Subgroups — #26226519

Order 12032.

1218 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	#20243573	C₂	1	8	yes	maximal, complemented
3	#20243368	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	complemented
7	#15689087	—	2	8	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15688878	C₂²	1	8	yes	complemented
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
15	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
16	C₂×#95×C₄₇	C₄	1	1	yes	characteristic
17	C₂×#95×C₄₇	C₄	1	1	yes	characteristic
18	C₂×#10625329	—	2	8	no	—
19	C₂×#10625329	—	4	8	no	—
20	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
21	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
24	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	—	4	4	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	D₄	1	4	yes	—
31	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
35	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₄₇	—	4	4	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
39	#95×C₄₇	C₂×C₄	1	4	yes	—
40	#95×C₄₇	C₂×C₄	1	4	yes	—
41	C₂⁴×C₄₇	—	2	4	no	—
42	C₂⁴×C₄₇	—	4	2	no	—
43	C₂⁴×C₄₇	—	4	1	no	—
44	C₂⁴×C₄₇	—	4	1	no	—
45	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	4	8	no	—
52	C₂²×C₄×C₄₇	—	4	8	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	M₁₆	1	8	yes	complemented
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²⋊C₄×C₄₇	—	2	16	no	—
72	C₄²×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	—	4	4	no	—
74	#10625329	—	4	16	no	—
75	C₂×C₄₇⋊C₈	—	2	8	no	—
76	C₂×C₄₇⋊C₈	—	2	16	no	—
77	C₂³×C₄₇	—	2	2	no	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	—	2	4	no	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	8	no	—
85	C₂³×C₄₇	—	4	4	no	—
86	C₂³×C₄₇	—	4	4	no	—
87	C₂³×C₄₇	—	4	8	no	—
88	C₂³×C₄₇	—	4	8	no	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	4	4	no	—
105	C₂×C₄×C₄₇	—	4	8	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	—	4	8	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
114	C₄₇⋊C₈	—	4	8	no	—
115	#21934	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	4	16	no	—
124	C₂²×C₄₇	—	4	4	no	—
125	C₂²×C₄₇	—	4	4	no	—
126	C₂²×C₄₇	#423	1	4	yes	—
127	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	#518	1	2	yes	—
129	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
130	C₂²×C₄₇	#568	1	1	yes	characteristic
131	C₂²×C₄₇	#568	1	1	yes	characteristic
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	—	2	8	no	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	#534	1	2	yes	—
137	C₄×C₄₇	#543	1	2	yes	—
138	C₂×#407	—	47	2	no	—
139	C₂×C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
140	#1851	—	47	8	no	—
141	#1797	—	47	4	no	—
142	C₂×C₄₇	—	2	4	no	—
143	C₂×C₄₇	—	4	4	no	—
144	C₂×C₄₇	#2315	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
146	C₂×C₄₇	#2905	1	1	yes	characteristic
147	C₂×C₄₇	#2973	1	2	yes	—
148	C₂×C₄₇	#2975	1	2	yes	—
149	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
150	C₂²×M₁₆	—	47	4	no	—
151	C₂²×C₂²⋊C₄	Dic₄₇	1	2	yes	—
152	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
153	C₂×C₄⋊C₈	—	47	8	no	—
154	#407	—	94	8	no	—
155	C₂×C₂²⋊C₈	—	47	4	no	—
156	C₂×C₂²⋊C₈	—	47	4	no	—
157	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
158	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
159	C₂×#95	Dic₄₇	1	1	yes	characteristic
160	C₂×#95	Dic₄₇	1	1	yes	characteristic
161	#336	—	47	8	no	—
162	C₄₇	#21934	1	1	yes	characteristic, complemented, Sylow
163	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
164	C₂³×C₄	—	4	4	no	—
165	C₂³×C₄	Dic₉₄	1	4	yes	—
166	C₂³×C₄	D₁₈₈	1	4	yes	—
167	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
168	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
169	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
171	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	94	8	no	—
174	C₂×M₁₆	—	188	8	no	—
175	C₂²×C₈	—	47	8	no	—
176	C₂×C₂²⋊C₄	—	2	8	no	—
177	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
178	C₂×C₄²	—	4	4	no	—
179	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
180	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
181	C₄⋊C₈	—	188	8	no	—
182	C₂²⋊C₈	—	94	8	no	—
183	C₂²⋊C₈	—	94	8	no	—
184	#95	C₂×Dic₄₇	1	4	yes	—
185	#95	C₂×Dic₄₇	1	4	yes	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	4	2	no	—
188	C₂⁴	—	4	1	no	—
189	C₂⁴	—	4	1	no	—
190	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
191	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
192	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	#10625329	1	8	yes	complemented
205	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
206	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
207	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
208	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
209	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
210	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
212	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
213	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
216	M₁₆	—	188	16	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	94	16	no	—
219	C₂²⋊C₄	—	2	16	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	4	4	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	4	8	no	—
233	C₂³	—	4	8	no	—
234	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
235	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
236	C₂³	C₂×#10625329	1	4	yes	—
237	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
255	C₂×C₄	C₂×#10625329	1	8	yes	—
256	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
257	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
258	C₂×C₄	#13094069	1	4	yes	—
259	C₈	—	188	8	no	—
260	C₂²	—	2	4	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	16	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	#15689147	1	4	yes	—
271	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
272	C₂²	#15689598	1	2	yes	—
273	C₂²	C₂²×#10625329	1	2	yes	—
274	C₂²	#15689890	1	1	yes	characteristic
275	C₂²	#15689890	1	1	yes	characteristic
276	C₄	—	2	8	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	4	no	—
280	C₄	#15689685	1	2	yes	—
281	C₄	#15689749	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	4	4	no	—
284	C₂	#20246002	1	1	yes	characteristic
285	C₂	C₂×#15689147	1	1	yes	characteristic
286	C₂	#20251952	1	1	yes	characteristic
287	C₂	#20252417	1	2	yes	—
288	C₂	#20252429	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄×C₂²⋊C₄×C₄₇	C₂
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#21934
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20243573	C₂	semidirect
3. #20243368	C₂	semidirect
6. C₂×C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
10. #15688878	C₂²	semidirect
61. C₂²×C₄×C₄₇	M₁₆	semidirect
162. C₄₇	#21934	semidirect
204. C₂²×C₄	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
#20243573	2
#20243368	2
C₂×C₄×C₂²⋊C₄×C₄₇	2
#21934	47