Subgroups — #25115011

Order 11008.

1062 conjugacy classes of subgroups.

The subgroup lattice has 336 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	2	yes	maximal
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19137506	C₂	1	2	yes	maximal, complemented
4	#19135328	C₂	1	2	yes	maximal, complemented
5	#19134859	C₂	1	4	yes	maximal, complemented
6	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134606	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	4	yes	—
11	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
12	C₄×D₃₄₄	—	2	8	no	—
13	C₄×D₃₄₄	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	2	yes	—
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×#13063257	C₂²	1	2	yes	—
17	C₂×#13063247	C₂²	1	4	yes	—
18	C₂×#13063247	C₂²	1	2	yes	—
19	#14582797	C₂²	1	4	yes	—
20	#14582770	—	2	4	no	—
21	#14582765	C₂²	1	2	yes	—
22	#14582758	—	2	2	no	—
23	#95×D₄₃	—	2	2	no	—
24	#14582662	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#336	C₂²	1	2	yes	—
30	C₂²×D₁₇₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₃	—	2	2	no	—
32	C₂²×C₄×D₄₃	—	2	4	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₂×D₃₄₄	D₄	1	4	yes	—
35	C₂×D₃₄₄	Q₈	1	4	yes	—
36	C₄×D₁₇₂	—	2	4	no	—
37	C₄×D₁₇₂	—	2	8	no	—
38	C₄×D₁₇₂	—	4	4	no	—
39	C₄×D₁₇₂	C₂×C₄	1	8	yes	complemented
40	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
42	C₂×Dic₄₃⋊C₄	—	2	4	no	—
43	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
44	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
45	#13063259	—	2	4	no	—
46	#13063259	C₂×C₄	1	4	yes	complemented
47	#13063257	—	4	2	no	—
48	#13063247	—	2	8	no	—
49	#13063247	C₂×C₄	1	8	yes	complemented
50	C₄₃⋊#95	—	2	4	no	—
51	C₄₃⋊#95	—	2	2	no	—
52	C₄₃⋊#95	—	2	2	no	—
53	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	—	2	2	no	—
60	C₄×C₈×C₄₃	—	2	4	no	—
61	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
62	#95×C₄₃	D₄	1	1	yes	characteristic
63	#95×C₄₃	D₄	1	1	yes	characteristic
64	C₂³×D₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	2	2	no	—
71	C₂²×Dic₄₃	—	2	4	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×D₁₇₂	C₂×D₄	1	4	yes	—
77	C₂×D₁₇₂	C₂×Q₈	1	4	yes	—
78	C₂×C₄×D₄₃	—	2	8	no	—
79	C₂×C₄×D₄₃	—	4	8	no	—
80	C₂×C₄×D₄₃	—	4	8	no	—
81	C₂×C₄×D₄₃	—	8	4	no	—
82	C₂×C₄×D₄₃	—	8	4	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
92	C₄⋊C₄×C₄₃	—	2	4	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	—	2	2	no	—
95	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₃	#41	1	2	yes	—
97	(C₂×D₄₃)⋊C₄	—	2	8	no	—
98	(C₂×D₄₃)⋊C₄	—	4	8	no	—
99	C₄⋊Dic₄₃	—	2	2	no	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	C₄⋊Dic₄₃	—	2	4	no	—
102	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
103	Dic₄₃⋊C₄	—	8	4	no	—
104	D₃₄₄	C₄⋊C₄	1	16	yes	complemented
105	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²×D₄₃	—	2	4	no	—
107	C₂²×D₄₃	—	2	4	no	—
108	C₂²×D₄₃	—	2	4	no	—
109	C₂²×D₄₃	—	4	4	no	—
110	C₂²×D₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	#135	1	4	yes	—
121	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×Dic₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	4	4	no	—
