	C₄×C₄⋊C₄	—	86	2	no	—
207	#340	—	43	4	no	—
208	#340	—	43	2	no	—
209	C₄₃	#22658	1	1	yes	characteristic, complemented, Sylow
210	C₂²×D₄	D₁₇₂	1	1	yes	characteristic
211	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
212	C₂²×C₈	—	2	4	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂×Dic₄₃	1	8	yes	—
215	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
216	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₄×D₄	Dic₈₆	1	4	yes	—
219	C₄×D₄	C₂×Dic₄₃	1	4	yes	—
220	#117	—	86	8	no	—
221	#117	—	172	4	no	—
222	C₂×C₄⋊C₄	—	43	2	no	—
223	C₂×C₄⋊C₄	—	43	4	no	—
224	C₂×C₄⋊C₄	—	43	2	no	—
225	C₂×C₄⋊C₄	—	86	2	no	—
226	C₂×C₄⋊C₄	—	86	2	no	—
227	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	86	4	no	—
229	C₂×C₂²⋊C₄	—	86	4	no	—
230	C₂×C₂²⋊C₄	D₁₇₂	1	2	yes	—
231	C₂×C₄²	—	86	4	no	—
232	C₂×C₄²	—	86	2	no	—
233	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
234	#107	—	43	4	no	—
235	#107	—	86	4	no	—
236	#107	—	86	2	no	—
237	#107	—	172	2	no	—
238	#106	—	86	8	no	—
239	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
240	C₂²⋊C₈	C₂×Dic₄₃	1	8	yes	—
241	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
242	#95	—	86	2	no	—
243	#95	—	86	4	no	—
244	C₂⁴	C₂×D₁₇₂	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₄⋊Dic₄₃	1	4	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	86	2	no	—
253	C₂²×C₄	—	86	2	no	—
254	C₂²×C₄	—	86	4	no	—
255	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
256	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
257	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
260	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	16	no	—
266	C₂×C₈	D₄×D₄₃	1	2	yes	—
267	C₂×C₈	#10621900	1	2	yes	—
268	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
269	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
270	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
271	C₄⋊C₄	—	43	4	no	—
272	C₄⋊C₄	—	86	2	no	—
273	C₄⋊C₄	—	86	4	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₄⋊C₄	—	86	4	no	—
276	C₄⋊C₄	—	86	2	no	—
277	C₄⋊C₄	—	344	2	no	—
278	C₄⋊C₄	—	172	4	no	—
279	C₄⋊C₄	C₄⋊Dic₄₃	1	4	yes	—
280	C₂²⋊C₄	—	344	4	no	—
281	C₂²⋊C₄	—	172	8	no	—
282	C₂²⋊C₄	C₄⋊Dic₄₃	1	8	yes	—
283	C₄²	—	172	8	no	—
284	C₄²	—	172	4	no	—
285	C₄²	C₂×Dic₈₆	1	2	yes	—
286	C₄²	C₂²×Dic₄₃	1	2	yes	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂×C₄⋊Dic₄₃	1	4	yes	—
294	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
295	D₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	86	4	no	—
305	C₂×C₄	—	172	2	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	—	172	4	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	—	172	4	no	—
310	C₂×C₄	—	172	4	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
316	C₂×C₄	#13063347	1	2	yes	—
317	C₂×C₄	#13063347	1	2	yes	—
318	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
320	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
321	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
322	C₂×C₄	#13063371	1	1	yes	characteristic
323	C₂×C₄	#13063372	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	2	no	—
326	C₈	D₄×Dic₄₃	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
335	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
336	C₂²	#14583553	1	1	yes	characteristic
337	C₂²	C₂×#13063347	1	1	yes	characteristic
338	C₂²	C₂×#13063347	1	1	yes	characteristic
339	C₂²	#14583704	1	1	yes	characteristic
340	C₂²	#14583716	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	344	2	no	—
344	C₄	—	344	4	no	—
345	C₄	—	172	4	no	—
346	C₄	#14582937	1	4	yes	complemented
347	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
348	C₄	#14583382	1	1	yes	characteristic
349	C₄	D₄×Dic₈₆	1	1	yes	characteristic
350	C₄	#14583606	1	1	yes	characteristic
351	C₂	—	4	4	no	—
352	C₂	D₄×C₄⋊Dic₄₃	1	1	yes	characteristic
353	C₂	C₂×#14582937	1	1	yes	characteristic
354	C₂	#19143195	1	1	yes	characteristic
355	C₂	#19146317	1	1	yes	characteristic
356	C₂	#19146355	1	1	yes	characteristic
357	C₂	#19147018	1	1	yes	characteristic
358	C₂	#19147026	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₈×D₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₄×C₄₃	D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₄₃
hypocenter	C₄₃	#22658
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14582937	C₂	semidirect
3. #19135359	C₂	semidirect
4. #19134796	C₂	semidirect
5. #19134755	C₂	semidirect
6. #19134624	C₂	semidirect
7. C₄×#13063259	C₂	semidirect
26. C₈×D₄×C₄₃	C₄	semidirect
44. #13063259	D₄	semidirect
164. C₄×C₄₃	#427	semidirect
191. C₈×D₄	Dic₄₃	semidirect
209. C₄₃	#22658	semidirect
346. C₄	#14582937	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582937	2
#19137544	2
#19135359	2
#19134796	2
#19134755	2
#19134624	2
C₄×#13063259	2
C₂×C₈×D₄×C₄₃	2
#22658	43