Subgroups — #25114735

Order 11008.

982 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582934	C₂	1	2	yes	maximal, complemented
3	#19137542	C₂	1	2	yes	maximal
4	#19135353	C₂	1	2	yes	maximal, complemented
5	#19134795	C₂	1	4	yes	maximal, complemented
6	#19134752	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134617	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₃	C₂²	1	2	yes	—
10	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
11	C₂×#13063283	C₂²	1	4	yes	—
12	C₈×D₁₇₂	C₄	1	8	yes	complemented
13	#14582934	—	2	8	no	—
14	C₂×#13063258	C₂²	1	4	yes	—
15	C₂×#13063257	C₂²	1	2	yes	—
16	C₂×#13063257	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
19	#14582801	—	2	2	no	—
20	#14582791	—	2	4	no	—
21	#14582769	C₂²	1	2	yes	—
22	#14582756	C₂²	1	4	yes	—
23	C₄×C₄⋊Dic₄₃	—	2	2	no	—
24	#14582661	C₂²	1	4	yes	—
25	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	2	yes	—
30	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	C₂³	1	2	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	C₂×C₈×D₄₃	—	2	4	no	—
34	C₂×C₈×D₄₃	—	2	4	no	—
35	C₂×C₈×D₄₃	C₂×C₄	1	8	yes	—
36	C₄×D₁₇₂	C₂×C₄	1	4	yes	—
37	C₄×D₁₇₂	Q₈	1	4	yes	—
38	#13063283	—	2	8	no	—
39	#13063283	—	4	4	no	—
40	C₂×C₄⋊Dic₄₃	—	2	2	no	—
41	C₂×C₄⋊Dic₄₃	—	2	2	no	—
42	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
44	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
46	C₂×(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
47	C₂×C₄×Dic₄₃	—	2	4	no	—
48	#13063258	—	2	8	no	—
49	#13063257	—	2	4	no	—
50	#13063257	—	4	2	no	—
51	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
52	(C₂×D₄₃)⋊C₈	C₂×C₄	1	8	yes	—
53	C₄₃⋊#95	—	2	4	no	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#106×C₄₃	—	2	2	no	—
61	#106×C₄₃	D₄	1	4	yes	complemented
62	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
63	#95×C₄₃	—	2	2	no	—
64	C₂³×D₄₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	2	4	no	—
71	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
72	C₂×D₁₇₂	—	2	4	no	—
73	C₂×D₁₇₂	C₄⋊C₄	1	4	yes	—
74	C₂×C₄×D₄₃	—	2	2	no	—
75	C₂×C₄×D₄₃	—	2	2	no	—
76	C₂×C₄×D₄₃	—	2	2	no	—
77	C₂×C₄×D₄₃	—	2	2	no	—
78	C₂×C₄×D₄₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₄₃	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₄₃	—	2	1	no	—
81	C₂×C₈×C₄₃	—	2	1	no	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₃	#41	1	2	yes	—
86	C₄⋊C₄×C₄₃	—	2	2	no	—
87	C₄⋊C₄×C₄₃	—	2	2	no	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	C₂×D₄	1	4	yes	—
90	C₄²×C₄₃	—	4	4	no	—
91	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
93	(C₂×D₄₃)⋊C₄	—	4	8	no	—
94	(C₂×D₄₃)⋊C₄	—	8	4	no	—
95	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₄⋊Dic₄₃	—	2	4	no	—
97	C₄⋊Dic₄₃	—	2	4	no	—
98	C₄⋊Dic₄₃	—	4	4	no	—
99	C₄⋊Dic₄₃	—	8	2	no	—
100	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
101	C₄×Dic₄₃	—	4	8	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
105	C₈×D₄₃	—	2	16	no	—
106	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	2	no	—
108	C₂²×D₄₃	—	2	2	no	—
109	C₂²×D₄₃	—	2	4	no	—
110	C₂²×D₄₃	—	2	4	no	—
