Subgroups — C₂²×#15688871 #26184570

Order 12032.

1334 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₉₄	C₄	1	2	yes	—
6	C₂³×C₄₇⋊C₈	—	2	1	no	—
7	C₂×#13094024	—	2	3	no	—
8	C₂×#13094024	—	2	3	no	—
9	C₂×#13094024	C₄	1	6	yes	—
10	C₂×#13094024	C₂²	1	6	yes	—
11	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
13	C₂×C₄⋊(C₄₇⋊C₈)	—	2	3	no	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
15	#15688871	C₂²	1	16	yes	direct product factor
16	C₂³×C₈×C₄₇	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₄₇	—	2	3	no	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	6	yes	—
20	C₂²×Dic₉₄	—	2	6	no	—
21	C₂²×Dic₉₄	—	4	8	no	—
22	C₂²×Dic₉₄	C₂×C₄	1	12	yes	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	6	no	—
25	C₂²×C₄₇⋊C₈	—	4	4	no	—
26	#13094024	—	2	12	no	—
27	#13094024	—	2	6	no	—
28	#13094024	—	2	6	no	—
29	#13094024	C₂×C₄	1	12	yes	—
30	#13094024	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	6	no	—
32	C₂×C₄⋊Dic₄₇	—	2	6	no	—
33	C₂×C₄⋊Dic₄₇	—	2	6	no	—
34	C₂×C₄⋊Dic₄₇	—	2	6	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
36	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	6	no	—
41	C₂²×C₈×C₄₇	—	4	4	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
43	C₂×C₄²×C₄₇	D₄	1	3	yes	—
44	C₂×C₄²×C₄₇	D₄	1	3	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
46	C₄⋊C₈×C₄₇	—	2	6	no	—
47	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
48	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
53	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
56	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
60	C₂²×Dic₄₇	—	2	12	no	—
61	C₂²×Dic₄₇	—	4	8	no	—
62	C₂×Dic₉₄	—	2	12	no	—
63	C₂×Dic₉₄	—	4	24	no	—
64	C₂×Dic₉₄	—	4	24	no	—
65	C₂×Dic₉₄	C₂²×C₄	1	8	yes	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	4	12	no	—
68	C₄²×C₄₇	C₂²⋊C₄	1	6	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
70	C₄²×C₄₇	C₂×D₄	1	3	yes	—
71	C₄²×C₄₇	C₂×D₄	1	3	yes	—
72	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊Dic₄₇	—	2	12	no	—
74	C₄⋊Dic₄₇	—	2	12	no	—
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	4	12	no	—
79	C₂³×C₄₇	#100	1	4	yes	—
80	C₂³×C₄₇	#103	1	4	yes	—
81	C₂³×C₄₇	#103	1	4	yes	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
83	C₂×C₄×C₄₇	—	2	6	no	—
84	C₂×C₄×C₄₇	—	2	6	no	—
85	C₂×C₄×C₄₇	#103	1	12	yes	—
86	C₂×C₄×C₄₇	#103	1	12	yes	—
87	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
90	C₂×C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
91	C₂×C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
92	C₂×C₄×C₄₇	C₂×Q₁₆	1	6	yes	—
93	C₂×C₄×C₄₇	C₂×Q₁₆	1	6	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	4	24	no	—
99	Dic₉₄	—	4	24	no	—
100	Dic₉₄	—	4	8	no	—
101	C₈×C₄₇	—	4	4	no	—
102	C₄₇⋊C₈	—	4	4	no	—
103	C₂²×#333	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	#333	1	16	yes	complemented
105	C₂²×C₄₇	C₂×#100	1	6	yes	—
106	C₂²×C₄₇	C₂×#103	1	6	yes	—
107	C₂²×C₄₇	C₂×#103	1	6	yes	—
108	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₄₇	—	2	3	no	—
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	C₂×#103	1	6	yes	—
112	C₄×C₄₇	C₂×#103	1	6	yes	—
113	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
114	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
115	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
116	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
117	Dic₄₇	—	4	8	no	—
118	C₂²×C₄⋊Q₈	—	47	1	no	—
119	C₂²×C₄⋊C₈	—	47	1	no	—
120	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
121	C₂×#333	—	47	12	no	—
122	C₂×C₄₇	C₂×#333	1	12	yes	complemented
123	C₂×C₄₇	C₂²×#100	1	1	yes	characteristic
124	C₂×C₄₇	C₂²×#103	1	1	yes	characteristic
125	C₂×C₄₇	C₂²×#103	1	1	yes	characteristic
126	C₂³×Q₈	—	47	2	no	—
127	C₂³×C₈	—	2	1	no	—
128	C₂³×C₈	—	94	1	no	—
129	C₂×C₄⋊Q₈	—	47	6	no	—
130	C₂×C₄⋊Q₈	—	47	6	no	—
131	C₂×C₄⋊Q₈	—	94	3	no	—
132	C₂×C₄⋊Q₈	—	94	3	no	—
133	C₂²×C₄⋊C₄	—	94	1	no	—
134	C₂²×C₄⋊C₄	—	94	1	no	—
135	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	2	3	no	—
137	C₂×C₄⋊C₈	—	47	6	no	—
138	C₂×C₄⋊C₈	—	94	3	no	—
139	C₂×C₄⋊C₈	C₂×D₄₇	1	6	yes	—
140	#333	—	47	16	no	—
