Subgroups — #25116357

Order 11008.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#19137558	C₂	1	2	yes	maximal
3	#19135456	C₂	1	2	yes	maximal
4	#19135230	C₂	1	4	yes	maximal
5	#19134759	C₂	1	1	yes	maximal, characteristic
6	#19134611	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	1	yes	characteristic
14	C₂×#13063257	C₂²	1	2	yes	—
15	C₂×#13063250	C₂²	1	4	yes	—
16	C₂×#13063250	C₂²	1	2	yes	—
17	#14582805	C₂²	1	4	yes	—
18	#14582780	—	2	4	no	—
19	#14582766	—	2	2	no	—
20	#14582765	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
22	#14582744	—	2	2	no	—
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	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
37	C₂×Dic₄₃⋊C₄	—	2	2	no	—
38	C₂×C₄×Dic₄₃	—	2	2	no	—
39	C₂×C₄×Dic₄₃	—	2	4	no	—
40	#13063259	—	2	4	no	—
41	#13063259	C₂×C₄	1	4	yes	—
42	#13063257	—	4	2	no	—
43	#13063250	—	4	4	no	—
44	#13063250	C₂×C₄	1	8	yes	—
45	C₄₃⋊#95	—	2	4	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	4	no	—
49	C₄₃⋊#95	—	2	2	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	2	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	—	2	2	no	—
57	C₄×C₈×C₄₃	—	2	4	no	—
58	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
59	#95×C₄₃	D₄	1	1	yes	characteristic
60	#95×C₄₃	D₄	1	1	yes	characteristic
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	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
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	4	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	2	2	no	—
87	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₃	#41	1	2	yes	—
89	C₄⋊Dic₄₃	—	2	2	no	—
90	C₄⋊Dic₄₃	—	2	4	no	—
91	C₄⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
93	Dic₄₃⋊C₄	—	2	8	no	—
94	C₄×Dic₄₃	—	2	8	no	—
95	C₄×Dic₄₃	—	4	8	no	—
96	Dic₁₇₂	—	2	8	no	—
97	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₃	#135	1	4	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×#41	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	—	4	2	no	—
117	C₂×Dic₄₃	—	4	2	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	—	4	2	no	—
125	Dic₈₆	—	4	2	no	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	—	4	2	no	—
128	C₈×C₄₃	#117	1	4	yes	—
129	#22809	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₃	#547	1	1	yes	characteristic
132	C₂²×C₄₃	#550	1	1	yes	characteristic
133	C₂²×C₄₃	C₂×#135	1	2	yes	—
134	C₂²×C₄₃	#578	1	2	yes	—
135	C₄×C₄₃	—	2	4	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	—	4	2	no	—
139	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₄₃	#520	1	1	yes	characteristic
141	C₄×C₄₃	#555	1	1	yes	characteristic
142	C₄×C₄₃	#556	1	1	yes	characteristic
143	Dic₄₃	—	4	4	no	—
144	Dic₄₃	—	4	4	no	—
145	Dic₄₃	—	8	4	no	—
146	C₂×C₄×Q₁₆	—	43	1	no	—
147	#2356	—	43	2	no	—
148	#1969	—	43	2	no	—
149	#1927	—	43	4	no	—
150	#1846	—	43	1	no	—
151	#1805	—	43	2	no	—
152	C₄×#107	—	43	2	no	—
153	C₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic
154	C₂×C₄₃	#2354	1	1	yes	characteristic
155	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
156	C₂×C₄₃	#2950	1	1	yes	characteristic
157	C₂×C₄₃	#3336	1	2	yes	—
158	C₂×C₄₃	#3393	1	2	yes	—
159	C₂²×Q₁₆	—	43	1	no	—
160	C₂×C₄×Q₈	—	43	2	no	—
161	C₄×Q₁₆	—	43	8	no	—
162	C₂×#107	—	43	1	no	—
163	C₂×#107	—	43	2	no	—
164	C₂×#106	—	43	2	no	—
165	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
166	C₂×#103	—	43	2	no	—
167	C₂×#103	—	43	4	no	—
168	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
169	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
170	#391	—	43	4	no	—
171	#387	—	86	4	no	—