111	C₂²×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	4	2	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	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₃	#135	1	4	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₄₃	—	2	2	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	4	4	no	—
131	C₂×Dic₄₃	—	4	4	no	—
132	C₂×Dic₄₃	—	4	4	no	—
133	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
134	D₁₇₂	—	2	8	no	—
135	C₄×D₄₃	—	2	4	no	—
136	C₄×D₄₃	—	2	4	no	—
137	C₄×D₄₃	—	2	4	no	—
138	C₄×D₄₃	—	2	4	no	—
139	C₈×C₄₃	—	2	2	no	—
140	C₈×C₄₃	C₄×D₄	1	4	yes	—
141	C₄₃⋊C₈	—	2	4	no	—
142	#22657	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₃	D₄²	1	1	yes	characteristic
145	C₂²×C₄₃	#547	1	1	yes	characteristic
146	C₂²×C₄₃	C₂×#135	1	2	yes	—
147	C₂²×C₄₃	#577	1	2	yes	—
148	C₂×D₄₃	—	2	4	no	—
149	C₂×D₄₃	—	2	4	no	—
150	C₂×D₄₃	—	2	4	no	—
151	C₂×D₄₃	—	4	4	no	—
152	C₂×D₄₃	—	4	4	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	4	4	no	—
155	C₄×C₄₃	—	8	2	no	—
156	C₄×C₄₃	#427	1	4	yes	complemented
157	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₄₃	#520	1	1	yes	characteristic
159	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
160	C₄×C₄₃	#554	1	1	yes	characteristic
161	Dic₄₃	—	2	4	no	—
162	Dic₄₃	—	8	4	no	—
163	C₂×C₈×D₄	—	43	1	no	—
164	C₂×#427	—	43	2	no	—
165	#2354	—	43	2	no	—
166	#1947	—	43	2	no	—
167	#1854	—	43	4	no	—
168	#1844	—	43	1	no	—
169	#1807	—	43	2	no	—
170	C₄×#106	D₄₃	1	1	yes	characteristic, complemented
171	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
173	C₂×C₄₃	#2912	1	1	yes	characteristic
174	C₂×C₄₃	#3284	1	2	yes	—
175	C₂×C₄₃	#3340	1	2	yes	—
176	D₄₃	—	4	4	no	—
177	C₂³×C₈	—	43	2	no	—
178	C₂×C₄×D₄	—	43	1	no	—
179	C₂×#117	—	43	4	no	—
180	C₈×D₄	—	43	8	no	—
181	#427	—	86	8	no	—
182	C₂×#107	—	43	4	no	—
183	C₂×#106	—	43	2	no	—
184	C₂×#106	—	43	2	no	—
185	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
186	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	43	1	no	—
188	C₂×C₂²⋊C₈	—	43	2	no	—
189	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
190	#390	—	86	2	no	—
191	#388	—	86	4	no	—
192	#384	—	43	2	no	—
193	#380	—	43	4	no	—
194	C₄×C₄⋊C₄	—	86	2	no	—
195	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
196	#340	—	43	4	no	—
197	#340	C₂×D₄₃	1	2	yes	—
198	C₄₃	#22657	1	1	yes	characteristic, complemented, Sylow
199	C₂²×D₄	—	43	1	no	—
200	C₂³×C₄	—	43	2	no	—
201	C₂²×C₈	—	43	2	no	—
202	C₂²×C₈	—	43	8	no	—
203	C₂²×C₈	—	86	4	no	—
204	C₂²×C₈	—	86	4	no	—
205	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₄×D₄	—	43	4	no	—
208	C₄×D₄	—	43	4	no	—
209	#117	—	86	8	no	—
210	#117	—	172	4	no	—
211	C₂×C₄⋊C₄	—	43	1	no	—
212	C₂×C₄⋊C₄	—	43	4	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₄⋊C₄	—	86	2	no	—
215	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
217	C₂×C₂²⋊C₄	—	43	2	no	—
218	C₂×C₂²⋊C₄	—	86	4	no	—
219	C₂×C₂²⋊C₄	—	86	4	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	86	4	no	—
222	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	#107	—	86	8	no	—
224	#106	—	2	2	no	—
225	#106	—	86	4	no	—
226	#106	—	172	2	no	—
227	#106	D₁₇₂	1	4	yes	complemented
228	C₄⋊C₈	—	43	4	no	—
229	C₂²⋊C₈	—	43	8	no	—
230	C₄×C₈	C₄×D₄₃	1	4	yes	—
231	#95	—	2	2	no	—
232	#95	—	86	4	no	—
233	C₂⁴	—	43	2	no	—
234	C₂×D₄	—	43	4	no	—
