Subgroups — #25116349

Order 11008.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19137550	C₂	1	2	yes	maximal
3	#19135456	C₂	1	2	yes	maximal
4	#19135237	C₂	1	4	yes	maximal
5	#19134755	C₂	1	1	yes	maximal, characteristic
6	#19134601	C₂	1	2	yes	maximal
7	C₄×#13063259	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₄×Dic₁₇₂	C₄	1	8	yes	complemented
12	C₂×#13063259	C₂²	1	2	yes	—
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063250	C₂²	1	4	yes	—
16	C₂×#13063250	C₂²	1	2	yes	—
17	#14582805	C₂²	1	4	yes	—
18	#14582786	—	2	4	no	—
19	#14582769	C₂²	1	2	yes	—
20	#14582764	—	2	2	no	—
21	C₄⋊C₄×Dic₄₃	—	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×Dic₈₆	C₂³	1	2	yes	—
30	C₂×Dic₁₇₂	—	2	2	no	—
31	C₂×Dic₁₇₂	—	2	2	no	—
32	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
33	C₄×Dic₈₆	—	2	4	no	—
34	C₄×Dic₈₆	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₄₃	—	2	2	no	—
36	C₂×C₄⋊Dic₄₃	—	2	2	no	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
39	C₂×Dic₄₃⋊C₄	—	2	2	no	—
40	C₂×C₄×Dic₄₃	—	2	2	no	—
41	C₂×C₄×Dic₄₃	—	2	4	no	—
42	#13063259	—	2	4	no	—
43	#13063259	—	4	2	no	—
44	#13063259	C₂×C₄	1	4	yes	—
45	#13063250	—	4	4	no	—
46	#13063250	C₂×C₄	1	8	yes	—
47	C₄₃⋊#95	—	2	4	no	—
48	C₄₃⋊#95	—	2	4	no	—
49	C₄₃⋊#95	—	2	4	no	—
50	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	2	no	—
58	C₄×C₈×C₄₃	—	2	4	no	—
59	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
60	#95×C₄₃	D₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₃	—	2	2	no	—
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	4	no	—
69	C₂×Dic₈₆	—	2	2	no	—
70	C₂×Dic₈₆	—	2	2	no	—
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	—	2	2	no	—
73	C₂×Dic₈₆	—	2	2	no	—
74	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₄₃	—	2	1	no	—
76	C₂×C₈×C₄₃	—	2	1	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₃	#41	1	2	yes	—
83	C₂×C₈×C₄₃	#41	1	2	yes	—
84	C₄⋊C₄×C₄₃	—	2	2	no	—
85	C₄⋊C₄×C₄₃	Q₁₆	1	4	yes	complemented
86	C₄²×C₄₃	—	2	2	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₃	#41	1	2	yes	—
90	C₄⋊Dic₄₃	—	2	2	no	—
91	C₄⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	—	2	4	no	—
93	C₄⋊Dic₄₃	—	4	4	no	—
94	C₄⋊Dic₄₃	—	8	2	no	—
95	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
96	Dic₄₃⋊C₄	—	2	8	no	—
97	C₄×Dic₄₃	—	2	8	no	—
98	C₄×Dic₄₃	—	8	4	no	—
99	Dic₁₇₂	—	2	8	no	—
100	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
111	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₄₃	#135	1	2	yes	—
113	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₄₃	—	2	4	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	4	4	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	C₂×Dic₄₃	—	4	4	no	—
128	Dic₈₆	—	2	4	no	—
129	Dic₈₆	—	2	4	no	—
130	Dic₈₆	—	4	2	no	—
131	Dic₈₆	—	4	2	no	—
132	C₈×C₄₃	—	2	2	no	—
133	C₈×C₄₃	—	4	2	no	—
134	C₈×C₄₃	#117	1	4	yes	—
135	#22808	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₃	#547	1	1	yes	characteristic
138	C₂²×C₄₃	#550	1	1	yes	characteristic
139	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
141	C₂²×C₄₃	#576	1	1	yes	characteristic
142	C₂²×C₄₃	#578	1	1	yes	characteristic
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	4	2	no	—
146	C₄×C₄₃	C₄×Q₁₆	1	4	yes	—
147	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
148	C₄×C₄₃	#520	1	1	yes	characteristic
149	C₄×C₄₃	#555	1	1	yes	characteristic
150	C₄×C₄₃	#556	1	1	yes	characteristic
151	Dic₄₃	—	4	4	no	—
152	Dic₄₃	—	4	4	no	—
153	Dic₄₃	—	8	4	no	—
154	C₂×C₄×Q₁₆	—	43	1	no	—
155	#2355	—	43	2	no	—
156	#1969	—	43	2	no	—
157	#1928	—	43	4	no	—
158	#1845	—	43	1	no	—
159	#1803	—	43	2	no	—
160	C₄×#107	—	43	2	no	—
161	C₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic
162	C₂×C₄₃	#2354	1	1	yes	characteristic
163	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
164	C₂×C₄₃	#2951	1	1	yes	characteristic
165	C₂×C₄₃	#3338	1	1	yes	characteristic
166	C₂×C₄₃	#3342	1	1	yes	characteristic
167	C₂×C₄₃	#3395	1	1	yes	characteristic
168	C₂×C₄₃	#3396	1	1	yes	characteristic
169	C₂²×Q₁₆	—	43	1	no	—
170	C₂×C₄×Q₈	—	43	2	no	—
171	C₄×Q₁₆	—	43	8	no	—
172	C₂×#107	—	43	1	no	—
173	C₂×#107	—	43	2	no	—
174	C₂×#107	—	43	2	no	—
175	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×#103	—	43	2	no	—
177	C₂×#103	—	43	4	no	—
178	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
180	#391	—	43	4	no	—
