Subgroups — C₂²×#15688939 #26369233

Order 12032.

1134 conjugacy classes of subgroups.

The subgroup lattice has 256 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
2	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688939	C₂	1	12	yes	maximal, direct product factor
4	C₂³×Q₁₆×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	—
8	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
9	C₂×#13093960	—	2	3	no	—
10	C₂×#13093960	C₂²	1	6	yes	—
11	#15688939	C₂²	1	16	yes	direct product factor
12	C₂³×Q₈×C₄₇	—	2	1	no	—
13	C₂²×Q₁₆×C₄₇	—	2	3	no	—
14	C₂²×Q₁₆×C₄₇	—	2	1	no	—
15	C₂²×Q₁₆×C₄₇	—	2	3	no	—
16	C₂²×Q₁₆×C₄₇	C₄	1	6	yes	—
17	C₂²×Q₁₆×C₄₇	C₄	1	2	yes	—
18	C₂²×Q₁₆×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	—
25	C₂×C₄₇⋊C₁₆	—	2	6	no	—
26	#13093960	—	2	6	no	—
27	#13093960	C₂³	1	4	yes	—
28	C₂²×Q₈×C₄₇	—	2	6	no	—
29	C₂²×Q₈×C₄₇	—	2	2	no	—
30	C₂²×Q₈×C₄₇	—	2	6	no	—
31	C₂²×Q₈×C₄₇	—	4	1	no	—
32	C₂²×Q₈×C₄₇	—	4	3	no	—
33	C₂²×Q₈×C₄₇	—	4	3	no	—
34	C₂³×C₄×C₄₇	—	4	1	no	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂×Q₁₆×C₄₇	—	2	6	no	—
37	C₂×Q₁₆×C₄₇	—	2	12	no	—
38	C₂×Q₁₆×C₄₇	—	2	6	no	—
39	C₂×Q₁₆×C₄₇	—	2	6	no	—
40	C₂×Q₁₆×C₄₇	—	2	6	no	—
41	C₂×Q₁₆×C₄₇	—	2	6	no	—
42	C₂×Q₁₆×C₄₇	C₂×C₄	1	12	yes	—
43	C₂×Q₁₆×C₄₇	C₂×C₄	1	12	yes	—
44	C₂×Q₁₆×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	SD₁₆	1	1	yes	characteristic
52	C₂×Q₈×C₄₇	—	2	12	no	—
53	C₂×Q₈×C₄₇	—	2	12	no	—
54	C₂×Q₈×C₄₇	—	2	4	no	—
55	C₂×Q₈×C₄₇	—	4	12	no	—
56	C₂×Q₈×C₄₇	—	4	6	no	—
57	C₂×Q₈×C₄₇	—	4	6	no	—
58	C₂×Q₈×C₄₇	—	4	6	no	—
59	C₂×Q₈×C₄₇	—	4	6	no	—
60	C₂×Q₈×C₄₇	—	4	6	no	—
61	C₂²×C₄×C₄₇	—	4	6	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	6	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	D₈	1	3	yes	—
68	C₂²×C₄×C₄₇	SD₁₆	1	3	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
70	Q₁₆×C₄₇	—	2	12	no	—
71	Q₁₆×C₄₇	—	2	12	no	—
72	Q₁₆×C₄₇	—	2	4	no	—
73	Q₁₆×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	6	yes	—
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	3	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
83	#10625329	—	2	4	no	—
84	#10625329	—	4	6	no	—
85	C₂×C₄₇⋊C₈	—	4	6	no	—
86	C₄₇⋊C₁₆	—	2	4	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	#102	1	3	yes	—
89	C₂³×C₄₇	#102	1	1	yes	characteristic
90	C₂³×C₄₇	C₂×SD₁₆	1	3	yes	—
91	Q₈×C₄₇	—	2	8	no	—
92	Q₈×C₄₇	—	4	4	no	—
93	Q₈×C₄₇	—	4	12	no	—
94	Q₈×C₄₇	—	4	12	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	12	no	—
97	C₂×C₄×C₄₇	—	4	12	no	—
98	C₂×C₄×C₄₇	#102	1	6	yes	—
99	C₂×C₄×C₄₇	#102	1	3	yes	—
100	C₂×C₄×C₄₇	#102	1	3	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
103	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
104	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
105	C₂×C₄×C₄₇	C₂×SD₁₆	1	3	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
108	C₈×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
109	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₄₇⋊C₈	—	4	4	no	—
112	C₂²×#362	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	6	no	—
114	C₂²×C₄₇	—	2	6	no	—
115	C₂²×C₄₇	#362	1	4	yes	complemented
116	C₂²×C₄₇	C₂×#102	1	3	yes	—
117	C₂²×C₄₇	C₂×#102	1	3	yes	—
118	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
119	C₄×C₄₇	—	4	8	no	—
120	C₄×C₄₇	C₂×#102	1	3	yes	—
121	C₄×C₄₇	C₂×#102	1	3	yes	—
122	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
124	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
125	C₂²×M₃₂	—	47	1	no	—
126	C₂²×#108	—	47	1	no	—
127	C₂×#362	—	47	12	no	—
128	C₂×C₄₇	—	2	1	no	—
129	C₂×C₄₇	—	2	3	no	—
130	C₂×C₄₇	C₂×#362	1	6	yes	complemented
131	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
132	C₂³×Q₈	—	2	1	no	—
133	C₂²×Q₁₆	—	2	3	no	—
134	C₂²×Q₁₆	—	2	1	no	—
135	C₂²×Q₁₆	—	2	3	no	—
136	C₂²×Q₁₆	Dic₄₇	1	6	yes	—
137	C₂²×Q₁₆	Dic₄₇	1	2	yes	—
138	C₂²×Q₁₆	C₂×D₄₇	1	6	yes	—
139	C₂²×M₁₆	—	94	1	no	—
140	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
141	C₂×M₃₂	—	47	6	no	—
142	C₂×M₃₂	—	94	3	no	—
143	C₂²×C₁₆	—	94	1	no	—
144	C₂×#108	—	47	6	no	—
145	C₂×#108	—	94	3	no	—
146	#362	—	47	16	no	—
147	C₄₇	C₂²×#362	1	1	yes	characteristic, complemented, Sylow