130	C₂×Dic₄₃	—	4	4	no	—
131	C₂×Dic₄₃	—	4	4	no	—
132	D₁₇₂	—	2	4	no	—
133	D₁₇₂	—	2	4	no	—
134	D₁₇₂	—	2	4	no	—
135	D₁₇₂	C₂×C₄⋊C₄	1	8	yes	—
136	C₄×D₄₃	—	8	8	no	—
137	C₄×D₄₃	—	8	16	no	—
138	C₈×C₄₃	—	2	2	no	—
139	C₈×C₄₃	—	4	2	no	—
140	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
141	#22682	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₃	D₄²	1	1	yes	characteristic
144	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
145	C₂²×C₄₃	C₂×#135	1	2	yes	—
146	C₂²×C₄₃	#576	1	2	yes	—
147	C₂×D₄₃	—	2	8	no	—
148	C₂×D₄₃	—	4	8	no	—
149	C₂×D₄₃	—	4	4	no	—
150	C₂×D₄₃	—	4	4	no	—
151	C₂×D₄₃	—	4	8	no	—
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	4	no	—
155	C₄×C₄₃	—	4	2	no	—
156	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₄₃	#520	1	1	yes	characteristic
158	C₄×C₄₃	#547	1	1	yes	characteristic
159	C₄×C₄₃	#554	1	1	yes	characteristic
160	Dic₄₃	—	4	4	no	—
161	Dic₄₃	—	8	4	no	—
162	C₂×C₄×D₈	—	43	1	no	—
163	C₂×C₄×D₈	—	43	2	no	—
164	#2349	—	43	2	no	—
165	#1942	—	43	2	no	—
166	#1864	—	43	4	no	—
167	#1846	—	43	1	no	—
168	#1804	—	43	2	no	—
169	C₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
170	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
172	C₂×C₄₃	#2948	1	1	yes	characteristic
173	C₂×C₄₃	#3280	1	2	yes	—
174	C₂×C₄₃	#3380	1	2	yes	—
175	D₄₃	—	4	8	no	—
176	C₂²×D₈	—	43	1	no	—
177	C₂×C₄×D₄	—	43	2	no	—
178	C₂×C₄×D₄	—	43	4	no	—
179	C₄×D₈	—	43	8	no	—
180	C₄×D₈	—	86	8	no	—
181	C₂×#107	—	43	1	no	—
182	C₂×#107	—	43	2	no	—
183	C₂×#106	—	43	2	no	—
184	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
185	C₂×#102	—	43	2	no	—
186	C₂×#102	—	43	4	no	—
187	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
188	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
189	#389	—	43	4	no	—
190	#385	—	86	4	no	—
191	#382	—	43	2	no	—
192	#380	—	86	2	no	—
193	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
194	C₂×#95	—	86	2	no	—
195	#340	—	43	4	no	—
196	#336	C₂×D₄₃	1	2	yes	—
197	C₄₃	#22682	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	43	2	no	—
199	C₂³×C₄	—	86	2	no	—
200	C₂³×C₄	—	86	4	no	—
201	C₂×D₈	—	43	4	no	—
202	C₂×D₈	—	43	4	no	—
203	C₂×D₈	—	43	4	no	—
204	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
205	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₄×D₄	—	43	8	no	—
208	C₄×D₄	—	86	8	no	—
209	C₄×D₄	—	86	4	no	—
210	C₄×D₄	—	172	4	no	—
211	C₂×C₄⋊C₄	—	43	2	no	—
212	C₂×C₄⋊C₄	—	43	4	no	—
213	C₂×C₄⋊C₄	—	86	4	no	—
214	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
215	C₂×C₂²⋊C₄	—	86	2	no	—
216	C₂×C₂²⋊C₄	—	86	4	no	—
217	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
218	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
219	#107	—	43	4	no	—
220	#107	—	86	4	no	—
221	#106	—	172	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	#102	—	43	8	no	—
224	#102	—	86	8	no	—
225	C₄×C₈	—	2	4	no	—
226	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
227	#95	—	86	2	no	—
228	#95	—	86	4	no	—
229	#95	—	86	2	no	—
230	#95	D₁₇₂	1	1	yes	characteristic
231	#95	D₁₇₂	1	1	yes	characteristic
232	C₂⁴	—	86	2	no	—
233	C₂×D₄	—	43	4	no	—
234	C₂×D₄	—	43	4	no	—
