Subgroups — C₂²×#15689132 #27079806

Order 12032.

5070 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093983	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689132	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093948	C₂	1	4	yes	maximal, complemented
5	C₂⁴×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	—	2	8	no	—
7	C₂²×#10625329	C₂²	1	12	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093983	—	2	6	no	—
10	C₂×#13093983	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₂²⋊Dic₄₇	—	2	1	no	—
13	C₂²×C₄×Dic₄₇	—	2	1	no	—
14	#15689132	C₂²	1	64	yes	direct product factor
15	C₂×#13093948	C₂²	1	48	yes	complemented
16	C₂³×Q₈×C₄₇	—	2	8	no	—
17	C₂³×Q₈×C₄₇	—	2	1	no	—
18	C₂³×Q₈×C₄₇	—	2	6	no	—
19	C₂³×Q₈×C₄₇	C₄	1	16	yes	complemented
20	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
21	C₂³×Q₈×C₄₇	C₂²	1	12	yes	—
22	C₂⁴×C₄×C₄₇	—	2	1	no	—
23	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	—	2	2	no	—
25	C₂×#10625329	—	2	48	no	—
26	C₂×#10625329	C₂³	1	16	yes	—
27	C₂²×C₄₇⋊C₈	—	2	8	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
29	#13093983	—	2	24	no	—
30	#13093983	C₂³	1	16	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	6	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
33	C₂×C₂²⋊Dic₄₇	—	2	12	no	—
34	C₂×C₄×Dic₄₇	—	2	12	no	—
35	#13093948	C₂³	1	64	yes	complemented
36	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×Q₈×C₄₇	—	2	16	no	—
38	C₂²×Q₈×C₄₇	—	2	96	no	—
39	C₂²×Q₈×C₄₇	—	2	8	no	—
40	C₂²×Q₈×C₄₇	—	2	48	no	—
41	C₂²×Q₈×C₄₇	—	2	12	no	—
42	C₂²×Q₈×C₄₇	—	2	6	no	—
43	C₂²×Q₈×C₄₇	—	2	24	no	—
44	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
45	C₂²×Q₈×C₄₇	C₂×C₄	1	96	yes	complemented
46	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
47	C₂²×Q₈×C₄₇	C₂³	1	16	yes	—
48	C₂³×C₄×C₄₇	—	2	16	no	—
49	C₂³×C₄×C₄₇	—	2	2	no	—
50	C₂³×C₄×C₄₇	—	2	12	no	—
51	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
52	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	D₄	1	6	yes	—
54	C₂³×C₄×C₄₇	D₄	1	8	yes	—
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
57	C₂⁴×C₄₇	—	2	8	no	—
58	C₂⁴×C₄₇	C₂²⋊C₄	1	8	yes	—
59	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₄₇	—	2	48	no	—
62	C₂×Q₈×C₄₇	—	2	96	no	—
63	C₂×Q₈×C₄₇	—	2	192	no	—
64	C₂×Q₈×C₄₇	—	2	24	no	—
65	C₂×Q₈×C₄₇	—	2	32	no	—
66	C₂×Q₈×C₄₇	—	2	16	no	—
67	C₂×Q₈×C₄₇	—	2	12	no	—
68	C₂×Q₈×C₄₇	C₂²×C₄	1	48	yes	—
69	C₂×Q₈×C₄₇	C₂²×C₄	1	64	yes	complemented
70	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₄₇	—	2	16	no	—
72	C₂²×C₄×C₄₇	—	2	96	no	—
73	C₂²×C₄×C₄₇	—	2	12	no	—
74	C₂²×C₄×C₄₇	—	2	16	no	—
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	48	yes	—
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
86	C₂²×Dic₄₇	—	2	12	no	—
87	C₂²×Dic₄₇	—	4	8	no	—
88	C₂²⋊Dic₄₇	—	2	16	no	—
89	C₄⋊Dic₄₇	—	2	12	no	—
90	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
91	C₄×Dic₄₇	—	2	16	no	—
92	#10625329	—	2	32	no	—
93	C₂×C₄₇⋊C₈	—	2	24	no	—
94	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
95	C₂³×C₄₇	—	2	24	no	—
96	C₂³×C₄₇	—	2	32	no	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂³×C₄₇	#137	1	8	yes	—
100	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
102	Q₈×C₄₇	—	2	48	no	—