148	C₂²×Q₈	—	2	6	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×Q₈	—	2	6	no	—
151	C₂²×Q₈	—	4	3	no	—
152	C₂²×Q₈	—	4	1	no	—
153	C₂²×Q₈	—	4	3	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
156	C₂×Q₁₆	—	2	6	no	—
157	C₂×Q₁₆	—	2	12	no	—
158	C₂×Q₁₆	—	2	6	no	—
159	C₂×Q₁₆	—	2	6	no	—
160	C₂×Q₁₆	—	2	6	no	—
161	C₂×Q₁₆	—	2	6	no	—
162	C₂×Q₁₆	C₂×Dic₄₇	1	12	yes	—
163	C₂×Q₁₆	C₂×Dic₄₇	1	12	yes	—
164	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
165	C₂×M₁₆	—	94	6	no	—
166	C₂×M₁₆	—	188	3	no	—
167	C₂²×C₈	—	188	1	no	—
168	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
169	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
171	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₄₇	1	3	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
174	M₃₂	—	47	4	no	—
175	M₃₂	—	94	6	no	—
176	C₂×C₁₆	—	94	6	no	—
177	#108	—	47	4	no	—
178	#108	—	94	6	no	—
179	C₂⁴	#10625335	1	1	yes	characteristic
180	C₂×Q₈	—	2	12	no	—
181	C₂×Q₈	—	2	12	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	12	no	—
184	C₂×Q₈	—	4	6	no	—
185	C₂×Q₈	—	4	6	no	—
186	C₂×Q₈	—	4	6	no	—
187	C₂×Q₈	—	4	6	no	—
188	C₂×Q₈	—	4	6	no	—
189	C₂²×C₄	—	4	6	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	6	no	—
192	C₂²×C₄	#10625334	1	3	yes	—
193	C₂²×C₄	#10625334	1	1	yes	characteristic
194	C₂²×C₄	#10625335	1	3	yes	—
195	C₂²×C₄	C₂²⋊Dic₄₇	1	3	yes	—
196	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
198	Q₁₆	—	2	12	no	—
199	Q₁₆	—	2	12	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	C₂²×Dic₄₇	1	8	yes	—
202	M₁₆	—	94	4	no	—
203	M₁₆	—	188	6	no	—
204	C₂×C₈	—	188	6	no	—
205	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
206	C₂×C₈	C₂²⋊Dic₄₇	1	6	yes	—
207	C₂×C₈	C₂²⋊Dic₄₇	1	3	yes	—
208	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
209	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
210	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
211	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
212	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
213	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
214	C₁₆	—	94	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	#13093945	1	3	yes	—
217	C₂³	#13093945	1	1	yes	characteristic
218	C₂³	C₂×#10625335	1	3	yes	—
219	Q₈	—	2	8	no	—
220	Q₈	—	4	12	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	4	no	—
223	C₂×C₄	—	4	12	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	#13093945	1	6	yes	—
227	C₂×C₄	#13093945	1	3	yes	—
228	C₂×C₄	#13093945	1	3	yes	—
229	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
230	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
231	C₂×C₄	C₂×#10625334	1	3	yes	—
232	C₂×C₄	C₂×#10625334	1	3	yes	—
233	C₂×C₄	C₂×#10625335	1	3	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₈	—	188	4	no	—
236	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
237	C₈	C₂×C₂²⋊Dic₄₇	1	3	yes	—
238	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
239	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂²	—	2	6	no	—
241	C₂²	—	2	6	no	—
242	C₂²	#15688939	1	4	yes	direct product factor
243	C₂²	C₂×#13093945	1	3	yes	—
244	C₂²	C₂×#13093945	1	3	yes	—
245	C₂²	C₂²×#10625335	1	1	yes	characteristic
246	C₄	—	4	8	no	—
247	C₄	C₂×#13093945	1	3	yes	—
248	C₄	C₂×#13093945	1	3	yes	—
249	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₄	C₂²×#10625334	1	1	yes	characteristic
251	C₂	—	2	3	no	—
252	C₂	—	2	1	no	—
253	C₂	C₂×#15688939	1	6	yes	direct product factor
254	C₂	C₂²×#13093945	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093945
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	#102
prime radical	C₂⁴×C₄₇	SD₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂²×#362
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#15688939	C₂	direct (N × H)
11. #15688939	C₂²	direct (N × H)
115. C₂²×C₄₇	#362	semidirect
130. C₂×C₄₇	C₂×#362	semidirect
147. C₄₇	C₂²×#362	semidirect
242. C₂²	#15688939	direct (N × H)
253. C₂	C₂×#15688939	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×#15688939	2
C₂³×Q₁₆×C₄₇	2
C₂²×#362	47

Subgroups — C22×#15688939 #26369233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688939 #26369233