Subgroups — #25196458

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142958	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063325	C₂	1	1	yes	maximal, characteristic
3	#19135747	C₂	1	4	yes	maximal
4	#19135397	C₂	1	2	yes	maximal
5	#19134723	C₂	1	2	yes	maximal
6	#19134611	C₂	1	4	yes	maximal
7	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	C₂²	1	1	yes	characteristic
9	C₂×#13063317	C₂²	1	2	yes	—
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₂×#13063250	C₂²	1	8	yes	—
12	#14582805	C₂²	1	2	yes	—
13	#14582797	—	2	2	no	—
14	#14582797	—	2	2	no	—
15	#14582797	—	2	2	no	—
16	#14582769	C₂²	1	1	yes	characteristic
17	#14582769	C₂²	1	2	yes	—
18	#14582765	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₃	—	2	2	no	—
20	#14582662	C₂²	1	8	yes	—
21	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
23	C₄³×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#336	C₂²	1	2	yes	—
25	C₂²×Dic₈₆	C₂³	1	2	yes	—
26	#13063325	—	2	2	no	—
27	#13063325	Q₈	1	4	yes	—
28	#13063317	—	4	4	no	—
29	C₄×Dic₈₆	—	2	8	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
36	C₂×Dic₄₃⋊C₄	—	2	2	no	—
37	C₂×Dic₄₃⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₄₃	—	2	2	no	—
39	#13063250	—	4	8	no	—
40	C₄₃⋊#95	—	2	2	no	—
41	C₄₃⋊#95	—	2	2	no	—
42	C₄₃⋊#95	—	2	4	no	—
43	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₃	D₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₃	—	2	4	no	—
50	C₄×C₈×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₃	—	2	2	no	—
57	C₂²×Dic₄₃	—	2	2	no	—
58	C₂²×Dic₄₃	—	2	4	no	—
59	C₂×Dic₈₆	—	2	4	no	—
60	C₂×Dic₈₆	—	2	4	no	—
61	C₂×Dic₈₆	—	2	2	no	—
62	C₂×Dic₈₆	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₄₃	—	2	4	no	—
64	C₂×C₈×C₄₃	—	2	4	no	—
65	C₂×C₈×C₄₃	—	2	4	no	—
66	C₄²×C₄₃	—	2	4	no	—
67	C₄²×C₄₃	—	2	4	no	—
68	C₄²×C₄₃	—	2	2	no	—
69	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
70	C₄²×C₄₃	#41	1	4	yes	—
71	C₄²×C₄₃	#41	1	2	yes	—
72	C₄⋊Dic₄₃	—	2	8	no	—
73	C₄⋊Dic₄₃	—	2	4	no	—
74	C₄⋊Dic₄₃	—	2	8	no	—
75	C₄⋊Dic₄₃	—	4	4	no	—
76	C₄⋊Dic₄₃	—	4	4	no	—
77	C₄⋊Dic₄₃	—	4	4	no	—
78	Dic₄₃⋊C₄	—	4	8	no	—
79	Dic₄₃⋊C₄	—	4	4	no	—
80	C₄×Dic₄₃	—	4	4	no	—
81	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
88	C₂×C₄×C₄₃	#124	1	2	yes	—
89	C₂×C₄×C₄₃	#124	1	2	yes	—
90	C₂×C₄×C₄₃	#135	1	4	yes	—
91	C₂×C₄×C₄₃	#137	1	4	yes	—
92	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
95	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
96	C₂×Dic₄₃	—	2	4	no	—
97	C₂×Dic₄₃	—	4	2	no	—
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	—	4	4	no	—
101	C₂×Dic₄₃	—	4	4	no	—
102	C₂×Dic₄₃	—	4	4	no	—
103	C₂×Dic₄₃	—	4	4	no	—
104	Dic₈₆	—	2	8	no	—
105	Dic₈₆	—	4	4	no	—
106	C₈×C₄₃	—	4	4	no	—
107	#29144	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₃	#535	1	1	yes	characteristic
111	C₂²×C₄₃	C₂×#135	1	2	yes	—
112	C₂²×C₄₃	C₂×#137	1	2	yes	—
113	C₄×C₄₃	—	2	2	no	—
114	C₄×C₄₃	—	2	2	no	—
115	C₄×C₄₃	—	4	2	no	—
116	C₄×C₄₃	#479	1	8	yes	—
117	C₄×C₄₃	#554	1	2	yes	—
118	C₄×C₄₃	#555	1	2	yes	—
119	Dic₄₃	—	4	4	no	—
120	Dic₄₃	—	8	2	no	—
121	Dic₄₃	—	8	4	no	—
122	#2859	—	43	1	no	—
123	C₄×C₄⋊Q₈	—	43	1	no	—
124	#2026	—	43	4	no	—
125	#1956	—	43	2	no	—
126	#1834	—	43	2	no	—
127	#1805	—	43	4	no	—
128	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic
129	C₂×C₄₃	#2563	1	1	yes	characteristic
130	C₂×C₄₃	C₂×#479	1	2	yes	—
131	C₂×C₄₃	#3131	1	1	yes	characteristic
132	C₂×C₄₃	#3139	1	1	yes	characteristic
