Subgroups — C₂²×#15688936 #26369229

Order 12032.

1390 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688936	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂×#13093964	—	2	3	no	—
7	C₂×#13093964	C₂²	1	6	yes	complemented
8	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
9	C₂×#13093960	—	2	3	no	—
10	C₂×#13093960	C₂²	1	6	yes	complemented
11	#15688936	C₂²	1	16	yes	direct product factor
12	C₂³×D₄×C₄₇	—	2	1	no	—
13	C₂²×D₈×C₄₇	—	2	3	no	—
14	C₂²×D₈×C₄₇	—	2	1	no	—
15	C₂²×D₈×C₄₇	—	2	3	no	—
16	C₂²×D₈×C₄₇	C₄	1	6	yes	—
17	C₂²×D₈×C₄₇	C₄	1	2	yes	—
18	C₂²×D₈×C₄₇	C₂²	1	6	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	6	no	—
21	C₂×#10625329	—	4	3	no	—
22	C₂²×C₄₇⋊C₈	—	4	1	no	—
23	#13093964	—	2	6	no	—
24	#13093964	C₂³	1	4	yes	complemented
25	C₂×C₄₇⋊C₁₆	—	2	6	no	—
26	#13093960	—	2	6	no	—
27	#13093960	C₂³	1	4	yes	complemented
28	C₂⁵×C₄₇	—	4	1	no	—
29	C₂²×D₄×C₄₇	—	2	6	no	—
30	C₂²×D₄×C₄₇	—	2	2	no	—
31	C₂²×D₄×C₄₇	—	2	6	no	—
32	C₂²×D₄×C₄₇	—	4	1	no	—
33	C₂²×D₄×C₄₇	—	4	3	no	—
34	C₂²×D₄×C₄₇	—	4	3	no	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂×D₈×C₄₇	—	2	6	no	—
37	C₂×D₈×C₄₇	—	2	12	no	—
38	C₂×D₈×C₄₇	—	2	6	no	—
39	C₂×D₈×C₄₇	—	2	6	no	—
40	C₂×D₈×C₄₇	—	2	6	no	—
41	C₂×D₈×C₄₇	—	2	6	no	—
42	C₂×D₈×C₄₇	C₂×C₄	1	12	yes	—
43	C₂×D₈×C₄₇	C₂×C₄	1	12	yes	—
44	C₂×D₈×C₄₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	D₄	1	3	yes	—
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	D₄	1	3	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
51	C₂⁴×C₄₇	—	4	6	no	—
52	C₂⁴×C₄₇	—	4	2	no	—
53	C₂⁴×C₄₇	—	4	6	no	—
54	C₂⁴×C₄₇	—	8	8	no	—
55	C₂⁴×C₄₇	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₄₇	—	2	12	no	—
57	C₂×D₄×C₄₇	—	2	12	no	—
58	C₂×D₄×C₄₇	—	2	4	no	—
59	C₂×D₄×C₄₇	—	4	12	no	—
60	C₂×D₄×C₄₇	—	4	6	no	—
61	C₂×D₄×C₄₇	—	4	6	no	—
62	C₂×D₄×C₄₇	—	4	6	no	—
63	C₂×D₄×C₄₇	—	4	6	no	—
64	C₂×D₄×C₄₇	—	4	6	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	D₈	1	3	yes	—
69	C₂²×C₄×C₄₇	SD₁₆	1	3	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
71	D₈×C₄₇	—	2	12	no	—
72	D₈×C₄₇	—	2	12	no	—
73	D₈×C₄₇	—	2	4	no	—
74	D₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	6	yes	—
77	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
81	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
83	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
84	#10625329	—	2	4	no	—
85	#10625329	—	4	6	no	—
86	C₂×C₄₇⋊C₈	—	4	6	no	—
87	C₄₇⋊C₁₆	—	2	4	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	—	4	4	no	—
90	C₂³×C₄₇	—	4	12	no	—
91	C₂³×C₄₇	—	4	12	no	—
92	C₂³×C₄₇	—	8	24	no	—
93	C₂³×C₄₇	—	8	24	no	—
94	C₂³×C₄₇	—	8	8	no	—
95	C₂³×C₄₇	#102	1	3	yes	—
96	C₂³×C₄₇	#102	1	1	yes	characteristic
97	C₂³×C₄₇	C₂×SD₁₆	1	3	yes	—
98	D₄×C₄₇	—	2	8	no	—
99	D₄×C₄₇	—	4	4	no	—
100	D₄×C₄₇	—	4	12	no	—
101	D₄×C₄₇	—	4	12	no	—
102	C₂×C₄×C₄₇	#102	1	6	yes	—
103	C₂×C₄×C₄₇	#102	1	3	yes	—
104	C₂×C₄×C₄₇	#102	1	3	yes	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
108	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	3	yes	—
