Subgroups — #25095263

Order 11008.

954 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583023	C₂	1	1	yes	maximal, characteristic
3	C₄×#13063290	C₂	1	1	yes	maximal, characteristic
4	#19134464	C₂	1	1	yes	maximal, characteristic
5	#19134463	C₂	1	2	yes	maximal
6	#19134461	C₂	1	2	yes	maximal
7	#19134450	C₂	1	1	yes	maximal, characteristic
8	#19134447	C₂	1	2	yes	maximal
9	C₄×#13063235	C₂	1	1	yes	maximal, characteristic
10	#97×Dic₄₃	C₂	1	2	yes	maximal
11	C₄₃×#1770	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
13	C₂×#13063290	C₄	1	2	yes	—
14	C₈×Dic₈₆	C₄	1	8	yes	complemented
15	#14583023	C₄	1	8	yes	complemented
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
19	C₂×#13063236	C₂²	1	2	yes	—
20	C₂×#13063235	C₂²	1	2	yes	—
21	C₂×#13063235	C₂²	1	1	yes	characteristic
22	C₂×#13063235	C₂²	1	1	yes	characteristic
23	C₂×#13063235	C₂²	1	1	yes	characteristic
24	C₂×#13063235	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
27	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
28	C₄⋊C₄×Dic₄₃	C₄	1	2	yes	—
29	C₄×Dic₄₃⋊C₄	C₂²	1	2	yes	—
30	#14582750	C₄	1	4	yes	—
31	C₄²×Dic₄₃	C₂²	1	2	yes	—
32	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
33	#14582654	C₂²	1	2	yes	—
34	#14582653	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₄₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#336	C₂²	1	2	yes	—
40	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
44	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
45	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
46	#13063290	—	2	8	no	—
47	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
54	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
56	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
58	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
59	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
60	#13063236	—	2	4	no	—
61	#13063235	—	2	2	no	—
62	#13063235	—	2	4	no	—
63	#13063235	—	2	2	no	—
64	#13063235	—	2	2	no	—
65	#13063235	Q₈	1	4	yes	complemented
66	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
67	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
68	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
69	C₄₃⋊#95	C₂×C₄	1	4	yes	—
70	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
74	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
77	#97×C₄₃	—	2	4	no	—
78	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
79	#95×C₄₃	C₂×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
87	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₈₆	C₄²	1	4	yes	—
89	C₂×Dic₈₆	M₁₆	1	8	yes	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	1	no	—
94	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₄₃	#41	1	2	yes	—
99	C₄⋊C₄×C₄₃	—	2	4	no	—
100	C₄²×C₄₃	—	2	2	no	—
101	C₄²×C₄₃	—	2	2	no	—
102	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₄₃	C₄²	1	4	yes	—
105	Dic₄₃⋊C₄	—	2	8	no	—
106	Dic₄₃⋊C₄	C₄²	1	8	yes	—
107	C₄×Dic₄₃	—	2	4	no	—
108	C₄×Dic₄₃	—	2	1	no	—
109	C₄×Dic₄₃	—	2	1	no	—
110	C₄×Dic₄₃	—	2	8	no	—
111	C₄×Dic₄₃	—	4	4	no	—
112	C₄×Dic₄₃	—	4	2	no	—
113	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
114	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
115	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
116	C₄×Dic₄₃	#41	1	2	yes	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	1	no	—