103	Q₈×C₄₇	—	2	64	no	—
104	Q₈×C₄₇	—	2	8	no	—
105	Q₈×C₄₇	C₂³×C₄	1	16	yes	—
106	C₂×C₄×C₄₇	—	2	48	no	—
107	C₂×C₄×C₄₇	—	2	64	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	48	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
118	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	4	24	no	—
121	C₄₇⋊C₈	—	2	8	no	—
122	C₂²×#419	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	48	no	—
124	C₂²×C₄₇	—	2	8	no	—
125	C₂²×C₄₇	#419	1	16	yes	complemented
126	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
127	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	6	yes	—
128	C₂²×C₄₇	C₂×#137	1	12	yes	—
129	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₄₇	—	2	16	no	—
131	C₄×C₄₇	C₂²×C₂²⋊C₄	1	6	yes	—
132	C₄×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
133	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
135	Dic₄₇	—	4	8	no	—
136	C₂⁴×Q₈	D₄₇	1	1	yes	characteristic
137	C₂³×M₁₆	—	47	1	no	—
138	C₂²×#117	—	47	1	no	—
139	C₂×#419	—	47	24	no	—
140	C₂²×#103	—	47	4	no	—
141	C₂×C₄₇	—	2	8	no	—
142	C₂×C₄₇	C₂×#419	1	12	yes	complemented
143	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄₇	C₂²×#137	1	2	yes	—
145	C₂³×Q₈	—	2	8	no	—
146	C₂³×Q₈	—	2	6	no	—
147	C₂³×Q₈	—	2	1	no	—
148	C₂³×Q₈	Dic₄₇	1	16	yes	complemented
149	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
150	C₂³×Q₈	C₂×D₄₇	1	12	yes	—
151	C₂⁴×C₄	—	2	1	no	—
152	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×M₁₆	—	47	12	no	—
154	C₂²×M₁₆	—	94	8	no	—
155	C₂³×C₈	—	47	2	no	—
156	C₂×#117	—	47	12	no	—
157	C₂×#117	—	94	6	no	—
158	C₂²×C₄⋊C₄	—	47	2	no	—
159	C₂²×C₂²⋊C₄	—	94	1	no	—
160	C₂²×C₄²	—	94	1	no	—
161	#419	—	47	64	no	—
162	C₂×#103	—	47	48	no	—
163	C₄₇	C₂²×#419	1	1	yes	characteristic, complemented, Sylow
164	C₂⁵	C₂²⋊D₄₇	1	1	yes	characteristic
165	C₂²×Q₈	—	2	16	no	—
166	C₂²×Q₈	—	2	96	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	48	no	—
169	C₂²×Q₈	—	2	12	no	—
170	C₂²×Q₈	—	2	6	no	—
171	C₂²×Q₈	—	2	24	no	—
172	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
173	C₂²×Q₈	C₂×Dic₄₇	1	96	yes	complemented
174	C₂²×Q₈	C₂²×D₄₇	1	12	yes	—
175	C₂²×Q₈	C₂²×D₄₇	1	16	yes	—
176	C₂³×C₄	—	2	16	no	—
177	C₂³×C₄	—	2	12	no	—
178	C₂³×C₄	—	2	2	no	—
179	C₂³×C₄	—	94	2	no	—
180	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
181	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
182	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
184	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×M₁₆	—	47	16	no	—
187	C₂×M₁₆	—	94	48	no	—
188	C₂²×C₈	—	47	12	no	—
189	C₂²×C₈	—	94	8	no	—
190	#117	—	47	16	no	—
191	#117	—	94	24	no	—
192	C₂×C₄⋊C₄	—	47	12	no	—
193	C₂×C₄⋊C₄	—	94	6	no	—
194	C₂×C₂²⋊C₄	—	94	12	no	—
195	C₂×C₄²	—	94	12	no	—
196	#103	—	47	64	no	—
197	C₂⁴	—	2	8	no	—
198	C₂⁴	C₂²⋊Dic₄₇	1	8	yes	—
199	C₂⁴	C₂×C₂²⋊D₄₇	1	6	yes	—
200	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂×Q₈	—	2	48	no	—
202	C₂×Q₈	—	2	96	no	—
203	C₂×Q₈	—	2	192	no	—
204	C₂×Q₈	—	2	24	no	—
205	C₂×Q₈	—	2	32	no	—
206	C₂×Q₈	—	2	16	no	—
207	C₂×Q₈	—	2	12	no	—
208	C₂×Q₈	C₂²×Dic₄₇	1	48	yes	—
209	C₂×Q₈	C₂²×Dic₄₇	1	64	yes	complemented
210	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	96	no	—
