Subgroups — C₂²×#15688930 #26369201

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688930	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	3	no	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	6	yes	complemented
9	C₂×#13093960	—	2	3	no	—
10	C₂×#13093960	C₂²	1	6	yes	complemented
11	#15688930	C₂²	1	16	yes	direct product factor
12	C₂²×#41×C₄₇	—	2	1	no	—
13	C₂×#135×C₄₇	—	2	3	no	—
14	C₂×#135×C₄₇	—	2	3	no	—
15	C₂×#135×C₄₇	C₄	1	6	yes	—
16	C₂×#135×C₄₇	C₂²	1	6	yes	—
17	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
18	C₂²×SD₁₆×C₄₇	—	2	1	no	—
19	C₂²×D₈×C₄₇	C₄	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#10625329	—	2	6	no	—
22	C₂×#10625329	—	4	3	no	—
23	C₂²×C₄₇⋊C₈	—	4	1	no	—
24	C₂×C₄₇⋊C₁₆	—	2	6	no	—
25	C₂×C₄₇⋊C₁₆	—	2	6	no	—
26	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	complemented
27	#13093960	—	2	6	no	—
28	#13093960	C₂³	1	4	yes	complemented
29	C₂×#41×C₄₇	—	2	6	no	—
30	C₂×#41×C₄₇	—	2	6	no	—
31	C₂×#41×C₄₇	—	4	3	no	—
32	C₂×#41×C₄₇	—	4	3	no	—
33	C₂²×Q₈×C₄₇	—	2	1	no	—
34	C₂²×D₄×C₄₇	—	2	1	no	—
35	C₂²×D₄×C₄₇	—	4	1	no	—
36	C₂³×C₄×C₄₇	—	4	1	no	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	#135×C₄₇	—	2	12	no	—
39	#135×C₄₇	—	2	6	no	—
40	#135×C₄₇	—	2	6	no	—
41	#135×C₄₇	C₂×C₄	1	12	yes	—
42	#135×C₄₇	C₂³	1	4	yes	—
43	C₂×Q₁₆×C₄₇	—	2	3	no	—
44	C₂×Q₁₆×C₄₇	C₂×C₄	1	6	yes	—
45	C₂×SD₁₆×C₄₇	—	2	6	no	—
46	C₂×SD₁₆×C₄₇	—	2	6	no	—
47	C₂×D₈×C₄₇	—	2	3	no	—
48	C₂×D₈×C₄₇	C₂×C₄	1	6	yes	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
51	C₂²×C₈×C₄₇	D₄	1	3	yes	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	3	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
55	C₂⁴×C₄₇	—	4	1	no	—
56	C₂⁴×C₄₇	SD₁₆	1	1	yes	characteristic
57	#41×C₄₇	—	2	12	no	—
58	#41×C₄₇	—	2	4	no	—
59	#41×C₄₇	—	4	12	no	—
60	#41×C₄₇	—	4	6	no	—
61	#41×C₄₇	—	4	6	no	—
62	C₂×Q₈×C₄₇	—	2	6	no	—
63	C₂×Q₈×C₄₇	—	4	3	no	—
64	C₂×D₄×C₄₇	—	2	6	no	—
65	C₂×D₄×C₄₇	—	4	6	no	—
66	C₂×D₄×C₄₇	—	4	6	no	—
67	C₂×D₄×C₄₇	—	4	3	no	—
68	C₂²×C₄×C₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	—	4	6	no	—
70	C₂²×C₄×C₄₇	—	4	6	no	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
73	C₂²×C₄×C₄₇	D₈	1	3	yes	—
74	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	SD₁₆	1	3	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
77	Q₁₆×C₄₇	—	2	6	no	—
78	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
79	SD₁₆×C₄₇	—	2	12	no	—
80	SD₁₆×C₄₇	—	2	4	no	—
81	D₈×C₄₇	—	2	6	no	—
82	D₈×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	6	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	#10625329	—	2	4	no	—
93	#10625329	—	4	6	no	—
94	C₂×C₄₇⋊C₈	—	4	6	no	—
95	C₄₇⋊C₁₆	—	2	4	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	4	6	no	—
98	C₂³×C₄₇	—	8	4	no	—
99	C₂³×C₄₇	#102	1	3	yes	—
100	C₂³×C₄₇	#102	1	1	yes	characteristic
101	C₂³×C₄₇	C₂×SD₁₆	1	3	yes	—
102	Q₈×C₄₇	—	2	4	no	—
103	Q₈×C₄₇	—	4	6	no	—
104	D₄×C₄₇	—	2	4	no	—
105	D₄×C₄₇	—	4	4	no	—
106	D₄×C₄₇	—	4	12	no	—
107	D₄×C₄₇	—	4	6	no	—
108	C₂×C₄×C₄₇	—	4	6	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	12	no	—
111	C₂×C₄×C₄₇	#102	1	3	yes	—
112	C₂×C₄×C₄₇	#102	1	3	yes	—