119	C₂×C₄₃⋊C₈	—	2	1	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
125	C₂×C₄₃⋊C₈	#41	1	2	yes	—
126	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
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	4	no	—
131	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	4	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	C₂×C₄²	1	4	yes	—
147	C₂×Dic₄₃	C₂×C₄²	1	4	yes	—
148	C₂×Dic₄₃	C₄×Q₈	1	4	yes	—
149	C₂×Dic₄₃	C₂×M₁₆	1	8	yes	—
150	Dic₈₆	#97	1	16	yes	complemented
151	C₈×C₄₃	—	2	4	no	—
152	C₈×C₄₃	—	2	2	no	—
153	C₈×C₄₃	C₄×Q₈	1	4	yes	complemented
154	C₄₃⋊C₈	—	2	2	no	—
155	C₄₃⋊C₈	—	2	4	no	—
156	C₄₃⋊C₈	C₄×Q₈	1	4	yes	complemented
157	Q₈×#97	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₂²×C₄₃	C₄×#41	1	1	yes	characteristic
161	C₂²×C₄₃	C₂²×M₁₆	1	2	yes	—
162	C₂²×C₄₃	#568	1	2	yes	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	4	4	no	—
165	C₄×C₄₃	C₂×#97	1	4	yes	—
166	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
169	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
170	Dic₄₃	—	2	4	no	—
171	Dic₄₃	—	2	4	no	—
172	Dic₄₃	—	4	2	no	—
173	Dic₄₃	C₂×#97	1	8	yes	—
174	C₂×C₈×Q₈	—	43	1	no	—
175	C₂×C₈×Q₈	—	43	1	no	—
176	C₄²×Q₈	—	43	1	no	—
177	#1770	—	43	2	no	—
178	#1770	—	43	1	no	—
179	#1770	—	43	2	no	—
180	#1770	D₄₃	1	1	yes	characteristic
181	#1767	—	43	1	no	—
182	#1767	—	43	2	no	—
183	C₄×#97	—	43	2	no	—
184	C₄×#97	—	43	1	no	—
185	C₂×C₄₃	C₄²×Q₈	1	1	yes	characteristic
186	C₂×C₄₃	C₂²×#97	1	1	yes	characteristic
187	C₂×C₄₃	#2875	1	1	yes	characteristic
188	C₂×C₄₃	Q₈×M₁₆	1	2	yes	—
189	C₂×C₄₃	#2971	1	2	yes	—
190	C₂×C₄×Q₈	—	43	2	no	—
191	C₂×C₄×Q₈	—	43	1	no	—
192	C₈×Q₈	—	43	8	no	—
193	C₈×Q₈	—	43	8	no	—
194	C₂×C₄⋊C₈	—	43	2	no	—
195	C₂×C₄⋊C₈	—	43	2	no	—
196	C₂×C₄⋊C₈	—	43	1	no	—
197	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
198	C₂×#97	—	43	1	no	—
199	C₂×#97	—	43	2	no	—
200	C₂×#97	—	43	1	no	—
201	C₂×#97	—	43	1	no	—
202	C₂×#97	—	43	1	no	—
203	C₂×#97	—	43	2	no	—
204	C₂×#97	C₂×D₄₃	1	2	yes	—
205	C₂×C₄×C₈	—	43	2	no	—
206	C₂×C₄×C₈	—	43	2	no	—
207	C₂×C₄×C₈	—	43	1	no	—
208	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
209	C₄×C₄⋊C₄	—	43	2	no	—
210	C₄×C₄⋊C₄	—	43	2	no	—
211	C₄×C₄⋊C₄	—	43	4	no	—
212	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
213	C₄³	—	43	2	no	—
214	C₄³	—	43	1	no	—
215	#336	—	43	2	no	—
216	#336	—	43	2	no	—
217	#336	C₂×D₄₃	1	2	yes	—
218	C₄₃	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	43	1	no	—
220	C₂²×C₈	—	43	1	no	—
221	C₂²×C₈	—	43	2	no	—
222	C₂²×C₈	—	43	1	no	—
223	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
224	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
226	C₄×Q₈	—	43	4	no	—
227	C₄×Q₈	—	43	4	no	—
228	C₄×Q₈	—	86	8	no	—
229	C₂×C₄⋊C₄	—	43	2	no	—
230	C₂×C₄⋊C₄	—	43	1	no	—
231	C₂×C₄⋊C₄	—	43	4	no	—
232	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
233	C₂×C₄²	—	43	2	no	—
234	C₂×C₄²	—	43	1	no	—
235	C₂×C₄²	—	43	4	no	—
236	C₂×C₄²	—	43	2	no	—
237	C₂×C₄²	—	43	1	no	—
238	C₂×C₄²	—	43	4	no	—
239	C₂×C₄²	—	43	2	no	—
240	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
241	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
242	C₄⋊C₈	—	43	8	no	—
243	C₄⋊C₈	—	43	8	no	—
244	C₄⋊C₈	—	43	4	no	—
245	C₄⋊C₈	C₄×D₄₃	1	4	yes	—
246	#97	—	2	4	no	—
247	#97	—	43	4	no	—
248	#97	—	86	4	no	—
249	#97	—	86	4	no	—
250	#97	—	86	2	no	—
251	#97	—	86	2	no	—
252	#97	—	86	2	no	—
253	C₄×C₈	—	43	8	no	—
