Subgroups — #25002467

Order 11008.

802 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	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
2	#19137002	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582670	C₂	1	4	yes	maximal
4	C₂×C₄₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂×#13063284	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	1	no	—
8	C₄×#10621877	—	2	1	no	—
9	C₄×#10621877	C₄	1	2	yes	complemented
10	C₄×#10621877	C₄	1	2	yes	complemented
11	C₂×#13063236	—	2	1	no	—
12	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
13	#14582769	—	2	1	no	—
14	#14582765	—	2	1	no	—
15	#14582759	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₄₃	—	2	2	no	—
17	#14582670	—	2	8	no	—
18	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#423	—	2	1	no	—
21	C₄₃×#423	—	2	1	no	—
22	C₄₃×#423	—	2	2	no	—
23	C₂×C₄⋊C₈×C₄₃	C₄	1	4	yes	complemented
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	—	2	2	no	—
27	C₂×#10621877	C₂×C₄	1	4	yes	—
28	C₂×#10621877	D₄	1	4	yes	—
29	C₂×#10621877	Q₈	1	4	yes	—
30	C₂²×C₄₃⋊C₈	—	2	2	no	—
31	#13063284	—	2	4	no	—
32	#13063284	—	2	4	no	—
33	C₂×C₄⋊Dic₄₃	—	2	4	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₃	—	2	2	no	—
37	#13063236	—	2	2	no	—
38	#13063236	—	2	2	no	—
39	C₄×C₄₃⋊C₈	—	2	2	no	—
40	C₄×C₄₃⋊C₈	—	2	2	no	—
41	C₄₃⋊#95	—	2	4	no	—
42	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₃	—	2	4	no	—
45	C₂×M₁₆×C₄₃	—	2	2	no	—
46	C₂×M₁₆×C₄₃	—	2	2	no	—
47	C₂×M₁₆×C₄₃	—	2	4	no	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	D₄	1	2	yes	—
57	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
58	C₄⋊C₈×C₄₃	—	2	4	no	—
59	C₄⋊C₈×C₄₃	—	2	4	no	—
60	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	4	4	no	—
62	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₃	#41	1	2	yes	—
77	C₂²×C₄×C₄₃	#41	1	2	yes	—
78	C₂²×Dic₄₃	—	2	4	no	—
79	M₁₆×C₄₃	—	4	4	no	—
80	M₁₆×C₄₃	—	4	2	no	—
81	M₁₆×C₄₃	—	4	2	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	2	8	no	—
84	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
94	C₂²⋊Dic₄₃	—	2	8	no	—
95	C₄⋊Dic₄₃	—	4	8	no	—
96	C₄⋊Dic₄₃	—	4	4	no	—
97	C₄×Dic₄₃	—	4	2	no	—
98	C₄×Dic₄₃	—	4	2	no	—
99	#10621877	—	2	4	no	—
100	#10621877	—	2	4	no	—
101	#10621877	—	2	4	no	—
102	#10621877	C₄⋊C₄	1	8	yes	complemented
103	C₂×C₄₃⋊C₈	—	2	4	no	—
104	C₂×C₄₃⋊C₈	—	2	4	no	—
105	C₂×C₄₃⋊C₈	—	2	4	no	—
106	C₂³×C₄₃	—	4	2	no	—
107	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₄₃	C₄×D₄	1	2	yes	—
111	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	#95	1	2	yes	—
119	C₂×C₄×C₄₃	#95	1	2	yes	—
120	C₂×C₄×C₄₃	#95	1	2	yes	—
121	C₂×C₄×C₄₃	#95	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#117	1	2	yes	—
128	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	#121	1	2	yes	—
131	C₂×C₄×C₄₃	#121	1	2	yes	—
132	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#125	1	2	yes	—
137	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
140	C₂×Dic₄₃	—	4	8	no	—
141	C₂×Dic₄₃	—	4	4	no	—
142	C₈×C₄₃	—	4	4	no	—
143	C₄₃⋊C₈	—	2	8	no	—
144	#10855	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	—	2	2	no	—
148	C₂²×C₄₃	—	4	2	no	—
149	C₂²×C₄₃	—	4	2	no	—
150	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
151	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
152	C₂²×C₄₃	#389	1	2	yes	—
153	C₂²×C₄₃	#390	1	1	yes	characteristic
154	C₂²×C₄₃	#391	1	1	yes	characteristic
155	C₄×C₄₃	—	2	2	no	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	#344	1	4	yes	complemented
159	C₄×C₄₃	#380	1	2	yes	—
160	C₄×C₄₃	#382	1	1	yes	characteristic
161	C₄×C₄₃	#384	1	1	yes	characteristic
162	C₄×C₄₃	#389	1	2	yes	—
163	C₄×C₄₃	#390	1	1	yes	characteristic
164	C₄×C₄₃	#391	1	1	yes	characteristic
165	Dic₄₃	—	8	4	no	—
166	C₂×#423	D₄₃	1	1	yes	characteristic
167	C₂×C₄×M₁₆	—	43	1	no	—
168	#2285	—	43	1	no	—
169	C₂×#344	—	43	4	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	2	2	no	—
172	C₂×C₄₃	—	2	2	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	C₄⋊#95	1	1	yes	characteristic
175	C₂×C₄₃	C₂×#344	1	2	yes	—
176	C₂²×M₁₆	—	43	2	no	—
177	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
178	C₂×#117	—	43	2	no	—
179	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
180	#423	—	2	2	no	—
181	#423	—	2	1	no	—
182	#423	—	2	1	no	—
183	C₂×C₄⋊C₈	Dic₄₃	1	4	yes	complemented
184	C₄×M₁₆	—	43	2	no	—
185	C₄×M₁₆	—	43	2	no	—
186	C₄×M₁₆	—	86	1	no	—
187	C₄×M₁₆	—	86	1	no	—
188	C₂×#97	—	86	1	no	—
189	C₂×C₄×C₈	—	86	1	no	—
190	#384	—	86	1	no	—
191	#382	—	86	1	no	—
192	#380	—	43	4	no	—
193	C₄×C₄⋊C₄	—	86	2	no	—
194	#344	—	86	8	no	—
195	C₄₃	#10855	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	C₄×D₄₃	1	2	yes	—