110	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
112	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
113	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₄₇⋊C₈	—	4	4	no	—
116	C₂²×#361	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	6	no	—
118	C₂²×C₄₇	—	2	6	no	—
119	C₂²×C₄₇	—	4	8	no	—
120	C₂²×C₄₇	—	8	8	no	—
121	C₂²×C₄₇	—	8	24	no	—
122	C₂²×C₄₇	—	8	24	no	—
123	C₂²×C₄₇	#361	1	4	yes	complemented
124	C₂²×C₄₇	C₂×#102	1	3	yes	—
125	C₂²×C₄₇	C₂×#102	1	3	yes	—
126	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
127	C₄×C₄₇	C₂×#102	1	3	yes	—
128	C₄×C₄₇	C₂×#102	1	3	yes	—
129	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
131	C₂³×D₈	D₄₇	1	1	yes	characteristic
132	C₂²×M₃₂	—	47	1	no	—
133	C₂²×#108	—	47	1	no	—
134	C₂×#361	—	47	12	no	—
135	C₂×C₄₇	—	2	1	no	—
136	C₂×C₄₇	—	2	3	no	—
137	C₂×C₄₇	—	8	8	no	—
138	C₂×C₄₇	C₂×#361	1	6	yes	complemented
139	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
140	C₂³×D₄	—	2	1	no	—
141	C₂²×D₈	—	2	3	no	—
142	C₂²×D₈	—	2	1	no	—
143	C₂²×D₈	—	2	3	no	—
144	C₂²×D₈	Dic₄₇	1	6	yes	—
145	C₂²×D₈	Dic₄₇	1	2	yes	—
146	C₂²×D₈	C₂×D₄₇	1	6	yes	—
147	C₂²×M₁₆	—	94	1	no	—
148	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂×M₃₂	—	47	6	no	—
150	C₂×M₃₂	—	94	3	no	—
151	C₂²×C₁₆	—	94	1	no	—
152	C₂×#108	—	47	6	no	—
153	C₂×#108	—	94	3	no	—
154	#361	—	47	16	no	—
155	C₄₇	C₂²×#361	1	1	yes	characteristic, complemented, Sylow
156	C₂⁵	—	4	1	no	—
157	C₂²×D₄	—	2	6	no	—
158	C₂²×D₄	—	2	2	no	—
159	C₂²×D₄	—	2	6	no	—
160	C₂²×D₄	—	4	3	no	—
161	C₂²×D₄	—	4	1	no	—
162	C₂²×D₄	—	4	3	no	—
163	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
164	C₂×D₈	—	2	6	no	—
165	C₂×D₈	—	2	12	no	—
166	C₂×D₈	—	2	6	no	—
167	C₂×D₈	—	2	6	no	—
168	C₂×D₈	—	2	6	no	—
169	C₂×D₈	—	2	6	no	—
170	C₂×D₈	C₂×Dic₄₇	1	12	yes	—
171	C₂×D₈	C₂×Dic₄₇	1	12	yes	—
172	C₂×D₈	C₂²×D₄₇	1	4	yes	—
173	C₂×M₁₆	—	94	6	no	—
174	C₂×M₁₆	—	188	3	no	—
175	C₂²×C₈	—	188	1	no	—
176	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
177	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
179	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
181	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
182	M₃₂	—	47	4	no	—
183	M₃₂	—	94	6	no	—
184	C₂×C₁₆	—	94	6	no	—
185	#108	—	47	4	no	—
186	#108	—	94	6	no	—
187	C₂⁴	—	4	6	no	—
188	C₂⁴	—	4	2	no	—
189	C₂⁴	—	4	6	no	—
190	C₂⁴	—	8	8	no	—
191	C₂⁴	#10625335	1	1	yes	characteristic
192	C₂×D₄	—	2	12	no	—
193	C₂×D₄	—	2	12	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	12	no	—
196	C₂×D₄	—	4	6	no	—
197	C₂×D₄	—	4	6	no	—
198	C₂×D₄	—	4	6	no	—
199	C₂×D₄	—	4	6	no	—
200	C₂×D₄	—	4	6	no	—
201	C₂²×C₄	#10625334	1	3	yes	—
202	C₂²×C₄	#10625334	1	1	yes	characteristic
203	C₂²×C₄	#10625335	1	3	yes	—
204	C₂²×C₄	C₂²⋊Dic₄₇	1	3	yes	—
205	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
207	D₈	—	2	12	no	—
208	D₈	—	2	12	no	—
209	D₈	—	2	4	no	—
210	D₈	C₂²×Dic₄₇	1	8	yes	—
211	M₁₆	—	94	4	no	—
212	M₁₆	—	188	6	no	—
213	C₂×C₈	—	188	6	no	—