181	#387	—	86	4	no	—
182	#384	—	43	2	no	—
183	#382	—	86	2	no	—
184	C₄×C₄⋊C₄	—	43	4	no	—
185	C₄×C₄⋊C₄	—	86	2	no	—
186	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
187	#340	—	43	4	no	—
188	#336	C₂×D₄₃	1	2	yes	—
189	C₄₃	#22808	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	43	2	no	—
191	C₂×Q₁₆	—	43	4	no	—
192	C₂×Q₁₆	—	86	2	no	—
193	C₂×Q₁₆	—	86	2	no	—
194	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
195	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₄×Q₈	—	43	8	no	—
198	C₄×Q₈	—	86	4	no	—
199	C₂×C₄⋊C₄	—	43	2	no	—
200	C₂×C₄⋊C₄	—	43	4	no	—
201	C₂×C₄⋊C₄	—	86	2	no	—
202	C₂×C₄⋊C₄	—	86	2	no	—
203	C₂×C₄⋊C₄	—	86	2	no	—
204	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
205	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
206	C₂×C₄²	—	86	2	no	—
207	C₂×C₄²	—	86	4	no	—
208	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
210	#107	—	43	4	no	—
211	#107	—	86	4	no	—
212	#107	—	172	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#103	—	43	8	no	—
215	#103	—	172	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₄×D₄₃	1	4	yes	—
218	#95	—	86	4	no	—
219	#95	—	86	4	no	—
220	#95	—	86	4	no	—
221	#95	D₁₇₂	1	2	yes	—
222	C₂×Q₈	—	43	4	no	—
223	C₂×Q₈	—	86	2	no	—
224	C₂×Q₈	—	86	2	no	—
225	C₂×Q₈	—	86	2	no	—
226	C₂×Q₈	—	86	2	no	—
227	C₂×Q₈	—	86	2	no	—
228	C₂²×C₄	—	86	2	no	—
229	C₂²×C₄	—	86	2	no	—
230	C₂²×C₄	—	86	4	no	—
231	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
232	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
234	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	Q₁₆	—	86	8	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
244	C₂×C₈	#10621900	1	2	yes	—
245	C₂×C₈	#10621902	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	43	4	no	—
248	C₄⋊C₄	—	86	8	no	—
249	C₄⋊C₄	—	86	2	no	—
250	C₄⋊C₄	—	86	4	no	—
251	C₄⋊C₄	—	86	4	no	—
252	C₄⋊C₄	—	344	2	no	—
253	C₄⋊C₄	—	172	4	no	—
254	C₄⋊C₄	Dic₁₇₂	1	4	yes	complemented
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	86	8	no	—
258	C₄²	—	344	4	no	—
259	C₄²	C₂×C₄×D₄₃	1	2	yes	—
260	C₄²	#10621898	1	2	yes	—
261	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
262	Q₈	—	86	4	no	—
263	Q₈	—	86	4	no	—
264	Q₈	—	172	2	no	—
265	Q₈	—	172	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	86	4	no	—
274	C₂×C₄	—	86	4	no	—
275	C₂×C₄	—	172	4	no	—
276	C₂×C₄	—	172	4	no	—
277	C₂×C₄	—	172	4	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	—	172	2	no	—
282	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
283	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
284	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
285	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
286	C₂×C₄	#13063347	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
289	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
290	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
291	C₂×C₄	#13063371	1	1	yes	characteristic
292	C₂×C₄	#13063372	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#13063283	1	4	yes	—
296	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
297	C₂²	#14583553	1	1	yes	characteristic
298	C₂²	#14583571	1	1	yes	characteristic
299	C₂²	C₂²×Dic₁₇₂	1	1	yes	characteristic
300	C₂²	C₂×#13063347	1	1	yes	characteristic
301	C₂²	#14583704	1	1	yes	characteristic
302	C₂²	#14583716	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	344	4	no	—
307	C₄	—	172	4	no	—
308	C₄	—	172	4	no	—
309	C₄	C₄×Dic₁₇₂	1	4	yes	—
310	C₄	C₂×#13063283	1	1	yes	characteristic
311	C₄	#14583379	1	1	yes	characteristic
312	C₄	#14583614	1	1	yes	characteristic
313	C₄	#14583622	1	1	yes	characteristic
314	C₂	#19137542	1	1	yes	characteristic
315	C₂	C₂×C₄×Dic₁₇₂	1	1	yes	characteristic
316	C₂	#19143472	1	1	yes	characteristic
317	C₂	#19146988	1	1	yes	characteristic
318	C₂	#19147026	1	1	yes	characteristic
319	C₂	#19147601	1	1	yes	characteristic
320	C₂	#19147608	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, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#22808
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
11. C₄×Dic₁₇₂	C₄	semidirect
85. C₄⋊C₄×C₄₃	Q₁₆	semidirect
189. C₄₃	#22808	semidirect
254. C₄⋊C₄	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇₂	2
#19137550	2
#19135456	2
#19135237	2
#19134755	2
#19134601	2
C₄×#13063259	2
C₈×C₄⋊C₄×C₄₃	2
#22808	43