Subgroups — #25067321

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145181	C₂	1	1	yes	maximal, characteristic
2	#19143529	C₂	1	2	yes	maximal
3	#19133447	C₂	1	8	yes	maximal, complemented
4	#19133354	C₂	1	1	yes	maximal, characteristic
5	#19133051	C₂	1	2	yes	maximal
6	C₂×C₄₃×#322	C₂	1	1	yes	maximal, characteristic
7	#14583456	C₂²	1	2	yes	—
8	#14583304	C₄	1	1	yes	characteristic
9	#14583279	C₄	1	1	yes	characteristic
10	#14583250	C₄	1	2	yes	—
11	#14583140	C₄	1	4	yes	—
12	#14583029	—	2	2	no	—
13	#14583029	C₂²	1	4	yes	—
14	C₈×Dic₈₆	—	2	2	no	—
15	C₈×Dic₈₆	C₂²	1	4	yes	—
16	C₄×Dic₁₇₂	C₂²	1	4	yes	—
17	#14582925	—	2	2	no	—
18	#14582923	C₂²	1	2	yes	—
19	#14582854	—	2	2	no	—
20	#14582652	C₂²	1	4	yes	—
21	#14582631	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#322	C₂²	1	4	yes	—
26	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
27	C₂×#10621877	—	2	2	no	—
28	#13063325	C₂×C₄	1	2	yes	—
29	#13063325	C₂×C₄	1	2	yes	—
30	#13063317	—	2	2	no	—
31	#13063304	—	2	2	no	—
32	#13063304	C₂×C₄	1	4	yes	—
33	C₄×Dic₈₆	—	2	2	no	—
34	C₄×Dic₈₆	C₂³	1	4	yes	—
35	#13063284	—	2	2	no	—
36	#13063259	C₂×C₄	1	4	yes	—
37	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
39	Dic₄₃⋊C₈	—	2	4	no	—
40	Dic₄₃⋊C₈	—	2	4	no	—
41	#13063250	C₂×C₄	1	8	yes	—
42	#13063235	—	2	4	no	—
43	C₈×Dic₄₃	—	2	4	no	—
44	C₂²×C₈×C₄₃	—	2	1	no	—
45	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₄₃	—	2	1	no	—
50	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₃	D₄	1	2	yes	—
53	C₄×C₈×C₄₃	C₂³	1	2	yes	—
54	C₄×C₈×C₄₃	C₂³	1	2	yes	—
55	C₄×C₈×C₄₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
59	C₂×Dic₈₆	—	2	2	no	—
60	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	—	2	2	no	—
63	C₂×C₈×C₄₃	—	4	2	no	—
64	C₂×C₈×C₄₃	—	4	2	no	—
65	C₂×C₈×C₄₃	Q₁₆	1	4	yes	complemented
66	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₃	#41	1	1	yes	characteristic
73	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊Dic₄₃	—	2	4	no	—
75	C₄⋊Dic₄₃	—	2	2	no	—
76	C₄⋊Dic₄₃	—	2	2	no	—
77	C₄⋊Dic₄₃	—	2	2	no	—
78	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
79	Dic₄₃⋊C₄	—	2	4	no	—
80	Dic₄₃⋊C₄	—	2	4	no	—
81	C₄×Dic₄₃	—	2	4	no	—
82	#10621877	—	4	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	Dic₁₇₂	—	4	4	no	—
85	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₄₃	#135	1	2	yes	—
93	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₄₃	—	2	4	no	—
97	C₂×Dic₄₃	—	2	4	no	—
98	Dic₈₆	—	2	8	no	—
99	Dic₈₆	—	4	4	no	—
100	C₈×C₄₃	—	2	2	no	—
101	C₈×C₄₃	—	2	2	no	—
102	C₈×C₄₃	—	2	2	no	—
103	C₈×C₄₃	—	4	2	no	—
104	C₈×C₄₃	C₂×Q₁₆	1	4	yes	—
105	C₄₃⋊C₈	—	4	4	no	—
106	#17977	—	43	1	no	maximal, Sylow
107	C₂²×C₄₃	—	2	1	no	—
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	C₄×Q₁₆	1	2	yes	—
110	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₄₃	—	2	1	no	—
112	C₄×C₄₃	—	2	1	no	—
113	C₄×C₄₃	C₄×Q₁₆	1	2	yes	—
114	C₄×C₄₃	#444	1	2	yes	—
115	C₄×C₄₃	#568	1	2	yes	—
116	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
117	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
118	Dic₄₃	—	4	4	no	—
119	Dic₄₃	—	4	4	no	—
120	#3170	—	43	1	no	—
121	#2960	—	43	2	no	—
122	#1584	—	43	8	no	—
123	#1567	—	43	1	no	—
124	#1493	—	43	2	no	—
125	C₂×#322	D₄₃	1	1	yes	characteristic
126	C₂×C₄₃	—	2	2	no	—
127	C₂×C₄₃	#2932	1	1	yes	characteristic
128	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
129	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
130	#533	—	43	2	no	—
131	#500	—	43	1	no	—
132	#494	—	43	1	no	—
133	#487	—	43	2	no	—
134	#467	—	43	4	no	—