141	C₄₇	C₂²×#333	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	47	12	no	—
143	C₂²×Q₈	—	94	6	no	—
144	C₂²×Q₈	—	188	8	no	—
145	C₂³×C₄	—	94	2	no	—
146	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
147	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
148	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂²×C₈	—	2	6	no	—
150	C₂²×C₈	—	4	4	no	—
151	C₂²×C₈	—	94	6	no	—
152	C₂²×C₈	—	188	4	no	—
153	C₄⋊Q₈	—	47	12	no	—
154	C₄⋊Q₈	—	47	4	no	—
155	C₄⋊Q₈	—	94	12	no	—
156	C₄⋊Q₈	—	94	6	no	—
157	C₄⋊Q₈	—	94	6	no	—
158	C₂×C₄⋊C₄	—	94	6	no	—
159	C₂×C₄⋊C₄	—	94	6	no	—
160	C₂×C₄⋊C₄	—	94	6	no	—
161	C₂×C₄⋊C₄	—	94	6	no	—
162	C₂×C₄²	C₄×D₄₇	1	3	yes	—
163	C₂×C₄²	D₁₈₈	1	3	yes	—
164	C₂×C₄²	C₂²⋊D₄₇	1	3	yes	—
165	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
166	C₄⋊C₈	—	2	6	no	—
167	C₄⋊C₈	—	47	4	no	—
168	C₄⋊C₈	—	94	6	no	—
169	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
170	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
171	C₂×Q₈	—	47	8	no	—
172	C₂×Q₈	—	94	12	no	—
173	C₂×Q₈	—	188	24	no	—
174	C₂×Q₈	—	188	24	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	94	12	no	—
177	C₂²×C₄	—	188	8	no	—
178	C₂²×C₄	#10625325	1	4	yes	—
179	C₂²×C₄	Dic₁₈₈	1	4	yes	—
180	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	3	yes	—
181	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	3	yes	—
182	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	3	yes	—
183	C₂²×C₄	#10625335	1	4	yes	—
184	C₂²×C₄	#10625337	1	4	yes	—
185	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
186	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
187	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	12	no	—
190	C₂×C₈	—	94	4	no	—
191	C₂×C₈	—	188	12	no	—
192	C₄⋊C₄	—	94	12	no	—
193	C₄⋊C₄	—	94	12	no	—
194	C₄⋊C₄	—	94	4	no	—
195	C₄⋊C₄	—	94	4	no	—
196	C₄²	(C₂×D₄₇)⋊C₄	1	6	yes	—
197	C₄²	C₂×C₄×D₄₇	1	3	yes	—
198	C₄²	C₂×D₁₈₈	1	3	yes	—
199	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
200	C₄²	C₂³×D₄₇	1	1	yes	characteristic
201	C₂³	#13093940	1	4	yes	—
202	C₂³	#13093947	1	4	yes	—
203	C₂³	#13093949	1	4	yes	—
204	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	3	yes	—
205	Q₈	—	188	8	no	—
206	Q₈	—	188	24	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	6	no	—
209	C₂×C₄	—	94	8	no	—
210	C₂×C₄	—	188	24	no	—
211	C₂×C₄	#13093947	1	12	yes	—
212	C₂×C₄	#13093949	1	12	yes	—
213	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	3	yes	—
214	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	3	yes	—
215	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	3	yes	—
216	C₂×C₄	C₂×#10625335	1	6	yes	—
217	C₂×C₄	C₂×#10625325	1	6	yes	—
218	C₂×C₄	C₂×#10625337	1	6	yes	—
219	C₂×C₄	C₂×Dic₁₈₈	1	6	yes	—
220	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
223	C₈	—	4	4	no	—
224	C₈	—	188	4	no	—
225	C₂²	#15688871	1	16	yes	direct product factor
226	C₂²	C₂×#13093940	1	6	yes	—
227	C₂²	C₂×#13093947	1	6	yes	—
228	C₂²	C₂×#13093949	1	6	yes	—
229	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
230	C₄	—	2	3	no	—
231	C₄	—	2	1	no	—
232	C₄	—	188	8	no	—
233	C₄	C₂×#13093947	1	6	yes	—
234	C₄	C₂×#13093949	1	6	yes	—
235	C₄	C₂²×#10625335	1	1	yes	characteristic
236	C₄	C₂²×#10625325	1	1	yes	characteristic
237	C₄	C₂²×#10625337	1	1	yes	characteristic
238	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
239	C₂	C₂×#15688871	1	12	yes	direct product factor
240	C₂	C₂²×#13093940	1	1	yes	characteristic
241	C₂	C₂²×#13093947	1	1	yes	characteristic
242	C₂	C₂²×#13093949	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂²×#333
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#15688871	C₂	direct (N × H)
15. #15688871	C₂²	direct (N × H)
104. C₂²×C₄₇	#333	semidirect
122. C₂×C₄₇	C₂×#333	semidirect
141. C₄₇	C₂²×#333	semidirect
225. C₂²	#15688871	direct (N × H)
239. C₂	C₂×#15688871	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×#13094024	2
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688871	2
C₂²×C₄⋊C₈×C₄₇	2
C₂²×#333	47

Subgroups — C22×#15688871 #26184570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688871 #26184570