Subgroups — #26226708

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20244164	C₂	1	8	yes	maximal, complemented
3	#20243426	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	2	no	—
6	#15689180	—	2	2	no	—
7	#15689176	—	2	2	no	—
8	#15689176	—	2	2	no	—
9	C₂×#13093960	C₂²	1	8	yes	—
10	#15689147	C₂²	1	4	yes	—
11	#15689087	C₂²	1	4	yes	—
12	C₄×#10625329	C₂²	1	4	yes	—
13	#15688888	C₂²	1	8	yes	—
14	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₄₇	C₄	1	4	yes	—
17	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
20	C₂×#10625329	C₂³	1	8	yes	—
21	#13093960	—	4	8	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
26	#13093935	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	2	4	no	—
28	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	4	4	no	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
43	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
45	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₄₇	—	2	8	no	—
65	C₂×C₈×C₄₇	—	4	4	no	—
66	C₂×C₈×C₄₇	—	4	4	no	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
68	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	C₂×D₄	1	2	yes	—
75	C₄²×C₄₇	C₂×D₄	1	2	yes	—
76	#10625329	—	2	8	no	—
77	#10625329	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂³×C₄₇	—	2	2	no	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	4	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	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	1	no	—
97	C₂×C₄×C₄₇	—	4	1	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	#108	1	4	yes	complemented
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₄₇	—	2	8	no	—
111	C₄₇⋊C₈	—	4	8	no	—
112	#21969	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	8	no	—
114	C₂²×C₄₇	—	2	1	no	—
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	—	2	1	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	—	2	1	no	—
120	C₂²×C₄₇	C₂×#108	1	2	yes	—
121	C₂²×C₄₇	#430	1	2	yes	—
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	#544	1	2	yes	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	C₂×#108	1	4	yes	—
130	C₄×C₄₇	#518	1	1	yes	characteristic
131	C₄×C₄₇	#519	1	1	yes	characteristic
132	C₄×C₄₇	#537	1	2	yes	—
133	#2920	—	47	2	no	—
134	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
135	#1950	—	47	8	no	—
136	#1810	—	47	4	no	—
137	C₂×C₄₇	—	2	2	no	—
138	C₂×C₄₇	—	2	2	no	—
139	C₂×C₄₇	—	2	2	no	—
140	C₂×C₄₇	#2316	1	1	yes	characteristic
141	C₂×C₄₇	C₂²×#108	1	1	yes	characteristic
142	C₂×C₄₇	C₂×#430	1	1	yes	characteristic
143	C₂³×C₈	C₂×D₄₇	1	2	yes	—
144	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
145	#432	—	94	2	no	—
146	#432	—	94	2	no	—
147	#431	—	94	2	no	—
148	#431	—	94	2	no	—
149	C₂×#108	—	47	8	no	—
150	#423	—	47	4	no	—
151	C₂×C₄⋊C₈	Dic₄₇	1	4	yes	—
152	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
153	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
154	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
155	#407	—	47	4	no	—
156	C₄×M₁₆	—	47	4	no	—
157	#341	—	47	8	no	—
158	C₄₇	#21969	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
160	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	47	8	no	—
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
165	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
166	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
167	C₂×C₄²	Dic₉₄	1	2	yes	—
168	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
169	C₂×C₄²	D₁₈₈	1	2	yes	—
170	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
172	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
175	#108	—	188	8	no	—
176	C₄⋊C₈	—	94	4	no	—
177	C₄⋊C₈	—	94	4	no	—
178	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
179	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
180	C₂²⋊C₈	—	94	4	no	—
181	C₂²⋊C₈	—	94	4	no	—
182	#97	—	94	4	no	—
183	C₄×C₈	—	94	4	no	—
184	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
191	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
192	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
198	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	M₁₆	—	94	8	no	—
204	M₁₆	—	94	8	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	94	8	no	—
209	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
210	C₄²	C₄⋊Dic₄₇	1	2	yes	—
211	C₄²	C₄⋊Dic₄₇	1	2	yes	—
212	C₄²	C₄⋊Dic₄₇	1	2	yes	—
213	C₄²	C₄⋊Dic₄₇	1	2	yes	—
214	C₄²	C₂×D₁₈₈	1	2	yes	—
215	C₄²	C₂×D₁₈₈	1	2	yes	—
216	C₄²	C₂²×Dic₄₇	1	2	yes	—
217	C₄²	C₂²×Dic₄₇	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
222	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
223	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#13093960	1	4	yes	complemented
239	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
245	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
247	C₂×C₄	#13094069	1	2	yes	—
248	C₂×C₄	#13094072	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₈	—	188	8	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	C₂×#13093960	1	2	yes	—
259	C₂²	#15689172	1	2	yes	—
260	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
261	C₂²	#15689754	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	C₂×#13093960	1	4	yes	—
268	C₄	#15689598	1	1	yes	characteristic
269	C₄	#15689601	1	1	yes	characteristic
270	C₄	#15689704	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	#20246006	1	1	yes	characteristic
275	C₂	C₂²×#13093960	1	1	yes	characteristic
276	C₂	C₂×#15689172	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#21969
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20244164	C₂	semidirect
99. C₂×C₄×C₄₇	#108	semidirect
158. C₄₇	#21969	semidirect
238. C₂×C₄	#13093960	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20244164	2
#20243426	2
C₂²×C₄⋊C₈×C₄₇	2
#21969	47