113	C₂×C₄×C₄₇	#102	1	6	yes	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	3	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
121	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
122	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₄₇⋊C₈	—	4	4	no	—
125	C₂²×#359	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	6	no	—
127	C₂²×C₄₇	—	2	6	no	—
128	C₂²×C₄₇	—	4	4	no	—
129	C₂²×C₄₇	—	8	12	no	—
130	C₂²×C₄₇	#359	1	4	yes	complemented
131	C₂²×C₄₇	C₂×#102	1	3	yes	—
132	C₂²×C₄₇	C₂×#102	1	3	yes	—
133	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	C₂×#102	1	3	yes	—
136	C₄×C₄₇	C₂×#102	1	3	yes	—
137	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
139	C₂²×#135	D₄₇	1	1	yes	characteristic
140	C₂³×C₁₆	—	47	1	no	—
141	C₂²×#108	—	47	1	no	—
142	C₂×#359	—	47	12	no	—
143	C₂×C₄₇	—	2	1	no	—
144	C₂×C₄₇	—	2	3	no	—
145	C₂×C₄₇	—	8	4	no	—
146	C₂×C₄₇	C₂×#359	1	6	yes	complemented
147	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
148	C₂²×#41	—	2	1	no	—
149	C₂×#135	—	2	3	no	—
150	C₂×#135	—	2	3	no	—
151	C₂×#135	Dic₄₇	1	6	yes	—
152	C₂×#135	C₂×D₄₇	1	6	yes	—
153	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
154	C₂²×SD₁₆	—	2	1	no	—
155	C₂²×D₈	Dic₄₇	1	1	yes	characteristic
156	C₂²×M₁₆	—	94	1	no	—
157	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₂²×C₁₆	—	47	6	no	—
159	C₂²×C₁₆	—	94	1	no	—
160	C₂²×C₁₆	—	94	3	no	—
161	C₂×#108	—	47	6	no	—
162	C₂×#108	—	94	3	no	—
163	#359	—	47	16	no	—
164	C₄₇	C₂²×#359	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	—	2	6	no	—
166	C₂×#41	—	2	6	no	—
167	C₂×#41	—	4	3	no	—
168	C₂×#41	—	4	3	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂³×C₄	—	4	1	no	—
173	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
174	#135	—	2	12	no	—
175	#135	—	2	6	no	—
176	#135	—	2	6	no	—
177	#135	C₂×Dic₄₇	1	12	yes	—
178	#135	C₂²×D₄₇	1	4	yes	—
179	C₂×Q₁₆	—	2	3	no	—
180	C₂×Q₁₆	C₂×Dic₄₇	1	6	yes	—
181	C₂×SD₁₆	—	2	6	no	—
182	C₂×SD₁₆	—	2	6	no	—
183	C₂×D₈	—	2	3	no	—
184	C₂×D₈	C₂×Dic₄₇	1	6	yes	—
185	C₂×M₁₆	—	94	6	no	—
186	C₂×M₁₆	—	188	3	no	—
187	C₂²×C₈	—	188	1	no	—
188	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
190	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
191	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
193	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
194	C₂×C₁₆	—	47	4	no	—
195	C₂×C₁₆	—	94	6	no	—
196	C₂×C₁₆	—	94	6	no	—
197	#108	—	47	4	no	—
198	#108	—	94	6	no	—
199	C₂⁴	—	4	1	no	—
200	C₂⁴	#10625335	1	1	yes	characteristic
201	#41	—	2	12	no	—
202	#41	—	2	4	no	—
203	#41	—	4	12	no	—
204	#41	—	4	6	no	—
205	#41	—	4	6	no	—
206	C₂×Q₈	—	2	6	no	—
207	C₂×Q₈	—	4	3	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	4	6	no	—
210	C₂×D₄	—	4	6	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	6	no	—
214	C₂²×C₄	—	4	6	no	—
215	C₂²×C₄	#10625334	1	3	yes	—
216	C₂²×C₄	#10625334	1	1	yes	characteristic
217	C₂²×C₄	#10625335	1	3	yes	—
218	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₄₇	1	3	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
221	Q₁₆	—	2	6	no	—
222	Q₁₆	C₂²×Dic₄₇	1	4	yes	—
223	SD₁₆	—	2	12	no	—
224	SD₁₆	—	2	4	no	—