133	C₂×C₄₃	#3210	1	2	yes	—
134	C₂×C₄⋊Q₈	—	43	1	no	—
135	C₂×#125	—	43	2	no	—
136	C₂×C₄×Q₈	—	43	2	no	—
137	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
138	C₂×#103	—	43	8	no	—
139	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
140	#391	—	43	2	no	—
141	#389	—	86	2	no	—
142	#389	—	86	2	no	—
143	#389	—	86	2	no	—
144	#384	—	43	1	no	—
145	#384	—	43	2	no	—
146	#382	—	43	2	no	—
147	C₄×C₄⋊C₄	—	86	2	no	—
148	C₄³	C₂×D₄₃	1	1	yes	characteristic
149	#340	—	43	8	no	—
150	#336	C₂×D₄₃	1	2	yes	—
151	C₄₃	#29144	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	—	43	2	no	—
153	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
154	C₄⋊Q₈	—	43	4	no	—
155	C₄⋊Q₈	—	86	2	no	—
156	#125	—	172	4	no	—
157	C₄×Q₈	—	86	8	no	—
158	C₂×C₄⋊C₄	—	43	2	no	—
159	C₂×C₄⋊C₄	—	43	4	no	—
160	C₂×C₄⋊C₄	—	86	2	no	—
161	C₂×C₄⋊C₄	—	86	2	no	—
162	C₂×C₄⋊C₄	—	86	4	no	—
163	C₂×C₄⋊C₄	—	86	2	no	—
164	C₂×C₄⋊C₄	—	86	2	no	—
165	C₂×C₄⋊C₄	—	86	2	no	—
166	C₂×C₄²	—	86	2	no	—
167	C₂×C₄²	D₁₇₂	1	2	yes	—
168	C₂×C₄²	D₁₇₂	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
172	C₄⋊C₈	—	2	4	no	—
173	#103	—	172	8	no	—
174	C₄×C₈	—	2	4	no	—
175	#95	—	86	4	no	—
176	#95	—	86	2	no	—
177	#95	—	86	2	no	—
178	C₂×Q₈	—	43	4	no	—
179	C₂×Q₈	—	86	2	no	—
180	C₂×Q₈	—	86	4	no	—
181	C₂×Q₈	—	86	4	no	—
182	C₂²×C₄	—	86	2	no	—
183	C₂²×C₄	—	86	2	no	—
184	C₂²×C₄	—	86	4	no	—
185	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
186	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
189	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₄⋊C₄	—	86	8	no	—
194	C₄⋊C₄	—	86	4	no	—
195	C₄⋊C₄	—	86	8	no	—
196	C₄⋊C₄	—	172	8	no	—
197	C₄⋊C₄	—	172	4	no	—
198	C₄⋊C₄	—	172	4	no	—
199	C₄⋊C₄	—	172	4	no	—
200	C₄⋊C₄	—	172	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	172	4	no	—
205	C₄²	#10621898	1	4	yes	—
206	C₄²	#10621900	1	2	yes	—
207	C₄²	Q₈×D₄₃	1	2	yes	—
208	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
209	Q₈	—	86	8	no	—
210	Q₈	—	172	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	86	4	no	—
217	C₂×C₄	—	172	2	no	—
218	C₂×C₄	—	172	4	no	—
219	C₂×C₄	—	172	4	no	—
220	C₂×C₄	—	172	4	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	—	172	2	no	—
223	C₂×C₄	—	172	2	no	—
224	C₂×C₄	#13063305	1	4	yes	—
225	C₂×C₄	#13063313	1	2	yes	—
226	C₂×C₄	#13063313	1	2	yes	—
227	C₂×C₄	#13063347	1	4	yes	—
228	C₂×C₄	#13063358	1	4	yes	—
229	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
231	C₂×C₄	C₂×#10621898	1	2	yes	—
232	C₂×C₄	#13063371	1	2	yes	—
233	C₈	—	4	4	no	—
234	C₂²	C₂×#13063305	1	1	yes	characteristic
235	C₂²	C₂×#13063313	1	1	yes	characteristic
236	C₂²	#14583468	1	1	yes	characteristic
237	C₂²	C₂×#13063347	1	2	yes	—
238	C₂²	C₂×#13063358	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	344	4	no	—
243	C₄	—	344	2	no	—
244	C₄	—	172	4	no	—
245	C₄	#14583207	1	8	yes	—
246	C₄	#14583609	1	2	yes	—
247	C₄	#14583614	1	2	yes	—
248	C₂	#19139764	1	1	yes	characteristic
249	C₂	C₂×#14583207	1	2	yes	—
250	C₂	#19144918	1	1	yes	characteristic
251	C₂	#19144976	1	1	yes	characteristic
252	C₂	#19145553	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#29144
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
151. C₄₃	#29144	semidirect

Maximal subgroups

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

subgroup	index
#19142958	2
C₄×#13063325	2
#19135747	2
#19135397	2
#19134723	2
#19134611	2
C₄×C₄⋊C₈×C₄₃	2
#29144	43