235	C₂×D₄	—	86	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	43	2	no	—
239	C₂²×C₄	—	43	4	no	—
240	C₂²×C₄	—	43	4	no	—
241	C₂²×C₄	—	86	2	no	—
242	C₂²×C₄	—	86	2	no	—
243	C₂²×C₄	—	86	2	no	—
244	C₂²×C₄	—	86	2	no	—
245	C₂²×C₄	—	86	4	no	—
246	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	43	4	no	—
252	C₂×C₈	—	86	2	no	—
253	C₂×C₈	—	86	2	no	—
254	C₂×C₈	—	86	16	no	—
255	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
257	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
258	C₂×C₈	#10621898	1	2	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	43	4	no	—
263	C₄⋊C₄	—	86	4	no	—
264	C₄⋊C₄	—	86	4	no	—
265	C₄⋊C₄	—	344	2	no	—
266	C₄⋊C₄	—	172	4	no	—
267	C₄⋊C₄	C₂×D₁₇₂	1	4	yes	—
268	C₂²⋊C₄	—	43	8	no	—
269	C₂²⋊C₄	—	344	4	no	—
270	C₂²⋊C₄	—	172	8	no	—
271	C₄²	—	4	4	no	—
272	C₄²	—	172	8	no	—
273	C₄²	C₂×C₄×D₄₃	1	2	yes	—
274	C₄²	Q₈×D₄₃	1	2	yes	—
275	C₂³	—	43	4	no	—
276	C₂³	—	86	2	no	—
277	C₂³	—	86	4	no	—
278	C₂³	—	86	4	no	—
279	C₂³	—	86	2	no	—
280	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
281	D₄	—	86	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	43	4	no	—
290	C₂×C₄	—	86	2	no	—
291	C₂×C₄	—	86	2	no	—
292	C₂×C₄	—	86	4	no	—
293	C₂×C₄	—	86	4	no	—
294	C₂×C₄	—	86	4	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	—	172	4	no	—
297	C₂×C₄	—	172	4	no	—
298	C₂×C₄	—	172	4	no	—
299	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
300	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
301	C₂×C₄	#13063346	1	4	yes	—
302	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
304	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
306	C₂×C₄	#13063371	1	1	yes	characteristic
307	C₂×C₄	#13063372	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	86	4	no	—
310	C₈	C₄×D₁₇₂	1	4	yes	—
311	C₂²	—	86	4	no	—
312	C₂²	—	86	4	no	—
313	C₂²	—	86	4	no	—
314	C₂²	—	172	4	no	—
315	C₂²	—	172	4	no	—
316	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
317	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
318	C₂²	#14583554	1	1	yes	characteristic
319	C₂²	C₂×#13063346	1	2	yes	—
320	C₂²	#14583706	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	86	4	no	—
325	C₄	—	344	4	no	—
326	C₄	#14582934	1	4	yes	complemented
327	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
328	C₄	#14583379	1	1	yes	characteristic
329	C₄	Q₈×D₁₇₂	1	1	yes	characteristic
330	C₄	#14583609	1	1	yes	characteristic
331	C₂	—	172	4	no	—
332	C₂	C₄⋊C₄×D₁₇₂	1	1	yes	characteristic
333	C₂	C₂×#14582934	1	1	yes	characteristic
334	C₂	#19143187	1	1	yes	characteristic
335	C₂	#19146326	1	2	yes	—
336	C₂	#19147010	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	#22657
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#14582934	C₂	semidirect
4. #19135353	C₂	semidirect
5. #19134795	C₂	semidirect
6. #19134752	C₂	semidirect
7. #19134617	C₂	semidirect
8. C₄×#106×C₄₃	C₂	semidirect
12. C₈×D₁₇₂	C₄	semidirect
61. #106×C₄₃	D₄	semidirect
156. C₄×C₄₃	#427	semidirect
170. C₄×#106	D₄₃	semidirect
198. C₄₃	#22657	semidirect
227. #106	D₁₇₂	semidirect
326. C₄	#14582934	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₇₂	2
C₂×#14582934	2
#19137542	2
#19135353	2
#19134795	2
#19134752	2
#19134617	2
C₄×#106×C₄₃	2
#22657	43