Subgroups — #25116341

Order 11008.

934 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	#19137542	C₂	1	2	yes	maximal, complemented
3	#19135461	C₂	1	2	yes	maximal
4	#19135216	C₂	1	4	yes	maximal, complemented
5	#19134755	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134596	C₂	1	2	yes	maximal, complemented
7	C₄×#13063259	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₂×#13063283	—	2	2	no	—
12	C₄×D₃₄₄	C₄	1	8	yes	complemented
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063259	C₂²	1	2	yes	—
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×#13063247	C₂²	1	4	yes	—
17	C₂×#13063247	C₂²	1	2	yes	—
18	#14582801	C₂²	1	4	yes	—
19	#14582791	—	2	4	no	—
20	#14582769	C₂²	1	2	yes	—
21	#14582756	—	2	2	no	—
22	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
23	#14582662	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#336	C₂²	1	2	yes	—
29	C₂²×D₁₇₂	C₂³	1	2	yes	—
30	C₂²×C₄×D₄₃	—	2	2	no	—
31	C₂×D₃₄₄	—	2	2	no	—
32	C₂×D₃₄₄	—	2	2	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₄×D₁₇₂	—	2	4	no	—
35	C₄×D₁₇₂	C₂×C₄	1	8	yes	—
36	#13063283	—	4	8	no	—
37	C₂×C₄⋊Dic₄₃	—	2	2	no	—
38	C₂×C₄⋊Dic₄₃	—	2	2	no	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
42	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
43	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
44	C₂×C₄×Dic₄₃	—	2	4	no	—
45	#13063259	—	2	4	no	—
46	#13063259	—	4	2	no	—
47	#13063259	C₂×C₄	1	4	yes	—
48	#13063247	—	4	4	no	—
49	#13063247	C₂×C₄	1	8	yes	—
50	C₄₃⋊#95	—	2	4	no	—
51	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	—	2	4	no	—
60	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
61	#95×C₄₃	D₄	1	2	yes	—
62	C₂³×D₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	—	2	4	no	—
70	C₂×D₁₇₂	—	2	2	no	—
71	C₂×D₁₇₂	—	2	2	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	—	2	2	no	—
74	C₂×D₁₇₂	—	2	2	no	—
75	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
76	C₂×C₄×D₄₃	—	2	8	no	—
77	C₂×C₄×D₄₃	—	4	4	no	—
78	C₂×C₄×D₄₃	—	4	4	no	—
79	C₂×C₈×C₄₃	—	2	1	no	—
80	C₂×C₈×C₄₃	—	2	1	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₃	#41	1	2	yes	—
87	C₂×C₈×C₄₃	#41	1	2	yes	—
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₄⋊C₄×C₄₃	D₈	1	4	yes	complemented
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	2	2	no	—
92	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₃	#41	1	2	yes	—
94	(C₂×D₄₃)⋊C₄	—	2	8	no	—
95	(C₂×D₄₃)⋊C₄	—	4	8	no	—
96	(C₂×D₄₃)⋊C₄	—	8	4	no	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	2	4	no	—
100	C₄⋊Dic₄₃	—	4	4	no	—
101	C₄⋊Dic₄₃	—	8	2	no	—
102	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
103	C₄×Dic₄₃	—	4	8	no	—
104	D₃₄₄	—	2	8	no	—
105	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²×D₄₃	—	2	4	no	—
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	—	4	2	no	—
109	C₂²×D₄₃	—	4	2	no	—
110	C₂²×D₄₃	—	4	2	no	—
111	C₂²×D₄₃	—	4	2	no	—
112	C₂²×D₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₃	#135	1	2	yes	—
125	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	4	no	—
132	C₂×Dic₄₃	—	4	2	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	C₂×Dic₄₃	—	4	4	no	—
135	C₂×Dic₄₃	—	4	4	no	—
136	C₂×Dic₄₃	—	4	4	no	—
137	D₁₇₂	—	2	4	no	—
138	D₁₇₂	—	2	4	no	—
139	D₁₇₂	—	4	2	no	—
140	D₁₇₂	—	4	2	no	—
141	C₄×D₄₃	—	4	16	no	—
142	C₈×C₄₃	—	2	2	no	—
143	C₈×C₄₃	—	4	2	no	—
144	C₈×C₄₃	#117	1	4	yes	complemented
145	#22807	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₄₃	#547	1	1	yes	characteristic
