Subgroups — #25097151

Order 11008.

886 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137002	C₂	1	1	yes	maximal, characteristic
2	#19136993	C₂	1	1	yes	maximal, characteristic
3	#19134763	C₂	1	2	yes	maximal, complemented
4	#19134729	C₂	1	2	yes	maximal, complemented
5	#19134628	C₂	1	4	yes	maximal
6	C₄×#13063259	C₂	1	2	yes	maximal, complemented
7	C₄×#13063257	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	—	2	1	no	—
10	C₂×#13063259	C₂²	1	4	yes	—
11	C₂×#13063257	C₂²	1	4	yes	—
12	#14582769	C₂²	1	2	yes	—
13	#14582767	—	2	1	no	—
14	#14582765	—	2	1	no	—
15	#14582765	C₂²	1	2	yes	—
16	#14582762	—	2	1	no	—
17	#14582759	—	2	1	no	—
18	#14582759	C₂²	1	2	yes	—
19	#14582759	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₃	—	2	1	no	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊Dic₄₃	—	2	1	no	—
25	#14582745	—	2	1	no	—
26	#14582662	C₂²	1	4	yes	—
27	#14582662	C₂²	1	4	yes	—
28	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₄₃	C₄	1	8	yes	complemented
31	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
34	#13063284	—	4	4	no	—
35	C₂×C₄⋊Dic₄₃	—	2	2	no	—
36	C₂×C₄⋊Dic₄₃	—	2	2	no	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
41	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
42	C₂×C₄×Dic₄₃	—	2	2	no	—
43	C₂×C₄×Dic₄₃	—	2	1	no	—
44	C₂×C₄×Dic₄₃	—	2	1	no	—
45	#13063259	—	2	8	no	—
46	#13063257	—	2	8	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	2	no	—
49	C₄₃⋊#95	—	2	2	no	—
50	C₄₃⋊#95	—	2	2	no	—
51	C₄₃⋊#95	—	2	2	no	—
52	C₄₃⋊#95	—	2	1	no	—
53	C₄₃⋊#95	—	2	1	no	—
54	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₄₃	—	2	8	no	—
58	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
59	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	#97×C₄₃	C₂×C₄	1	4	yes	—
68	#97×C₄₃	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₄₃	C₂×C₄	1	8	yes	—
70	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	—	2	4	no	—
74	C₂²×C₄×C₄₃	—	4	2	no	—
75	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₃	—	2	2	no	—
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	—	2	4	no	—
88	M₁₆×C₄₃	—	2	16	no	—
89	C₂×C₈×C₄₃	—	2	4	no	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₄₃	#41	1	8	yes	—
93	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
94	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
95	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
98	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
99	C₂²⋊Dic₄₃	—	4	4	no	—
100	C₂²⋊Dic₄₃	—	4	4	no	—
101	C₄⋊Dic₄₃	—	2	4	no	—
102	C₄⋊Dic₄₃	—	2	4	no	—
103	C₄⋊Dic₄₃	—	2	4	no	—
104	C₄⋊Dic₄₃	—	2	4	no	—
105	C₄⋊Dic₄₃	—	4	4	no	—
106	C₄⋊Dic₄₃	—	4	4	no	—
107	C₄×Dic₄₃	—	4	2	no	—
108	C₄×Dic₄₃	—	4	4	no	—
109	C₄×Dic₄₃	—	4	2	no	—
110	C₂³×C₄₃	—	2	2	no	—
111	C₂³×C₄₃	—	2	1	no	—
112	C₂³×C₄₃	—	2	1	no	—
113	C₂³×C₄₃	—	2	2	no	—
114	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	4	no	—
123	C₂×C₄×C₄₃	—	2	8	no	—
124	C₂×C₄×C₄₃	—	4	4	no	—
125	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₄₃	#137	1	2	yes	—
132	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
135	C₂×Dic₄₃	—	4	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	2	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	C₈×C₄₃	—	2	4	no	—
145	C₈×C₄₃	#117	1	8	yes	complemented
146	#20819	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	—	2	2	no	—
148	C₂²×C₄₃	—	2	2	no	—
149	C₂²×C₄₃	—	2	2	no	—
150	C₂²×C₄₃	—	4	2	no	—
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₄₃	#532	1	2	yes	—
154	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
156	C₂²×C₄₃	#575	1	2	yes	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	#428	1	4	yes	—
162	C₄×C₄₃	C₂×#117	1	2	yes	—
163	C₄×C₄₃	#533	1	2	yes	—
164	Dic₄₃	—	8	4	no	—
165	Dic₄₃	—	8	2	no	—
166	Dic₄₃	—	8	2	no	—
167	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
168	#2285	—	43	1	no	—
169	#2283	—	43	1	no	—
170	#1847	—	43	2	no	—
171	#1836	—	43	2	no	—
172	#1809	—	43	4	no	—
173	C₄×#107	—	43	2	no	—
174	C₄×#106	—	43	2	no	—
175	C₂×C₄₃	—	4	2	no	—
176	C₂×C₄₃	#2285	1	1	yes	characteristic
177	C₂×C₄₃	C₂×#428	1	1	yes	characteristic
178	C₂×C₄₃	#2911	1	1	yes	characteristic
179	C₂×C₄₃	#3171	1	2	yes	—
180	C₂×C₄₃	#3172	1	2	yes	—
181	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
182	C₂×#117	—	86	1	no	—
183	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
184	C₂×#107	—	43	4	no	—
185	C₂×#106	—	43	4	no	—
186	C₄×M₁₆	Dic₄₃	1	8	yes	complemented
187	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