225	D₈	—	2	6	no	—
226	D₈	C₂²×Dic₄₇	1	4	yes	—
227	M₁₆	—	94	4	no	—
228	M₁₆	—	188	6	no	—
229	C₂×C₈	—	188	6	no	—
230	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
231	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
232	C₂×C₈	C₂²⋊Dic₄₇	1	6	yes	—
233	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
234	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
235	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
236	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
237	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
238	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
239	C₁₆	—	94	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#13093945	1	3	yes	—
244	C₂³	#13093945	1	1	yes	characteristic
245	C₂³	C₂×#10625335	1	3	yes	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	6	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	12	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	6	no	—
252	C₂×C₄	—	4	6	no	—
253	C₂×C₄	—	4	12	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	#13093945	1	3	yes	—
256	C₂×C₄	#13093945	1	3	yes	—
257	C₂×C₄	#13093945	1	6	yes	—
258	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
259	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
260	C₂×C₄	C₂×#10625334	1	3	yes	—
261	C₂×C₄	C₂×#10625334	1	3	yes	—
262	C₂×C₄	C₂×#10625335	1	3	yes	—
263	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
264	C₈	—	188	4	no	—
265	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
266	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
267	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
268	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂²	—	2	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	12	no	—
273	C₂²	#15688930	1	4	yes	direct product factor
274	C₂²	C₂×#13093945	1	3	yes	—
275	C₂²	C₂×#13093945	1	3	yes	—
276	C₂²	C₂²×#10625335	1	1	yes	characteristic
277	C₄	—	4	4	no	—
278	C₄	C₂×#13093945	1	3	yes	—
279	C₄	C₂×#13093945	1	3	yes	—
280	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
281	C₄	C₂²×#10625334	1	1	yes	characteristic
282	C₂	—	2	3	no	—
283	C₂	—	2	1	no	—
284	C₂	—	8	4	no	—
285	C₂	C₂×#15688930	1	6	yes	direct product factor
286	C₂	C₂²×#13093945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂²×#135×C₄₇	C₂
socle	C₂³×C₄₇	#102
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×#135	D₄₇
hypocenter	C₄₇	C₂²×#359
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄₇⋊C₁₆	C₂	semidirect
2. C₂²×#13093960	C₂	semidirect
3. C₂×#15688930	C₂	direct (N × H)
8. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
10. C₂×#13093960	C₂²	semidirect
11. #15688930	C₂²	direct (N × H)
26. C₂×C₄₇⋊C₁₆	C₂³	semidirect
28. #13093960	C₂³	semidirect
130. C₂²×C₄₇	#359	semidirect
146. C₂×C₄₇	C₂×#359	semidirect
164. C₄₇	C₂²×#359	semidirect
273. C₂²	#15688930	direct (N × H)
285. C₂	C₂×#15688930	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×C₄₇⋊C₁₆	2
C₂²×#13093960	2
C₂×#15688930	2
C₂²×#135×C₄₇	2
C₂²×#359	47

Subgroups — C22×#15688930 #26369201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688930 #26369201