Subgroups — #25114924

Order 11008.

898 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583007	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19137515	C₂	1	2	yes	maximal
4	#19135333	C₂	1	2	yes	maximal
5	#19134852	C₂	1	4	yes	maximal
6	#19134759	C₂	1	1	yes	maximal, characteristic
7	#19134611	C₂	1	2	yes	maximal
8	C₄₃×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	4	yes	—
11	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
12	#14583007	—	2	8	no	—
13	C₄×Dic₁₇₂	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×#13063250	C₂²	1	4	yes	—
18	C₂×#13063250	C₂²	1	2	yes	—
19	#14582797	C₂²	1	4	yes	—
20	#14582794	—	2	4	no	—
21	#14582765	C₂²	1	2	yes	—
22	#14582763	—	2	2	no	—
23	#14582750	—	2	2	no	—
24	#14582662	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#97×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×Dic₈₆	C₂³	1	2	yes	—
31	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₁₇₂	D₄	1	4	yes	—
33	C₂×Dic₁₇₂	Q₈	1	4	yes	—
34	C₄×Dic₈₆	—	2	4	no	—
35	C₄×Dic₈₆	—	2	8	no	—
36	C₄×Dic₈₆	—	4	4	no	—
37	C₄×Dic₈₆	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
40	C₂×Dic₄₃⋊C₄	—	2	2	no	—
41	C₂×Dic₄₃⋊C₄	—	2	4	no	—
42	C₂×Dic₄₃⋊C₄	—	2	4	no	—
43	C₂×C₄×Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	—	2	4	no	—
45	#13063259	—	4	2	no	—
46	#13063259	C₂×C₄	1	4	yes	—
47	#13063257	—	2	4	no	—
48	#13063250	—	2	8	no	—
49	#13063250	C₂×C₄	1	8	yes	—
50	C₄₃⋊#95	—	2	2	no	—
51	C₄₃⋊#95	—	2	4	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	#97×C₄₃	—	2	4	no	—
61	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
62	#95×C₄₃	D₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	—	2	4	no	—
70	C₂×Dic₈₆	—	2	2	no	—
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	—	2	2	no	—
73	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
74	C₂×Dic₈₆	C₂×D₄	1	4	yes	—
75	C₂×Dic₈₆	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₄₃	—	2	1	no	—
77	C₂×C₈×C₄₃	—	2	1	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
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	C₂²×C₄	1	4	yes	—
94	Dic₄₃⋊C₄	—	2	8	no	—
95	Dic₄₃⋊C₄	—	4	8	no	—
96	Dic₄₃⋊C₄	—	4	8	no	—
97	C₄×Dic₄₃	—	2	8	no	—
98	C₄×Dic₄₃	—	4	8	no	—
99	Dic₁₇₂	C₄⋊C₄	1	16	yes	complemented
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	4	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₄₃	—	2	4	no	—
115	C₂×Dic₄₃	—	2	4	no	—
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	2	4	no	—
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	Dic₈₆	—	2	4	no	—
123	Dic₈₆	—	2	4	no	—
124	Dic₈₆	—	2	4	no	—
125	Dic₈₆	C₂×C₄⋊C₄	1	8	yes	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	—	4	2	no	—
128	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
129	#22675	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₃	D₄²	1	1	yes	characteristic
132	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₄₃	#575	1	2	yes	—
134	C₂²×C₄₃	#578	1	2	yes	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	4	4	no	—
138	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₄₃	#520	1	1	yes	characteristic
140	C₄×C₄₃	#547	1	1	yes	characteristic
141	C₄×C₄₃	#554	1	1	yes	characteristic
142	Dic₄₃	—	2	8	no	—
143	Dic₄₃	—	4	4	no	—
144	Dic₄₃	—	8	4	no	—
145	C₂×#441	—	43	2	no	—
146	C₂×C₄×Q₁₆	—	43	1	no	—
147	#2351	—	43	2	no	—
148	#1943	—	43	2	no	—
149	#1863	—	43	4	no	—
150	#1846	—	43	1	no	—
151	#1805	—	43	2	no	—
152	#1770	D₄₃	1	1	yes	characteristic
153	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄₃	#2943	1	1	yes	characteristic
155	C₂×C₄₃	#2950	1	1	yes	characteristic
156	C₂×C₄₃	#3277	1	2	yes	—
157	C₂×C₄₃	#3390	1	2	yes	—
158	C₂²×Q₁₆	—	43	1	no	—
159	C₂×C₄×Q₈	—	43	2	no	—
160	C₂×C₄×Q₈	—	43	4	no	—
161	#441	—	86	8	no	—
162	C₄×Q₁₆	—	43	8	no	—
163	C₂×#107	—	43	1	no	—
164	C₂×#107	—	43	2	no	—
165	C₂×#106	—	43	2	no	—
166	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