254	C₄×C₈	—	43	8	no	—
255	C₄×C₈	—	43	4	no	—
256	C₄×C₈	C₄×D₄₃	1	4	yes	—
257	#95	—	43	4	no	—
258	#95	C₄×D₄₃	1	2	yes	—
259	C₂×Q₈	—	43	4	no	—
260	C₂×Q₈	—	43	8	no	—
261	C₂²×C₄	—	43	2	no	—
262	C₂²×C₄	—	43	2	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
266	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	43	2	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	43	2	no	—
276	C₂×C₈	—	43	2	no	—
277	C₂×C₈	—	43	4	no	—
278	C₂×C₈	—	86	1	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	2	no	—
281	C₂×C₈	—	86	1	no	—
282	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
283	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
284	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
285	C₂×C₈	#10621900	1	2	yes	—
286	C₂×C₈	Q₈×D₄₃	1	2	yes	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	43	8	no	—
289	C₄⋊C₄	—	43	4	no	—
290	C₄⋊C₄	—	86	8	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	43	4	no	—
294	C₄²	—	43	4	no	—
295	C₄²	—	43	2	no	—
296	C₄²	—	43	2	no	—
297	C₄²	—	86	8	no	—
298	C₄²	—	86	4	no	—
299	C₄²	—	86	1	no	—
300	C₄²	—	86	1	no	—
301	C₄²	—	172	2	no	—
302	C₄²	—	172	4	no	—
303	C₄²	C₂×C₄×D₄₃	1	2	yes	—
304	C₄²	C₂×C₄×D₄₃	1	2	yes	—
305	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
306	Q₈	—	43	16	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	43	4	no	—
312	C₂×C₄	—	43	8	no	—
313	C₂×C₄	—	43	4	no	—
314	C₂×C₄	—	43	4	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	86	4	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	C₄²×D₄₃	1	2	yes	—
322	C₂×C₄	C₄²×D₄₃	1	2	yes	—
323	C₂×C₄	M₁₆×D₄₃	1	4	yes	—
324	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
326	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
327	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
328	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
329	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	—	43	4	no	—
333	C₈	—	86	2	no	—
334	C₈	—	86	4	no	—
335	C₈	#13063290	1	4	yes	complemented
336	C₂²	C₂×C₄²×D₄₃	1	1	yes	characteristic
337	C₂²	C₄×Q₈×D₄₃	1	1	yes	characteristic
338	C₂²	C₄×#10621900	1	1	yes	characteristic
339	C₂²	C₂×M₁₆×D₄₃	1	2	yes	—
340	C₂²	#14583664	1	2	yes	—
341	C₄	—	2	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	43	8	no	—
344	C₄	—	86	4	no	—
345	C₄	—	86	4	no	—
346	C₄	—	172	2	no	—
347	C₄	#97×D₄₃	1	4	yes	—
348	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
349	C₄	C₂×#13063290	1	1	yes	characteristic
350	C₄	C₄×#10621900	1	1	yes	characteristic
351	C₄	#14583367	1	1	yes	characteristic
352	C₂	C₄×#13063290	1	1	yes	characteristic
353	C₂	C₂×#97×D₄₃	1	1	yes	characteristic
354	C₂	#19143022	1	1	yes	characteristic
355	C₂	#19143547	1	2	yes	—
356	C₂	#19143611	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₄₃×#1770	C₂
socle, prime radical	C₂³×C₄₃	C₂³×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1770	D₄₃
hypocenter	C₄₃	Q₈×#97
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
14. C₈×Dic₈₆	C₄	semidirect
15. #14583023	C₄	semidirect
65. #13063235	Q₈	semidirect
150. Dic₈₆	#97	semidirect
153. C₈×C₄₃	C₄×Q₈	semidirect
156. C₄₃⋊C₈	C₄×Q₈	semidirect
218. C₄₃	Q₈×#97	semidirect
335. C₈	#13063290	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₈₆	2
C₂×#14583023	2
C₄×#13063290	2
#19134464	2
#19134463	2
#19134461	2
#19134450	2
#19134447	2
C₄×#13063235	2
#97×Dic₄₃	2
C₄₃×#1770	2
Q₈×#97	43