214	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
215	C₂×C₈	C₂²⋊Dic₄₇	1	6	yes	—
216	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
217	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
218	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
219	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
220	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
221	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
222	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
223	C₁₆	—	94	4	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	4	12	no	—
226	C₂³	—	4	12	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	8	24	no	—
229	C₂³	—	8	24	no	—
230	C₂³	—	8	8	no	—
231	C₂³	#13093945	1	3	yes	—
232	C₂³	#13093945	1	1	yes	characteristic
233	C₂³	C₂×#10625335	1	3	yes	—
234	D₄	—	2	8	no	—
235	D₄	—	4	12	no	—
236	D₄	—	4	12	no	—
237	D₄	—	4	4	no	—
238	C₂×C₄	#13093945	1	6	yes	—
239	C₂×C₄	#13093945	1	3	yes	—
240	C₂×C₄	#13093945	1	3	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
242	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
243	C₂×C₄	C₂×#10625334	1	3	yes	—
244	C₂×C₄	C₂×#10625334	1	3	yes	—
245	C₂×C₄	C₂×#10625335	1	3	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₈	—	188	4	no	—
248	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
249	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
250	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
251	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂²	—	2	6	no	—
253	C₂²	—	2	6	no	—
254	C₂²	—	4	8	no	—
255	C₂²	—	8	24	no	—
256	C₂²	—	8	8	no	—
257	C₂²	—	8	24	no	—
258	C₂²	#15688936	1	4	yes	direct product factor
259	C₂²	C₂×#13093945	1	3	yes	—
260	C₂²	C₂×#13093945	1	3	yes	—
261	C₂²	C₂²×#10625335	1	1	yes	characteristic
262	C₄	C₂×#13093945	1	3	yes	—
263	C₄	C₂×#13093945	1	3	yes	—
264	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
265	C₄	C₂²×#10625334	1	1	yes	characteristic
266	C₂	—	2	3	no	—
267	C₂	—	2	1	no	—
268	C₂	—	8	8	no	—
269	C₂	C₂×#15688936	1	6	yes	direct product factor
270	C₂	C₂²×#13093945	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093945
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂³×D₈×C₄₇	C₂
socle	C₂³×C₄₇	#102
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂³×D₈	D₄₇
hypocenter	C₄₇	C₂²×#361
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093964	C₂	semidirect
2. C₂²×#13093960	C₂	semidirect
3. C₂×#15688936	C₂	direct (N × H)
7. C₂×#13093964	C₂²	semidirect
10. C₂×#13093960	C₂²	semidirect
11. #15688936	C₂²	direct (N × H)
24. #13093964	C₂³	semidirect
27. #13093960	C₂³	semidirect
123. C₂²×C₄₇	#361	semidirect
138. C₂×C₄₇	C₂×#361	semidirect
155. C₄₇	C₂²×#361	semidirect
258. C₂²	#15688936	direct (N × H)
269. C₂	C₂×#15688936	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
C₂²×#13093960	2
C₂×#15688936	2
C₂³×D₈×C₄₇	2
C₂²×#361	47

Subgroups — C22×#15688936 #26369229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688936 #26369229