Subgroups — #26132579

Order 12032.

404 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20243426	C₂	1	1	yes	maximal, characteristic
3	#20241483	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093960	C₂²	1	2	yes	—
7	#15689087	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
10	#15688888	—	2	1	no	—
11	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₇	—	2	1	no	—
13	C₄₇×#427	—	2	1	no	—
14	C₄₇×#427	—	2	1	no	—
15	C₄₇×#427	C₄	1	2	yes	—
16	C₄₇×#427	C₂²	1	2	yes	—
17	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
18	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
19	C₂×#106×C₄₇	—	2	1	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	4	no	—
23	#13093960	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
26	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	—	2	4	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	#117×C₄₇	—	2	2	no	—
35	#117×C₄₇	—	2	2	no	—
36	#117×C₄₇	—	4	1	no	—
37	#117×C₄₇	—	4	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
41	C₂×C₄²×C₄₇	—	4	1	no	—
42	#107×C₄₇	—	2	1	no	—
43	#107×C₄₇	—	2	1	no	—
44	#107×C₄₇	C₂×C₄	1	2	yes	—
45	#107×C₄₇	C₂×C₄	1	2	yes	—
46	#106×C₄₇	—	2	2	no	—
47	#106×C₄₇	—	2	2	no	—
48	C₂⁴×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	4	1	no	—
52	C₂²×C₄×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	#41	1	4	yes	—
69	C₄⋊C₄×C₄₇	—	2	2	no	—
70	C₄⋊C₄×C₄₇	—	2	2	no	—
71	C₄⋊C₄×C₄₇	—	4	1	no	—
72	C₄⋊C₄×C₄₇	—	4	1	no	—
73	C₂²⋊C₄×C₄₇	—	4	2	no	—
74	C₂²⋊C₄×C₄₇	—	4	2	no	—
75	C₄²×C₄₇	—	4	2	no	—
76	C₄²×C₄₇	—	4	2	no	—
77	#10625329	—	4	2	no	—
78	#10625329	—	4	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₄₇⋊C₁₆	—	4	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	4	1	no	—
84	C₂³×C₄₇	#102	1	1	yes	characteristic
85	C₂³×C₄₇	#102	1	1	yes	characteristic
86	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	8	2	no	—
97	C₂×C₄×C₄₇	—	8	2	no	—
98	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#123	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#137	1	2	yes	—
104	C₈×C₄₇	—	2	1	no	—
105	C₈×C₄₇	—	2	1	no	—
106	C₈×C₄₇	#117	1	2	yes	—
107	C₈×C₄₇	#123	1	2	yes	—
108	C₄₇⋊C₈	—	8	2	no	—
109	#12869	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	1	no	—
111	C₂²×C₄₇	—	2	1	no	—
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	2	1	no	—
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	—	4	4	no	—
118	C₂²×C₄₇	#359	1	2	yes	—
119	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
120	C₂²×C₄₇	#483	1	2	yes	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	8	2	no	—
124	C₄×C₄₇	—	8	2	no	—
125	C₄×C₄₇	#381	1	1	yes	characteristic
126	C₄×C₄₇	#419	1	1	yes	characteristic
127	C₄×C₄₇	#480	1	2	yes	—
128	C₂×#427	D₄₇	1	1	yes	characteristic
129	C₂×C₂²⋊C₁₆	—	47	1	no	—
130	#1810	—	47	1	no	—
131	#1382	—	47	4	no	—
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	—	2	2	no	—
134	C₂×C₄₇	—	4	1	no	—
135	C₂×C₄₇	#1823	1	1	yes	characteristic
136	C₂×C₄₇	C₂×#359	1	1	yes	characteristic
137	C₂×C₄₇	#2143	1	1	yes	characteristic
138	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
139	C₂×#117	—	2	1	no	—
140	C₂²×C₁₆	—	47	2	no	—
141	C₂×#108	—	47	2	no	—
142	#427	—	2	1	no	—
143	#427	—	2	1	no	—
144	#427	Dic₄₇	1	2	yes	—
145	#427	C₂×D₄₇	1	2	yes	—
146	C₂×#107	Dic₄₇	1	1	yes	characteristic
147	C₂×#107	Dic₄₇	1	1	yes	characteristic
148	C₂×#106	—	2	1	no	—
149	#407	—	94	1	no	—
150	C₂²⋊C₁₆	—	94	1	no	—
151	C₂²⋊C₁₆	—	94	1	no	—
152	#341	—	94	1	no	—
153	C₄₇	#12869	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
155	C₂×M₁₆	—	94	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	4	1	no	—
166	#117	—	4	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	—	4	1	no	—
171	C₂×C₁₆	—	94	4	no	—
172	C₂×C₁₆	—	94	2	no	—
173	#108	—	94	4	no	—
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	#107	C₂×Dic₄₇	1	2	yes	—
177	#107	C₂×Dic₄₇	1	2	yes	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	C₂²⋊C₈	—	188	1	no	—
181	C₂²⋊C₈	—	188	1	no	—
182	C₂⁴	#10625335	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	#10625334	1	1	yes	characteristic
188	C₂²×C₄	#10625334	1	1	yes	characteristic
189	C₂²×C₄	#10625335	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	M₁₆	—	188	2	no	—
194	M₁₆	—	188	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	188	2	no	—
201	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
203	C₂×C₈	#10625350	1	4	yes	—
204	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
205	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	188	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	#13093945	1	1	yes	characteristic
219	C₂³	#13093945	1	1	yes	characteristic
220	C₂³	C₂×#10625335	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#13093945	1	1	yes	characteristic
233	C₂×C₄	#13093945	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
235	C₂×C₄	#13094007	1	2	yes	—
236	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
237	C₂×C₄	#13094055	1	2	yes	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	376	2	no	—
241	C₈	#13093983	1	2	yes	—
242	C₈	#13094007	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	#15688930	1	2	yes	—
252	C₂²	C₂×#13093945	1	1	yes	characteristic
253	C₂²	#15689453	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	#15688986	1	1	yes	characteristic
259	C₄	#15689132	1	1	yes	characteristic
260	C₄	#15689436	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	#20243474	1	1	yes	characteristic
265	C₂	C₂×#15688930	1	1	yes	characteristic
266	C₂	#20245116	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂⁴×C₄₇	SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#12869
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20241483	C₂	semidirect
153. C₄₇	#12869	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20243426	2
#20241483	2
C₂×C₄₇×#427	2
#12869	47