Subgroups — C₂²×#15689330 #27083077

Order 12032.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625337	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689330	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13093948	C₂	1	2	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂²×#10625337	—	2	16	no	—
9	C₂²×#10625337	C₂²	1	24	yes	—
10	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×Dic₉₄	—	2	6	no	—
12	C₂×C₄×Dic₉₄	C₂²	1	12	yes	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
15	C₂²×C₄×Dic₄₇	—	2	1	no	—
16	#15689330	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	12	yes	—
18	C₂×#13093948	C₂²	1	24	yes	—
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₄₇	—	2	6	no	—
21	C₂×C₄⋊Q₈×C₄₇	C₂²	1	12	yes	—
22	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
23	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	—	2	3	no	—
26	C₂²×Dic₉₄	—	2	4	no	—
27	C₂²×Dic₉₄	D₄	1	8	yes	—
28	C₂²×Dic₉₄	C₂³	1	6	yes	—
29	C₂³×Dic₄₇	—	2	1	no	—
30	C₂³×Dic₄₇	—	2	1	no	—
31	C₂×#10625337	—	2	96	no	—
32	C₂×#10625337	C₂³	1	32	yes	—
33	C₂²×C₄₇⋊C₈	—	2	8	no	—
34	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
35	C₄×Dic₉₄	—	2	24	no	—
36	C₄×Dic₉₄	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₄₇	—	2	3	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
39	C₂×Dic₄₇⋊C₄	—	2	12	no	—
40	C₂×C₄×Dic₄₇	—	2	12	no	—
41	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
42	#13093948	C₂³	1	32	yes	—
43	C₂²×Q₈×C₄₇	—	2	6	no	—
44	C₂²×Q₈×C₄₇	—	2	8	no	—
45	C₂²×Q₈×C₄₇	—	2	8	no	—
46	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
47	C₂³×C₄×C₄₇	—	2	2	no	—
48	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	—	2	24	no	—
52	C₄⋊Q₈×C₄₇	C₂³	1	16	yes	—
53	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
55	C₂×C₄²×C₄₇	D₄	1	6	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₄₇	—	2	12	no	—
59	C₂×Q₈×C₄₇	—	2	24	no	—
60	C₂×Q₈×C₄₇	—	2	24	no	—
61	C₂×Q₈×C₄₇	—	4	24	no	—
62	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	12	no	—
65	C₂²×C₄×C₄₇	—	4	8	no	—
66	C₂²×C₄×C₄₇	Q₁₆	1	8	yes	complemented
67	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₄₇	#41	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
73	C₂²×Dic₄₇	—	2	8	no	—
74	C₂²×Dic₄₇	—	2	6	no	—
75	C₂²×Dic₄₇	—	2	6	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂×Dic₉₄	—	2	24	no	—
78	C₂×Dic₉₄	—	2	6	no	—
79	C₂×Dic₉₄	—	2	12	no	—
80	C₂×Dic₉₄	C₂×D₄	1	24	yes	—
81	C₂×Dic₉₄	C₂⁴	1	4	yes	—
82	C₄⋊C₄×C₄₇	—	2	16	no	—
83	C₄⋊C₄×C₄₇	—	2	16	no	—
84	C₄²×C₄₇	C₂×D₄	1	12	yes	—
85	C₄²×C₄₇	C₂⁴	1	4	yes	—
86	#10625337	—	2	64	no	—
87	C₄⋊Dic₄₇	—	2	6	no	—
88	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
89	Dic₄₇⋊C₄	—	2	16	no	—
90	C₄×Dic₄₇	—	2	16	no	—
91	C₂×C₄₇⋊C₈	—	2	24	no	—
92	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
93	C₂³×C₄₇	#121	1	4	yes	—
94	C₂³×C₄₇	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₄₇	#137	1	4	yes	—
96	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