235	C₂×D₄	—	43	4	no	—
236	C₂×D₄	—	86	2	no	—
237	C₂×D₄	—	86	2	no	—
238	C₂×D₄	—	86	2	no	—
239	C₂²×C₄	—	86	2	no	—
240	C₂²×C₄	—	86	8	no	—
241	C₂²×C₄	—	86	4	no	—
242	C₂²×C₄	—	344	4	no	—
243	C₂²×C₄	—	344	4	no	—
244	C₂²×C₄	—	172	8	no	—
245	C₂²×C₄	—	172	8	no	—
246	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
247	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	D₈	—	43	16	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
259	C₂×C₈	D₄×D₄₃	1	2	yes	—
260	C₂×C₈	Q₈×D₄₃	1	2	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	43	4	no	—
263	C₄⋊C₄	—	86	2	no	—
264	C₄⋊C₄	—	86	4	no	—
265	C₄⋊C₄	—	86	4	no	—
266	C₄⋊C₄	—	344	4	no	—
267	C₂²⋊C₄	—	86	8	no	—
268	C₂²⋊C₄	—	172	8	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂×C₄×D₄₃	1	2	yes	—
272	C₄²	#10621898	1	2	yes	—
273	C₂³	—	86	4	no	—
274	C₂³	—	86	4	no	—
275	C₂³	—	86	4	no	—
276	C₂³	—	172	4	no	—
277	C₂³	—	172	4	no	—
278	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
279	D₄	—	43	8	no	—
280	D₄	—	86	4	no	—
281	D₄	—	86	4	no	—
282	D₄	—	86	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	86	4	no	—
291	C₂×C₄	—	344	16	no	—
292	C₂×C₄	—	344	8	no	—
293	C₂×C₄	—	172	4	no	—
294	C₂×C₄	—	172	4	no	—
295	C₂×C₄	—	172	4	no	—
296	C₂×C₄	—	172	4	no	—
297	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
298	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
299	C₂×C₄	#13063347	1	4	yes	—
300	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
304	C₂×C₄	#13063371	1	1	yes	characteristic
305	C₂×C₄	#13063372	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
309	C₂²	—	86	8	no	—
310	C₂²	—	172	4	no	—
311	C₂²	—	172	8	no	—
312	C₂²	—	172	8	no	—
313	C₂²	—	172	4	no	—
314	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
315	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
316	C₂²	Q₈×D₁₇₂	1	1	yes	characteristic
317	C₂²	C₂×#13063347	1	2	yes	—
318	C₂²	#14583704	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	4	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	344	4	no	—
324	C₄	—	172	4	no	—
325	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
326	C₄	#14583379	1	1	yes	characteristic
327	C₄	#14583554	1	1	yes	characteristic
328	C₄	#14583609	1	1	yes	characteristic
329	C₂	—	172	8	no	—
330	C₂	C₄⋊C₄×D₁₇₂	1	1	yes	characteristic
331	C₂	C₄×#13063347	1	1	yes	characteristic
332	C₂	#19143448	1	1	yes	characteristic
333	C₂	#19146282	1	2	yes	—
334	C₂	#19147463	1	2	yes	—

	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	#22682
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
3. #19137506	C₂	semidirect
4. #19135328	C₂	semidirect
5. #19134859	C₂	semidirect
6. #19134759	C₂	semidirect
7. #19134606	C₂	semidirect
8. C₈×C₄⋊C₄×C₄₃	C₂	semidirect
13. C₄×D₃₄₄	C₄	semidirect
39. C₄×D₁₇₂	C₂×C₄	semidirect
46. #13063259	C₂×C₄	semidirect
49. #13063247	C₂×C₄	semidirect
61. C₄×C₈×C₄₃	C₂×C₄	semidirect
104. D₃₄₄	C₄⋊C₄	semidirect
169. C₈×C₄⋊C₄	D₄₃	semidirect
197. C₄₃	#22682	semidirect
226. C₄×C₈	C₄×D₄₃	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
#19137506	2
#19135328	2
#19134859	2
#19134759	2
#19134606	2
C₈×C₄⋊C₄×C₄₃	2
#22682	43