148	C₂²×C₄₃	#550	1	1	yes	characteristic
149	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
150	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
151	C₂²×C₄₃	#576	1	1	yes	characteristic
152	C₂²×C₄₃	#578	1	1	yes	characteristic
153	C₂×D₄₃	—	4	4	no	—
154	C₂×D₄₃	—	4	4	no	—
155	C₂×D₄₃	—	4	4	no	—
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₂×D₄₃	—	8	2	no	—
160	C₄×C₄₃	—	2	4	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	4	2	no	—
163	C₄×C₄₃	C₄×D₈	1	4	yes	—
164	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
165	C₄×C₄₃	#520	1	1	yes	characteristic
166	C₄×C₄₃	#555	1	1	yes	characteristic
167	C₄×C₄₃	#556	1	1	yes	characteristic
168	Dic₄₃	—	4	4	no	—
169	Dic₄₃	—	8	4	no	—
170	C₂×C₄×D₈	—	43	1	no	—
171	#2354	—	43	2	no	—
172	#1970	—	43	2	no	—
173	#1925	—	43	4	no	—
174	#1845	—	43	1	no	—
175	#1802	—	43	2	no	—
176	C₄×#107	—	43	2	no	—
177	C₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
178	C₂×C₄₃	#2354	1	1	yes	characteristic
179	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
180	C₂×C₄₃	#2951	1	1	yes	characteristic
181	C₂×C₄₃	#3334	1	1	yes	characteristic
182	C₂×C₄₃	#3341	1	1	yes	characteristic
183	C₂×C₄₃	#3391	1	1	yes	characteristic
184	C₂×C₄₃	#3393	1	1	yes	characteristic
185	D₄₃	—	8	4	no	—
186	C₂²×D₈	—	43	1	no	—
187	C₂×C₄×D₄	—	43	2	no	—
188	C₂×#117	—	86	2	no	—
189	C₄×D₈	—	43	8	no	—
190	C₂×#107	—	43	1	no	—
191	C₂×#107	—	43	2	no	—
192	C₂×#107	—	43	2	no	—
193	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
194	C₂×#102	—	43	2	no	—
195	C₂×#102	—	43	4	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
197	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
198	#390	—	43	4	no	—
199	#388	—	86	4	no	—
200	#384	—	43	2	no	—
201	#380	—	86	2	no	—
202	C₄×C₄⋊C₄	—	43	4	no	—
203	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
204	#340	—	43	4	no	—
205	#336	C₂×D₄₃	1	2	yes	—
206	C₄₃	#22807	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₄	—	43	2	no	—
208	C₂³×C₄	—	86	2	no	—
209	C₂×D₈	—	43	4	no	—
210	C₂×D₈	—	86	2	no	—
211	C₂×D₈	—	86	2	no	—
212	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
213	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
215	C₄×D₄	—	43	8	no	—
216	C₄×D₄	—	86	4	no	—
217	#117	—	172	8	no	—
218	C₂×C₄⋊C₄	—	43	2	no	—
219	C₂×C₄⋊C₄	—	43	4	no	—
220	C₂×C₄⋊C₄	—	86	2	no	—
221	C₂×C₄⋊C₄	—	86	2	no	—
222	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
223	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	—	86	2	no	—
225	C₂×C₂²⋊C₄	—	86	4	no	—
226	C₂×C₂²⋊C₄	—	86	4	no	—
227	C₂×C₄²	—	86	4	no	—
228	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
229	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
230	#107	—	43	4	no	—
231	#107	—	86	4	no	—
232	#107	—	172	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	#102	—	43	8	no	—
235	#102	—	172	4	no	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	C₄×D₄₃	1	4	yes	—
238	#95	—	86	4	no	—
239	#95	D₁₇₂	1	2	yes	—
240	C₂⁴	—	86	2	no	—
241	C₂×D₄	—	43	4	no	—
242	C₂×D₄	—	86	2	no	—
243	C₂×D₄	—	86	2	no	—
244	C₂×D₄	—	86	2	no	—
245	C₂×D₄	—	86	2	no	—
246	C₂×D₄	—	86	2	no	—
247	C₂²×C₄	—	86	2	no	—
248	C₂²×C₄	—	86	8	no	—
249	C₂²×C₄	—	86	4	no	—
250	C₂²×C₄	—	172	4	no	—
251	C₂²×C₄	—	172	4	no	—
252	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
253	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
255	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