97	Q₈×C₄₇	—	2	8	no	—
98	Q₈×C₄₇	—	2	8	no	—
99	Q₈×C₄₇	—	4	24	no	—
100	C₂×C₄×C₄₇	—	2	12	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	4	24	no	—
103	C₂×C₄×C₄₇	#121	1	12	yes	—
104	C₂×C₄×C₄₇	C₂×Q₁₆	1	24	yes	complemented
105	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
110	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
111	C₂×Dic₄₇	—	2	24	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	4	12	no	—
115	Dic₉₄	—	2	24	no	—
116	Dic₉₄	—	2	4	no	—
117	Dic₉₄	C₂²×D₄	1	8	yes	—
118	C₄₇⋊C₈	—	2	8	no	—
119	C₂²×#462	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	#462	1	16	yes	complemented
121	C₂²×C₄₇	C₂×#121	1	6	yes	—
122	C₂²×C₄₇	C₂²×Q₁₆	1	6	yes	—
123	C₂²×C₄₇	C₂×#137	1	6	yes	—
124	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	4	8	no	—
127	C₄×C₄₇	C₂×#121	1	6	yes	—
128	C₄×C₄₇	C₂²×Q₁₆	1	8	yes	complemented
129	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
131	Dic₄₇	—	2	8	no	—
132	Dic₄₇	—	4	4	no	—
133	C₂³×Q₁₆	—	47	2	no	—
134	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
135	C₂²×C₄×Q₈	—	47	1	no	—
136	C₂×#462	—	47	24	no	—
137	C₂²×C₄⋊C₈	—	47	1	no	—
138	C₂²×#103	—	47	2	no	—
139	C₂×C₄₇	C₂×#462	1	12	yes	complemented
140	C₂×C₄₇	C₂²×#121	1	1	yes	characteristic
141	C₂×C₄₇	C₂³×Q₁₆	1	1	yes	characteristic
142	C₂×C₄₇	C₂²×#137	1	1	yes	characteristic
143	C₂³×Q₈	—	47	1	no	—
144	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
145	C₂²×Q₁₆	—	47	24	no	—
146	C₂²×Q₁₆	—	94	16	no	—
147	C₂³×C₈	—	47	2	no	—
148	C₂×C₄⋊Q₈	—	2	6	no	—
149	C₂×C₄⋊Q₈	C₂×D₄₇	1	12	yes	—
150	C₂×C₄×Q₈	—	47	12	no	—
151	C₂×C₄×Q₈	—	94	6	no	—
152	C₂²×C₄⋊C₄	—	2	1	no	—
153	C₂²×C₄⋊C₄	—	2	1	no	—
154	C₂²×C₄⋊C₄	—	47	1	no	—
155	C₂²×C₄⋊C₄	—	94	1	no	—
156	C₂²×C₄²	—	94	1	no	—
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#462	—	47	64	no	—
159	C₂×C₄⋊C₈	—	47	12	no	—
160	C₂×#103	—	47	24	no	—
161	C₄₇	C₂²×#462	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	2	6	no	—
163	C₂²×Q₈	—	2	8	no	—
164	C₂²×Q₈	—	2	8	no	—
165	C₂²×Q₈	—	47	8	no	—
166	C₂²×Q₈	—	47	6	no	—
167	C₂²×Q₈	—	94	4	no	—
168	C₂²×Q₈	—	94	3	no	—
169	C₂²×Q₈	C₂²×D₄₇	1	12	yes	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	—	94	1	no	—
172	C₂³×C₄	—	94	1	no	—
173	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₂×Q₁₆	—	47	32	no	—
177	C₂×Q₁₆	—	94	96	no	—
178	C₂²×C₈	—	47	12	no	—
179	C₂²×C₈	—	94	8	no	—
180	C₄⋊Q₈	—	2	24	no	—
181	C₄⋊Q₈	C₂²×D₄₇	1	16	yes	—
182	C₄×Q₈	—	47	16	no	—
183	C₄×Q₈	—	94	24	no	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	47	6	no	—
187	C₂×C₄⋊C₄	—	94	3	no	—
188	C₂×C₄⋊C₄	—	94	12	no	—
189	C₂×C₄²	—	94	12	no	—
190	C₂×C₄²	C₂²⋊D₄₇	1	6	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
192	C₄⋊C₈	—	47	16	no	—
193	#103	—	47	32	no	—
194	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂×Q₈	—	2	12	no	—
196	C₂×Q₈	—	2	24	no	—
197	C₂×Q₈	—	2	24	no	—
198	C₂×Q₈	—	4	24	no	—
199	C₂×Q₈	—	47	24	no	—
200	C₂×Q₈	—	47	4	no	—
201	C₂×Q₈	—	94	24	no	—