197	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	43	4	no	—
203	C₂×M₁₆	—	43	4	no	—
204	C₂×M₁₆	—	43	4	no	—
205	C₂×M₁₆	—	86	2	no	—
206	C₂×M₁₆	—	86	2	no	—
207	C₂×M₁₆	—	86	2	no	—
208	C₂²×C₈	—	86	2	no	—
209	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
210	#117	—	86	4	no	—
211	#117	—	86	4	no	—
212	C₂×C₄⋊C₄	—	86	4	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₂²⋊C₄	—	43	4	no	—
215	C₂×C₄²	—	86	2	no	—
216	C₂×C₄²	Dic₈₆	1	2	yes	—
217	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
219	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
220	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
221	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
222	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
223	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	4	no	—
227	#97	—	86	2	no	—
228	#97	—	86	2	no	—
229	C₄×C₈	—	86	2	no	—
230	C₄×C₈	—	86	2	no	—
231	#95	—	86	4	no	—
232	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	86	4	no	—
235	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
236	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
237	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
238	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
239	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
240	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
241	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₄₃	1	2	yes	—
245	C₂²×C₄	#10621900	1	2	yes	—
246	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
247	C₂²×C₄	#10621902	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	43	8	no	—
255	M₁₆	—	86	4	no	—
256	M₁₆	—	86	4	no	—
257	M₁₆	—	86	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	8	no	—
260	C₂×C₈	—	86	4	no	—
261	C₂×C₈	—	86	4	no	—
262	C₂×C₈	—	86	4	no	—
263	C₄⋊C₄	—	172	8	no	—
264	C₄⋊C₄	—	172	4	no	—
265	C₂²⋊C₄	—	86	8	no	—
266	C₄²	—	172	2	no	—
267	C₄²	—	172	2	no	—
268	C₄²	Dic₄₃⋊C₄	1	2	yes	—
269	C₄²	Dic₄₃⋊C₄	1	2	yes	—
270	C₄²	C₄⋊Dic₄₃	1	2	yes	—
271	C₄²	C₄⋊Dic₄₃	1	2	yes	—
272	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
273	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
274	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
275	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
276	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
277	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
280	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
281	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
282	C₂³	#13063287	1	2	yes	—
283	C₂³	#13063290	1	2	yes	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	172	8	no	—
291	C₂×C₄	—	172	4	no	—
292	C₂×C₄	C₄₃⋊#95	1	2	yes	—
293	C₂×C₄	C₄₃⋊#95	1	2	yes	—
294	C₂×C₄	C₄₃⋊#95	1	2	yes	—
295	C₂×C₄	C₄₃⋊#95	1	2	yes	—
296	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
299	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
301	C₂×C₄	#13063283	1	2	yes	—
302	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
303	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
304	C₂×C₄	#13063296	1	2	yes	—
305	C₂×C₄	#13063300	1	2	yes	—
306	C₂×C₄	#13063301	1	2	yes	—
307	C₂×C₄	#13063305	1	2	yes	—
308	C₂×C₄	#13063312	1	1	yes	characteristic
309	C₂×C₄	#13063314	1	1	yes	characteristic
310	C₂×C₄	#13063316	1	2	yes	—
311	C₂×C₄	#13063321	1	1	yes	characteristic
312	C₂×C₄	#13063323	1	1	yes	characteristic
313	C₂×C₄	#13063324	1	2	yes	—
314	C₈	—	4	4	no	—
315	C₈	—	86	8	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
322	C₂²	C₄⋊C₄×Dic₄₃	1	2	yes	—
323	C₂²	#14582795	1	2	yes	—
324	C₂²	#14582800	1	1	yes	characteristic
325	C₂²	#14582804	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	344	4	no	—
330	C₄	#14582670	1	4	yes	complemented
331	C₄	#14582756	1	2	yes	—
332	C₄	#14582764	1	1	yes	characteristic
333	C₄	#14582768	1	1	yes	characteristic
334	C₄	#14582798	1	2	yes	—
335	C₄	#14582802	1	1	yes	characteristic
336	C₄	#14582806	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	#19132824	1	1	yes	characteristic
342	C₂	C₂×#14582670	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₃
commutator	C₂³×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#423	C₂
socle	C₂²×C₄₃	C₂×#95
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#423	D₄₃
hypocenter	C₄₃	#10855
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
9. C₄×#10621877	C₄	semidirect
10. C₄×#10621877	C₄	semidirect
23. C₂×C₄⋊C₈×C₄₃	C₄	semidirect
102. #10621877	C₄⋊C₄	semidirect
158. C₄×C₄₃	#344	semidirect
183. C₂×C₄⋊C₈	Dic₄₃	semidirect
195. C₄₃	#10855	semidirect
330. C₄	#14582670	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621877	2
#19137002	2
C₂×#14582670	2
C₂×C₄₃×#423	2
#10855	43