135	#446	—	43	4	no	—
136	#446	—	86	2	no	—
137	C₈×Q₈	—	43	4	no	—
138	C₈×Q₈	—	86	2	no	—
139	C₄×Q₁₆	—	43	4	no	—
140	#424	—	86	2	no	—
141	#423	—	43	2	no	—
142	#405	—	86	2	no	—
143	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
146	#335	—	43	4	no	—
147	Q₈⋊C₈	—	43	8	no	—
148	#322	C₂×D₄₃	1	4	yes	—
149	C₄₃	#17977	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₁₆	—	43	4	no	—
151	C₂×M₁₆	—	86	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₄×D₄₃	1	2	yes	—
154	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
155	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
156	C₄⋊Q₈	—	43	2	no	—
157	C₄⋊Q₈	—	43	2	no	—
158	#125	—	86	2	no	—
159	C₂²⋊Q₈	—	43	4	no	—
160	C₂²⋊Q₈	—	86	2	no	—
161	C₄×Q₈	—	43	4	no	—
162	C₄×Q₈	—	86	2	no	—
163	#117	—	86	2	no	—
164	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
165	#107	—	43	4	no	—
166	C₄⋊C₈	—	43	4	no	—
167	C₄⋊C₈	—	86	4	no	—
168	C₄⋊C₈	—	86	4	no	—
169	C₄⋊C₈	—	86	2	no	—
170	#103	—	43	8	no	—
171	#97	—	86	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	86	4	no	—
174	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
175	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
176	C₄×C₈	D₁₇₂	1	2	yes	—
177	C₄×C₈	C₂²×D₄₃	1	2	yes	—
178	C₄×C₈	C₂²×D₄₃	1	2	yes	—
179	C₄×C₈	C₂²×D₄₃	1	2	yes	—
180	C₂×Q₈	—	43	4	no	—
181	C₂×Q₈	—	86	2	no	—
182	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
184	C₂²×C₄	#10621898	1	1	yes	characteristic
185	Q₁₆	—	172	4	no	—
186	M₁₆	—	172	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	86	4	no	—
192	C₂×C₈	Dic₁₇₂	1	4	yes	complemented
193	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
195	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
196	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
197	C₄⋊C₄	—	43	4	no	—
198	C₄⋊C₄	—	86	2	no	—
199	C₄⋊C₄	—	86	2	no	—
200	C₄⋊C₄	—	86	4	no	—
201	C₄⋊C₄	—	86	2	no	—
202	C₄⋊C₄	—	86	4	no	—
203	C₂²⋊C₄	—	86	4	no	—
204	C₄²	—	86	4	no	—
205	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
207	C₄²	#10621898	1	1	yes	characteristic
208	C₄²	C₂³×D₄₃	1	1	yes	characteristic
209	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
210	Q₈	—	86	8	no	—
211	Q₈	—	172	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	86	4	no	—
215	C₂×C₄	—	86	4	no	—
216	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
217	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
218	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
220	C₂×C₄	#13063347	1	2	yes	—
221	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
223	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	172	4	no	—
229	C₈	C₂×Dic₁₇₂	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Dic₁₇₂	1	2	yes	—
233	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	172	4	no	—
237	C₄	—	172	4	no	—
238	C₄	C₄×Dic₁₇₂	1	2	yes	—
239	C₄	#14583022	1	2	yes	—
240	C₄	#14583665	1	2	yes	—
241	C₄	C₂²×Dic₁₇₂	1	1	yes	characteristic
242	C₄	C₂×#13063347	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#19143314	1	1	yes	characteristic
245	C₂	C₂×C₄×Dic₁₇₂	1	1	yes	characteristic
246	C₂	C₂×#14583022	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#322	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₂×#322	D₄₃
hypocenter	C₄₃	#17977
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #19133447	C₂	semidirect
65. C₂×C₈×C₄₃	Q₁₆	semidirect
149. C₄₃	#17977	semidirect
192. C₂×C₈	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
#19145181	2
#19143529	2
#19133447	2
#19133354	2
#19133051	2
C₂×C₄₃×#322	2
#17977	43