202	C₂×Q₈	—	94	6	no	—
203	C₂×Q₈	—	94	12	no	—
204	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	12	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	94	8	no	—
209	C₂²×C₄	—	94	6	no	—
210	C₂²×C₄	—	94	6	no	—
211	C₂²×C₄	—	188	4	no	—
212	C₂²×C₄	#10625337	1	8	yes	complemented
213	C₂²×C₄	D₄×D₄₇	1	4	yes	—
214	C₂²×C₄	#10625352	1	4	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
218	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
219	Q₁₆	—	94	64	no	—
220	C₂×C₈	—	47	8	no	—
221	C₂×C₈	—	94	24	no	—
222	C₄⋊C₄	—	2	16	no	—
223	C₄⋊C₄	—	2	16	no	—
224	C₄⋊C₄	—	47	4	no	—
225	C₄⋊C₄	—	94	16	no	—
226	C₄⋊C₄	—	94	6	no	—
227	C₄²	—	94	16	no	—
228	C₄²	C₂×C₂²⋊D₄₇	1	12	yes	—
229	C₄²	C₂³×D₄₇	1	4	yes	—
230	C₂³	#13093995	1	4	yes	—
231	C₂³	C₂×#10625337	1	4	yes	—
232	C₂³	#13094055	1	4	yes	—
233	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
234	Q₈	—	2	8	no	—
235	Q₈	—	2	8	no	—
236	Q₈	—	4	24	no	—
237	Q₈	—	47	8	no	—
238	Q₈	—	94	24	no	—
239	Q₈	—	94	4	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	24	no	—
243	C₂×C₄	—	94	24	no	—
244	C₂×C₄	—	94	4	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	188	12	no	—
247	C₂×C₄	#13093995	1	12	yes	—
248	C₂×C₄	C₂×#10625337	1	24	yes	complemented
249	C₂×C₄	C₂×D₄×D₄₇	1	6	yes	—
250	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
252	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
253	C₂×C₄	C₂×#10625352	1	6	yes	—
254	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
255	C₈	—	94	8	no	—
256	C₂²	#15689330	1	16	yes	direct product factor
257	C₂²	C₂×#13093995	1	6	yes	—
258	C₂²	C₂²×#10625337	1	6	yes	—
259	C₂²	C₂×#13094055	1	6	yes	—
260	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	4	8	no	—
263	C₄	—	94	8	no	—
264	C₄	—	188	4	no	—
265	C₄	C₂×#13093995	1	6	yes	—
266	C₄	C₂²×#10625337	1	8	yes	complemented
267	C₄	C₂²×D₄×D₄₇	1	1	yes	characteristic
268	C₄	C₂²×#10625352	1	1	yes	characteristic
269	C₂	C₂×#15689330	1	12	yes	direct product factor
270	C₂	C₂²×#13093995	1	1	yes	characteristic
271	C₂	C₂³×#10625337	1	1	yes	characteristic
272	C₂	C₂²×#13094055	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#462
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#15689330	C₂	direct (N × H)
16. #15689330	C₂²	direct (N × H)
66. C₂²×C₄×C₄₇	Q₁₆	semidirect
104. C₂×C₄×C₄₇	C₂×Q₁₆	semidirect
120. C₂²×C₄₇	#462	semidirect
128. C₄×C₄₇	C₂²×Q₁₆	semidirect
139. C₂×C₄₇	C₂×#462	semidirect
161. C₄₇	C₂²×#462	semidirect
212. C₂²×C₄	#10625337	semidirect
248. C₂×C₄	C₂×#10625337	semidirect
256. C₂²	#15689330	direct (N × H)
266. C₄	C₂²×#10625337	semidirect
269. C₂	C₂×#15689330	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#10625337	2
C₂²×C₄×Dic₉₄	2
C₂×#15689330	2
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₂²×#13093948	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×#462	47

Subgroups — C22×#15689330 #27083077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689330 #27083077