213	C₂²×C₄	—	2	12	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	94	12	no	—
216	C₂²×C₄	—	188	8	no	—
217	C₂²×C₄	C₂²⋊Dic₄₇	1	48	yes	—
218	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
219	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
226	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
227	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
228	M₁₆	—	94	32	no	—
229	C₂×C₈	—	47	8	no	—
230	C₂×C₈	—	94	24	no	—
231	C₄⋊C₄	—	47	8	no	—
232	C₄⋊C₄	—	94	12	no	—
233	C₂²⋊C₄	—	94	16	no	—
234	C₄²	—	94	16	no	—
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	C₂×C₂²⋊Dic₄₇	1	24	yes	—
238	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
239	C₂³	#13094055	1	8	yes	—
240	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
241	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
242	Q₈	—	2	48	no	—
243	Q₈	—	2	64	no	—
244	Q₈	—	2	8	no	—
245	Q₈	C₂³×Dic₄₇	1	16	yes	—
246	C₂×C₄	—	2	48	no	—
247	C₂×C₄	—	2	64	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	—	188	24	no	—
251	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	48	yes	—
252	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
253	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	24	yes	—
254	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
255	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
256	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
257	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
258	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
260	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
261	C₈	—	94	8	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	48	no	—
264	C₂²	#15689132	1	16	yes	direct product factor
265	C₂²	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
266	C₂²	C₂²×C₂²⋊Dic₄₇	1	6	yes	—
267	C₂²	C₂×#13094055	1	12	yes	—
268	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₄	—	2	16	no	—
270	C₄	—	188	8	no	—
271	C₄	C₂²×C₂²⋊Dic₄₇	1	6	yes	—
272	C₄	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
273	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
274	C₄	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂	—	2	8	no	—
276	C₂	C₂×#15689132	1	12	yes	direct product factor
277	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
278	C₂	C₂²×#13094055	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂⁴×Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂⁴×Q₈	D₄₇
hypocenter	C₄₇	C₂²×#419
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#15689132	C₂	direct (N × H)
4. C₂²×#13093948	C₂	semidirect
14. #15689132	C₂²	direct (N × H)
15. C₂×#13093948	C₂²	semidirect
19. C₂³×Q₈×C₄₇	C₄	semidirect
35. #13093948	C₂³	semidirect
45. C₂²×Q₈×C₄₇	C₂×C₄	semidirect
69. C₂×Q₈×C₄₇	C₂²×C₄	semidirect
125. C₂²×C₄₇	#419	semidirect
142. C₂×C₄₇	C₂×#419	semidirect
148. C₂³×Q₈	Dic₄₇	semidirect
163. C₄₇	C₂²×#419	semidirect
173. C₂²×Q₈	C₂×Dic₄₇	semidirect
209. C₂×Q₈	C₂²×Dic₄₇	semidirect
264. C₂²	#15689132	direct (N × H)
276. C₂	C₂×#15689132	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×#10625329	2
C₂²×#13093983	2
C₂×#15689132	2
C₂²×#13093948	2
C₂⁴×Q₈×C₄₇	2
C₂²×#419	47

Subgroups — C22×#15689132 #27079806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689132 #27079806