172	#383	—	86	2	no	—
173	#382	—	43	2	no	—
174	C₄×C₄⋊C₄	—	43	4	no	—
175	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
176	#376	—	86	2	no	—
177	#340	—	43	4	no	—
178	#336	C₂×D₄₃	1	2	yes	—
179	C₄₃	#22809	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	43	2	no	—
181	C₂×Q₁₆	—	43	4	no	—
182	C₂×Q₁₆	—	86	2	no	—
183	C₂×Q₁₆	—	86	2	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	4	no	—
189	C₂×C₄⋊C₄	—	43	2	no	—
190	C₂×C₄⋊C₄	—	43	4	no	—
191	C₂×C₄⋊C₄	—	86	2	no	—
192	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
193	C₂×C₄²	—	86	2	no	—
194	C₂×C₄²	—	86	4	no	—
195	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
196	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
197	#107	—	43	4	no	—
198	#107	—	86	4	no	—
199	#106	—	172	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	#103	—	43	8	no	—
202	#103	—	172	4	no	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₄×D₄₃	1	4	yes	—
205	#95	—	86	2	no	—
206	#95	—	86	4	no	—
207	#95	—	86	4	no	—
208	#95	—	86	2	no	—
209	#95	—	86	4	no	—
210	#95	D₁₇₂	1	1	yes	characteristic
211	#95	D₁₇₂	1	1	yes	characteristic
212	C₂×Q₈	—	43	4	no	—
213	C₂×Q₈	—	86	2	no	—
214	C₂×Q₈	—	86	2	no	—
215	C₂×Q₈	—	86	2	no	—
216	C₂×Q₈	—	86	2	no	—
217	C₂×Q₈	—	86	2	no	—
218	C₂²×C₄	—	86	2	no	—
219	C₂²×C₄	—	86	2	no	—
220	C₂²×C₄	—	86	4	no	—
221	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
222	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
225	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
226	Q₁₆	—	86	8	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	#10621900	1	2	yes	—
235	C₂×C₈	#10621902	1	2	yes	—
236	C₄⋊C₄	—	2	4	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	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	86	8	no	—
245	C₄²	—	172	8	no	—
246	C₄²	C₂×C₄×D₄₃	1	2	yes	—
247	C₄²	#10621898	1	2	yes	—
248	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
249	Q₈	—	86	4	no	—
250	Q₈	—	86	4	no	—
251	Q₈	—	172	2	no	—
252	Q₈	—	172	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	86	4	no	—
261	C₂×C₄	—	86	4	no	—
262	C₂×C₄	—	172	4	no	—
263	C₂×C₄	—	172	4	no	—
264	C₂×C₄	—	172	4	no	—
265	C₂×C₄	—	172	2	no	—
266	C₂×C₄	—	172	2	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	#13063347	1	4	yes	—
271	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
273	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
274	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
275	C₂×C₄	#13063371	1	1	yes	characteristic
276	C₂×C₄	#13063372	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	#13063283	1	4	yes	—
280	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
281	C₂²	#14583553	1	1	yes	characteristic
282	C₂²	#14583571	1	1	yes	characteristic
283	C₂²	C₂×#13063347	1	2	yes	—
284	C₂²	#14583716	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	344	4	no	—
290	C₄	—	172	4	no	—
291	C₄	—	172	4	no	—
292	C₄	C₂×#13063283	1	1	yes	characteristic
293	C₄	#14583379	1	1	yes	characteristic
294	C₄	#14583614	1	1	yes	characteristic
295	C₄	#14583622	1	1	yes	characteristic
296	C₂	#19137542	1	1	yes	characteristic
297	C₂	C₄×#13063347	1	1	yes	characteristic
298	C₂	#19143464	1	1	yes	characteristic
299	C₂	#19146964	1	2	yes	—
300	C₂	#19147584	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, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#22809
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
179. C₄₃	#22809	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
#19137558	2
#19135456	2
#19135230	2
#19134759	2
#19134611	2
C₄×#13063259	2
C₈×C₄⋊C₄×C₄₃	2
#22809	43