Subgroups — C₂²×#15689028 #26374643

Order 12032.

6174 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689028	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄₇⋊#95	C₂	1	4	yes	maximal
5	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₉₄	—	2	8	no	—
7	C₂³×Dic₉₄	—	2	8	no	—
8	C₂³×Dic₉₄	C₄	1	16	yes	complemented
9	C₂³×Dic₉₄	C₂²	1	12	yes	—
10	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
11	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₄₇	—	2	8	no	—
13	C₂²×C₄×Dic₄₇	C₂²	1	12	yes	—
14	#15689028	C₂²	1	64	yes	direct product factor
15	C₂×C₄₇⋊#95	C₂²	1	48	yes	—
16	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₇	—	2	8	no	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	12	yes	—
20	C₂²×Dic₉₄	—	2	128	no	—
21	C₂²×Dic₉₄	—	2	48	no	—
22	C₂²×Dic₉₄	—	2	48	no	—
23	C₂²×Dic₉₄	C₂×C₄	1	96	yes	complemented
24	C₂²×Dic₉₄	C₂³	1	16	yes	—
25	C₂³×Dic₄₇	—	2	8	no	—
26	C₂³×Dic₄₇	—	2	8	no	—
27	C₂³×Dic₄₇	—	2	8	no	—
28	C₂³×Dic₄₇	—	2	8	no	—
29	C₂³×Dic₄₇	C₂×C₄	1	16	yes	—
30	C₂³×Dic₄₇	Q₈	1	16	yes	complemented
31	C₂³×Dic₄₇	C₂³	1	12	yes	—
32	C₂³×Dic₄₇	C₂³	1	12	yes	—
33	C₂×C₄×Dic₄₇	—	2	48	no	—
34	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	64	yes	—
36	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	—	2	8	no	—
38	C₂³×C₄×C₄₇	—	2	16	no	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	D₄	1	8	yes	—
41	C₂³×C₄×C₄₇	D₄	1	8	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
44	C₂×C₄⋊C₄×C₄₇	—	2	48	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
46	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
48	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
49	C₂⁴×C₄₇	C₂×Q₈	1	8	yes	—
50	C₂⁴×C₄₇	#41	1	8	yes	—
51	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
52	C₂²×C₄×C₄₇	—	2	24	no	—
53	C₂²×C₄×C₄₇	—	2	48	no	—
54	C₂²×C₄×C₄₇	—	4	32	no	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	64	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
61	C₂²×Dic₄₇	—	2	64	no	—
62	C₂²×Dic₄₇	—	2	64	no	—
63	C₂²×Dic₄₇	—	2	24	no	—
64	C₂²×Dic₄₇	—	2	24	no	—
65	C₂²×Dic₄₇	—	2	24	no	—
66	C₂²×Dic₄₇	—	2	24	no	—
67	C₂²×Dic₄₇	C₂²×C₄	1	48	yes	—
68	C₂²×Dic₄₇	C₂×Q₈	1	48	yes	complemented
69	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
70	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
71	C₂×Dic₉₄	—	2	384	no	—
72	C₂×Dic₉₄	—	2	32	no	—
73	C₂×Dic₉₄	—	2	32	no	—
74	C₂×Dic₉₄	C₂²×C₄	1	64	yes	complemented
75	C₄⋊C₄×C₄₇	—	2	32	no	—
76	C₄×Dic₄₇	—	2	32	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
78	C₂³×C₄₇	C₄×Q₈	1	32	yes	—
79	C₂³×C₄₇	C₂²⋊Q₈	1	32	yes	—
80	C₂³×C₄₇	#124	1	16	yes	—
81	C₂³×C₄₇	C₄⋊Q₈	1	16	yes	—
82	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
83	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
84	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
85	C₂³×C₄₇	C₂²×Q₈	1	12	yes	—
86	C₂³×C₄₇	C₂×#41	1	12	yes	—
87	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	16	no	—
90	C₂×C₄×C₄₇	—	4	48	no	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	96	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