167	C₂×#103	—	43	2	no	—
168	C₂×#103	—	43	4	no	—
169	C₂×#97	C₂×D₄₃	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
171	#389	—	43	4	no	—
172	#389	—	86	4	no	—
173	#382	—	43	2	no	—
174	#382	—	86	2	no	—
175	C₄×C₄⋊C₄	—	86	2	no	—
176	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
177	#340	—	43	4	no	—
178	#336	C₂×D₄₃	1	2	yes	—
179	C₄₃	#22675	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	43	2	no	—
181	C₂×Q₁₆	—	43	4	no	—
182	C₂×Q₁₆	—	43	4	no	—
183	C₂×Q₁₆	—	43	4	no	—
184	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
187	C₄×Q₈	—	43	8	no	—
188	C₄×Q₈	—	86	8	no	—
189	C₄×Q₈	—	86	4	no	—
190	C₄×Q₈	—	172	4	no	—
191	C₂×C₄⋊C₄	—	43	2	no	—
192	C₂×C₄⋊C₄	—	43	4	no	—
193	C₂×C₄⋊C₄	—	86	2	no	—
194	C₂×C₄⋊C₄	—	86	4	no	—
195	C₂×C₄⋊C₄	—	86	4	no	—
196	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
197	C₂×C₄²	—	86	2	no	—
198	C₂×C₄²	—	86	4	no	—
199	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
200	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
201	#107	—	43	4	no	—
202	#107	—	172	2	no	—
203	#106	—	86	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	43	8	no	—
206	#103	—	86	8	no	—
207	#97	—	2	4	no	—
208	C₄×C₈	C₄×D₄₃	1	4	yes	—
209	#95	—	86	2	no	—
210	#95	—	86	2	no	—
211	#95	—	86	4	no	—
212	#95	D₁₇₂	1	2	yes	—
213	C₂×Q₈	—	43	4	no	—
214	C₂×Q₈	—	43	4	no	—
215	C₂×Q₈	—	43	4	no	—
216	C₂×Q₈	—	86	2	no	—
217	C₂×Q₈	—	86	2	no	—
218	C₂×Q₈	—	86	2	no	—
219	C₂²×C₄	—	86	2	no	—
220	C₂²×C₄	—	86	2	no	—
221	C₂²×C₄	—	86	4	no	—
222	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
223	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
226	Q₁₆	—	43	16	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
234	C₂×C₈	D₄×D₄₃	1	2	yes	—
235	C₂×C₈	Q₈×D₄₃	1	2	yes	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	43	4	no	—
238	C₄⋊C₄	—	86	8	no	—
239	C₄⋊C₄	—	86	2	no	—
240	C₄⋊C₄	—	86	4	no	—
241	C₄⋊C₄	—	86	4	no	—
242	C₄⋊C₄	—	172	8	no	—
243	C₄⋊C₄	—	172	8	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	86	8	no	—
247	C₄²	—	172	8	no	—
248	C₄²	C₂×C₄×D₄₃	1	2	yes	—
249	C₄²	#10621898	1	2	yes	—
250	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
251	Q₈	—	43	8	no	—
252	Q₈	—	86	4	no	—
253	Q₈	—	86	4	no	—
254	Q₈	—	86	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	86	4	no	—
261	C₂×C₄	—	86	4	no	—
262	C₂×C₄	—	86	4	no	—
263	C₂×C₄	—	86	4	no	—
264	C₂×C₄	—	172	4	no	—
265	C₂×C₄	—	172	4	no	—
266	C₂×C₄	—	172	4	no	—
267	C₂×C₄	—	172	4	no	—
268	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
269	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
274	C₂×C₄	#13063371	1	1	yes	characteristic
275	C₂×C₄	#13063372	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	2	no	—
278	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
279	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
280	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
281	C₂²	Q₈×D₁₇₂	1	1	yes	characteristic
282	C₂²	#14583699	1	2	yes	—
283	C₂²	#14583716	1	2	yes	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	86	8	no	—
288	C₄	—	344	4	no	—
289	C₄	—	172	4	no	—
290	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
291	C₄	#14583379	1	1	yes	characteristic
292	C₄	#14583554	1	1	yes	characteristic
293	C₄	#14583609	1	1	yes	characteristic
294	C₂	C₄⋊C₄×D₁₇₂	1	1	yes	characteristic
295	C₂	#19143397	1	1	yes	characteristic
296	C₂	#19143464	1	1	yes	characteristic
297	C₂	#19146253	1	2	yes	—
298	C₂	#19147564	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1770	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1770	D₄₃
hypocenter	C₄₃	#22675
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₁₇₂	C₄	semidirect
99. Dic₁₇₂	C₄⋊C₄	semidirect
179. C₄₃	#22675	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583007	2
C₂×C₄×Dic₁₇₂	2
#19137515	2
#19135333	2
#19134852	2
#19134759	2
#19134611	2
C₄₃×#1770	2
#22675	43