257	D₈	—	86	8	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
265	C₂×C₈	#10621900	1	2	yes	—
266	C₂×C₈	#10621902	1	2	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	43	4	no	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	86	4	no	—
271	C₄⋊C₄	—	86	4	no	—
272	C₄⋊C₄	—	344	2	no	—
273	C₄⋊C₄	—	172	4	no	—
274	C₄⋊C₄	D₃₄₄	1	4	yes	complemented
275	C₂²⋊C₄	—	86	8	no	—
276	C₂²⋊C₄	—	344	4	no	—
277	C₂²⋊C₄	—	172	8	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	172	8	no	—
281	C₄²	C₂×C₄×D₄₃	1	2	yes	—
282	C₄²	#10621898	1	2	yes	—
283	C₂³	—	86	4	no	—
284	C₂³	—	172	2	no	—
285	C₂³	—	172	2	no	—
286	C₂³	—	172	2	no	—
287	C₂³	—	172	2	no	—
288	C₂³	—	172	2	no	—
289	C₂³	—	172	2	no	—
290	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
291	D₄	—	86	4	no	—
292	D₄	—	86	4	no	—
293	D₄	—	172	2	no	—
294	D₄	—	172	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	86	4	no	—
303	C₂×C₄	—	172	4	no	—
304	C₂×C₄	—	172	4	no	—
305	C₂×C₄	—	172	4	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	16	no	—
309	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
310	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
311	C₂×C₄	C₂×D₃₄₄	1	4	yes	—
312	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
313	C₂×C₄	#13063347	1	2	yes	—
314	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
315	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
316	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
317	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
318	C₂×C₄	#13063371	1	1	yes	characteristic
319	C₂×C₄	#13063372	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	4	2	no	—
322	C₈	#13063283	1	4	yes	complemented
323	C₂²	—	344	2	no	—
324	C₂²	—	344	2	no	—
325	C₂²	—	344	2	no	—
326	C₂²	—	344	2	no	—
327	C₂²	—	172	4	no	—
328	C₂²	—	172	4	no	—
329	C₂²	—	172	4	no	—
330	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
331	C₂²	#14583553	1	1	yes	characteristic
332	C₂²	#14583571	1	1	yes	characteristic
333	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
334	C₂²	C₂×#13063347	1	1	yes	characteristic
335	C₂²	#14583704	1	1	yes	characteristic
336	C₂²	#14583716	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	344	4	no	—
341	C₄	—	172	4	no	—
342	C₄	C₄×D₃₄₄	1	4	yes	—
343	C₄	C₂×#13063283	1	1	yes	characteristic
344	C₄	#14583379	1	1	yes	characteristic
345	C₄	#14583614	1	1	yes	characteristic
346	C₄	#14583622	1	1	yes	characteristic
347	C₂	—	344	4	no	—
348	C₂	#19137542	1	1	yes	characteristic
349	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic
350	C₂	#19143472	1	1	yes	characteristic
351	C₂	#19146943	1	1	yes	characteristic
352	C₂	#19147018	1	1	yes	characteristic
353	C₂	#19147570	1	1	yes	characteristic
354	C₂	#19147584	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₈×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#22807
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #19137542	C₂	semidirect
4. #19135216	C₂	semidirect
5. #19134755	C₂	semidirect
6. #19134596	C₂	semidirect
7. C₄×#13063259	C₂	semidirect
8. C₈×C₄⋊C₄×C₄₃	C₂	semidirect
12. C₄×D₃₄₄	C₄	semidirect
89. C₄⋊C₄×C₄₃	D₈	semidirect
144. C₈×C₄₃	#117	semidirect
177. C₈×C₄⋊C₄	D₄₃	semidirect
206. C₄₃	#22807	semidirect
274. C₄⋊C₄	D₃₄₄	semidirect
322. C₈	#13063283	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₄₄	2
#19137542	2
#19135461	2
#19135216	2
#19134755	2
#19134596	2
C₄×#13063259	2
C₈×C₄⋊C₄×C₄₃	2
#22807	43