93	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
95	C₂×Dic₄₇	—	2	96	no	—
96	C₂×Dic₄₇	—	2	96	no	—
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	2	8	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	2	8	no	—
101	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
102	C₂×Dic₄₇	C₂²×Q₈	1	16	yes	complemented
103	Dic₉₄	—	2	128	no	—
104	C₂²×#391	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	#391	1	64	yes	complemented
106	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	12	yes	—
107	C₂²×C₄₇	C₂×C₄×Q₈	1	24	yes	—
108	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	24	yes	—
109	C₂²×C₄₇	C₂×#124	1	12	yes	—
110	C₂²×C₄₇	C₂×C₄⋊Q₈	1	12	yes	—
111	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
112	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
113	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
115	C₂²×C₄₇	C₂²×#41	1	2	yes	—
116	C₄×C₄₇	—	4	8	no	—
117	C₄×C₄₇	C₂²×C₂²⋊C₄	1	16	yes	—
118	Dic₄₇	—	2	16	no	—
119	Dic₄₇	—	2	16	no	—
120	C₂⁴×Q₈	—	47	1	no	—
121	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic
122	C₂³×C₄²	—	47	1	no	—
123	C₂×#391	—	47	24	no	—
124	C₂²×#95	—	47	4	no	—
125	C₂×C₄₇	C₂×#391	1	24	yes	complemented
126	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄₇	C₂²×C₄×Q₈	1	2	yes	—
128	C₂×C₄₇	C₂²×C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
130	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
131	C₂³×Q₈	—	47	16	no	—
132	C₂³×Q₈	—	47	12	no	—
133	C₂³×Q₈	—	94	8	no	—
134	C₂³×Q₈	—	94	8	no	—
135	C₂⁴×C₄	—	47	2	no	—
136	C₂⁴×C₄	—	47	2	no	—
137	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
138	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	2	8	no	—
140	C₂²×C₄⋊C₄	C₂×D₄₇	1	12	yes	—
141	C₂²×C₄²	—	47	12	no	—
142	C₂²×C₄²	—	94	8	no	—
143	#391	—	47	64	no	—
144	C₂×#95	—	47	48	no	—
145	C₄₇	C₂²×#391	1	1	yes	characteristic, complemented, Sylow
146	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
147	C₂²×Q₈	—	47	96	no	—
148	C₂²×Q₈	—	47	16	no	—
149	C₂²×Q₈	—	94	128	no	—
150	C₂²×Q₈	—	94	48	no	—
151	C₂²×Q₈	—	94	48	no	—
152	C₂³×C₄	—	2	16	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	47	16	no	—
155	C₂³×C₄	—	47	16	no	—
156	C₂³×C₄	—	47	12	no	—
157	C₂³×C₄	—	47	12	no	—
158	C₂³×C₄	—	94	8	no	—
159	C₂³×C₄	—	94	8	no	—
160	C₂³×C₄	—	94	8	no	—
161	C₂³×C₄	—	94	8	no	—
162	C₂³×C₄	C₄×D₄₇	1	8	yes	—
163	C₂³×C₄	D₁₈₈	1	8	yes	—
164	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
165	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
167	C₂×C₄⋊C₄	—	2	48	no	—
168	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
169	C₂×C₄²	—	47	16	no	—
170	C₂×C₄²	—	94	48	no	—
171	#95	—	47	64	no	—
172	C₂⁴	C₂×C₄×D₄₇	1	4	yes	—
173	C₂⁴	C₂×D₁₈₈	1	4	yes	—
174	C₂⁴	#10625352	1	8	yes	—
175	C₂⁴	Q₈×D₄₇	1	8	yes	—
176	C₂⁴	C₂×C₂²⋊D₄₇	1	4	yes	—
177	C₂⁴	C₂³×D₄₇	1	3	yes	—
178	C₂×Q₈	—	47	64	no	—
179	C₂×Q₈	—	94	384	no	—
180	C₂×Q₈	—	94	32	no	—
181	C₂×Q₈	—	94	32	no	—
182	C₂²×C₄	—	2	24	no	—
183	C₂²×C₄	—	2	48	no	—
184	C₂²×C₄	—	4	32	no	—
185	C₂²×C₄	—	47	48	no	—
186	C₂²×C₄	—	47	48	no	—