188	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
190	#384	—	43	2	no	—
191	#383	—	86	1	no	—
192	#382	—	43	2	no	—
193	#382	—	86	1	no	—
194	#381	—	86	1	no	—
195	#380	—	43	2	no	—
196	#380	—	43	2	no	—
197	#380	—	86	1	no	—
198	C₄×C₄⋊C₄	—	43	1	no	—
199	C₄×C₄⋊C₄	—	43	1	no	—
200	C₄×C₄⋊C₄	—	43	2	no	—
201	C₄×C₄⋊C₄	—	86	1	no	—
202	C₄×C₂²⋊C₄	—	86	1	no	—
203	#376	—	86	1	no	—
204	#340	—	43	4	no	—
205	#340	—	43	4	no	—
206	C₄₃	#20819	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
208	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
209	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
210	C₂×M₁₆	—	2	8	no	—
211	C₂×M₁₆	C₂×Dic₄₃	1	8	yes	—
212	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
213	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
214	#117	—	172	4	no	—
215	C₂×C₄⋊C₄	—	43	2	no	—
216	C₂×C₄⋊C₄	—	43	2	no	—
217	C₂×C₄⋊C₄	—	43	4	no	—
218	C₂×C₄⋊C₄	—	86	2	no	—
219	C₂×C₄⋊C₄	—	86	2	no	—
220	C₂×C₂²⋊C₄	—	86	2	no	—
221	C₂×C₂²⋊C₄	—	86	2	no	—
222	C₂×C₄²	—	86	2	no	—
223	C₂×C₄²	—	86	1	no	—
224	C₂×C₄²	—	86	1	no	—
225	C₂×C₄²	Dic₈₆	1	4	yes	—
226	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
227	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
228	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
229	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
231	#107	—	86	8	no	—
232	#106	—	86	8	no	—
233	#97	C₂×Dic₄₃	1	4	yes	—
234	#97	C₂×Dic₄₃	1	4	yes	—
235	C₄×C₈	C₂×Dic₄₃	1	8	yes	—
236	#95	—	86	2	no	—
237	#95	—	86	2	no	—
238	#95	—	86	2	no	—
239	#95	—	86	2	no	—
240	#95	—	86	1	no	—
241	#95	—	86	1	no	—
242	#95	—	86	2	no	—
243	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	86	2	no	—
249	C₂²×C₄	—	86	2	no	—
250	C₂²×C₄	—	86	4	no	—
251	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
252	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
253	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
254	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
260	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
261	M₁₆	—	2	16	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	#10621898	1	8	yes	—
264	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
265	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
266	C₄⋊C₄	—	86	4	no	—
267	C₄⋊C₄	—	86	4	no	—
268	C₄⋊C₄	—	86	4	no	—
269	C₄⋊C₄	—	86	4	no	—
270	C₄⋊C₄	—	172	4	no	—
271	C₄⋊C₄	—	172	4	no	—
272	C₂²⋊C₄	—	172	4	no	—
273	C₂²⋊C₄	—	172	4	no	—
274	C₄²	—	172	4	no	—
275	C₄²	—	172	2	no	—
276	C₄²	—	172	2	no	—
277	C₄²	C₄⋊Dic₄₃	1	4	yes	—
278	C₄²	C₄⋊Dic₄₃	1	4	yes	—
279	C₄²	C₂×Dic₈₆	1	2	yes	—
280	C₄²	C₂×Dic₈₆	1	2	yes	—
281	C₄²	C₂²×Dic₄₃	1	2	yes	—
282	C₄²	C₂²×Dic₄₃	1	2	yes	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
288	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	172	4	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	—	172	4	no	—
304	C₂×C₄	—	172	4	no	—
305	C₂×C₄	—	172	2	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
308	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
309	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
312	C₂×C₄	#13063353	1	2	yes	—
313	C₂×C₄	#13063358	1	2	yes	—
314	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
316	C₂×C₄	C₂×#10621898	1	4	yes	—
317	C₈	—	2	4	no	—
318	C₈	#13063284	1	8	yes	complemented
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
325	C₂²	#14583449	1	2	yes	—
326	C₂²	C₂×#13063353	1	1	yes	characteristic
327	C₂²	C₂×#13063358	1	1	yes	characteristic
328	C₂²	#14583699	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	344	4	no	—
334	C₄	—	344	2	no	—
335	C₄	—	344	2	no	—
336	C₄	#14582941	1	4	yes	—
337	C₄	C₂×#13063284	1	2	yes	—
338	C₄	#14583456	1	2	yes	—
339	C₂	—	4	2	no	—
340	C₂	#19137002	1	1	yes	characteristic
341	C₂	C₂×#14582941	1	1	yes	characteristic
342	C₂	#19143185	1	1	yes	characteristic
343	C₂	#19145190	1	2	yes	—
344	C₂	#19145201	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	#20819
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #19134763	C₂	semidirect
4. #19134729	C₂	semidirect
6. C₄×#13063259	C₂	semidirect
7. C₄×#13063257	C₂	semidirect
30. C₄×M₁₆×C₄₃	C₄	semidirect
145. C₈×C₄₃	#117	semidirect
186. C₄×M₁₆	Dic₄₃	semidirect
206. C₄₃	#20819	semidirect
318. C₈	#13063284	semidirect

Maximal subgroups

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

subgroup	index
#19137002	2
#19136993	2
#19134763	2
#19134729	2
#19134628	2
C₄×#13063259	2
C₄×#13063257	2
C₂×C₄×M₁₆×C₄₃	2
#20819	43