187	C₂²×C₄	—	47	8	no	—
188	C₂²×C₄	—	47	8	no	—
189	C₂²×C₄	—	94	64	no	—
190	C₂²×C₄	—	94	64	no	—
191	C₂²×C₄	—	94	24	no	—
192	C₂²×C₄	—	94	24	no	—
193	C₂²×C₄	—	94	24	no	—
194	C₂²×C₄	—	94	24	no	—
195	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	64	yes	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
197	C₂²×C₄	C₂×D₁₈₈	1	24	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
200	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
201	C₄⋊C₄	—	2	32	no	—
202	C₄²	—	94	32	no	—
203	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	16	yes	—
204	C₂³	#13093989	1	32	yes	—
205	C₂³	#13094004	1	32	yes	—
206	C₂³	#13094011	1	16	yes	—
207	C₂³	#13094022	1	16	yes	—
208	C₂³	C₂²×C₄×D₄₇	1	6	yes	—
209	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
210	C₂³	C₂²×D₁₈₈	1	6	yes	—
211	C₂³	C₂×Q₈×D₄₇	1	12	yes	—
212	C₂³	C₂×#10625352	1	12	yes	—
213	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
214	Q₈	—	94	128	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	4	48	no	—
218	C₂×C₄	—	47	16	no	—
219	C₂×C₄	—	47	16	no	—
220	C₂×C₄	—	94	96	no	—
221	C₂×C₄	—	94	96	no	—
222	C₂×C₄	—	94	8	no	—
223	C₂×C₄	—	94	8	no	—
224	C₂×C₄	—	94	8	no	—
225	C₂×C₄	—	94	8	no	—
226	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	96	yes	—
227	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
228	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
229	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
230	C₂²	#15689028	1	64	yes	direct product factor
231	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	12	yes	—
232	C₂²	C₂×#13093989	1	24	yes	—
233	C₂²	C₂×#13094004	1	24	yes	—
234	C₂²	C₂×#13094011	1	12	yes	—
235	C₂²	C₂×#13094022	1	12	yes	—
236	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
237	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
239	C₂²	C₂²×Q₈×D₄₇	1	2	yes	—
240	C₂²	C₂²×#10625352	1	2	yes	—
241	C₄	—	4	8	no	—
242	C₄	—	94	16	no	—
243	C₄	—	94	16	no	—
244	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	16	yes	—
245	C₂	C₂×#15689028	1	24	yes	direct product factor
246	C₂	C₂³×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
247	C₂	C₂²×#13093989	1	2	yes	—
248	C₂	C₂²×#13094004	1	2	yes	—
249	C₂	C₂²×#13094011	1	1	yes	characteristic
250	C₂	C₂²×#13094022	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle, prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#391
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#15689028	C₂	direct (N × H)
8. C₂³×Dic₉₄	C₄	semidirect
14. #15689028	C₂²	direct (N × H)
23. C₂²×Dic₉₄	C₂×C₄	semidirect
30. C₂³×Dic₄₇	Q₈	semidirect
68. C₂²×Dic₄₇	C₂×Q₈	semidirect
74. C₂×Dic₉₄	C₂²×C₄	semidirect
102. C₂×Dic₄₇	C₂²×Q₈	semidirect
105. C₂²×C₄₇	#391	semidirect
125. C₂×C₄₇	C₂×#391	semidirect
145. C₄₇	C₂²×#391	semidirect
230. C₂²	#15689028	direct (N × H)
245. C₂	C₂×#15689028	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₉₄	2
C₂³×C₄×Dic₄₇	2
C₂×#15689028	2
C₂²×C₄₇⋊#95	2
C₂³×C₄⋊C₄×C₄₇	2
C₂²×#391	47

Subgroups — C22×#15689028 #26374643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689028 #26374643