Subgroups — C₂²×#15689405 #27083752

Order 12032.

2358 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689405	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093955	C₂	1	2	yes	maximal
4	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13093947	C₂	1	2	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×Dic₉₄	—	2	6	no	—
10	C₂×C₄×Dic₉₄	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
13	C₂²×C₄×Dic₄₇	—	2	1	no	—
14	#15689405	C₂²	1	64	yes	direct product factor
15	C₂×#13093955	C₂²	1	24	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	12	yes	—
17	C₂×#13093947	C₂²	1	24	yes	—
18	C₂³×Q₈×C₄₇	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₄₇	—	2	6	no	—
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	12	yes	—
21	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₉₄	—	2	3	no	—
25	C₂²×Dic₉₄	—	2	4	no	—
26	C₂²×Dic₉₄	Q₈	1	8	yes	complemented
27	C₂²×Dic₉₄	C₂³	1	6	yes	—
28	C₂³×Dic₄₇	—	2	1	no	—
29	C₂³×Dic₄₇	—	2	1	no	—
30	C₂²×C₄₇⋊C₈	—	2	8	no	—
31	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
32	C₄×Dic₉₄	—	2	24	no	—
33	C₄×Dic₉₄	C₂³	1	16	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	3	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
36	C₂×Dic₄₇⋊C₄	—	2	12	no	—
37	C₂×C₄×Dic₄₇	—	2	12	no	—
38	#13093955	C₂³	1	32	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
40	#13093947	C₂³	1	32	yes	—
41	C₂²×Q₈×C₄₇	—	2	12	no	—
42	C₂²×Q₈×C₄₇	—	4	8	no	—
43	C₂³×C₄×C₄₇	—	2	2	no	—
44	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₇	—	2	24	no	—
48	C₄⋊Q₈×C₄₇	C₂³	1	16	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
52	C₂×C₄²×C₄₇	D₄	1	6	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
54	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₇	—	2	16	no	—
56	C₂×Q₈×C₄₇	—	4	48	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	12	no	—
59	C₂²×C₄×C₄₇	—	4	8	no	—
60	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₄₇	#41	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
67	C₂²×Dic₄₇	—	2	8	no	—
68	C₂²×Dic₄₇	—	2	6	no	—
69	C₂²×Dic₄₇	—	2	6	no	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂×Dic₉₄	—	2	24	no	—
72	C₂×Dic₉₄	—	2	6	no	—
73	C₂×Dic₉₄	—	2	12	no	—
74	C₂×Dic₉₄	C₂×Q₈	1	24	yes	complemented
75	C₂×Dic₉₄	C₂⁴	1	4	yes	—
76	C₄⋊C₄×C₄₇	—	2	16	no	—
77	C₄⋊C₄×C₄₇	—	2	12	no	—
78	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
79	C₄²×C₄₇	C₂×D₄	1	12	yes	—
80	C₄²×C₄₇	C₂⁴	1	4	yes	—
81	C₄⋊Dic₄₇	—	2	6	no	—
82	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
83	Dic₄₇⋊C₄	—	2	16	no	—
84	C₄×Dic₄₇	—	2	16	no	—
85	C₂×C₄₇⋊C₈	—	2	24	no	—
86	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
87	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
88	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
89	C₂³×C₄₇	#137	1	4	yes	—
90	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
91	Q₈×C₄₇	—	4	32	no	—
92	C₂×C₄×C₄₇	—	2	12	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	4	24	no	—
95	C₂×C₄×C₄₇	C₂²⋊Q₈	1	12	yes	—
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	24	yes	—
97	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
99	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂²×Q₈	1	6	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
102	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
103	C₂×Dic₄₇	—	2	24	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	4	12	no	—
107	Dic₉₄	—	2	24	no	—
108	Dic₉₄	—	2	4	no	—
109	Dic₉₄	C₂²×Q₈	1	8	yes	complemented
110	C₄₇⋊C₈	—	2	8	no	—
111	C₂²×#475	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	#475	1	16	yes	complemented
113	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	6	yes	—
114	C₂²×C₄₇	C₂²×SD₁₆	1	6	yes	—
115	C₂²×C₄₇	C₂×#137	1	6	yes	—
116	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	4	8	no	—
119	C₄×C₄₇	C₂×C₂²⋊Q₈	1	6	yes	—
120	C₄×C₄₇	C₂²×SD₁₆	1	8	yes	—
121	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
122	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
123	Dic₄₇	—	2	8	no	—
124	Dic₄₇	—	4	4	no	—
125	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
126	C₂²×C₄×Q₈	—	47	1	no	—
127	C₂×#475	—	47	24	no	—
128	C₂²×#106	—	47	2	no	—
129	C₂²×C₄⋊C₈	—	47	1	no	—
130	C₂²×#103	—	47	2	no	—
131	C₂×C₄₇	C₂×#475	1	12	yes	complemented
132	C₂×C₄₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄₇	C₂³×SD₁₆	1	1	yes	characteristic
134	C₂×C₄₇	C₂²×#137	1	1	yes	characteristic
135	C₂³×Q₈	—	2	1	no	—
136	C₂³×Q₈	—	47	1	no	—
137	C₂³×C₈	—	47	2	no	—
138	C₂×C₄⋊Q₈	—	2	6	no	—
139	C₂×C₄⋊Q₈	C₂×D₄₇	1	12	yes	—
140	C₂×C₄×Q₈	—	47	12	no	—
141	C₂×C₄×Q₈	—	94	6	no	—
142	C₂²×C₄⋊C₄	—	2	1	no	—
143	C₂²×C₄⋊C₄	—	47	1	no	—
144	C₂²×C₄⋊C₄	—	94	1	no	—
145	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
146	C₂²×C₄²	—	94	1	no	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	#475	—	47	64	no	—
149	C₂×#106	—	47	24	no	—
150	C₂×C₄⋊C₈	—	47	12	no	—
151	C₂×#103	—	47	24	no	—
152	C₄₇	C₂²×#475	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	2	12	no	—
154	C₂²×Q₈	—	4	8	no	—
155	C₂²×Q₈	—	47	8	no	—
156	C₂²×Q₈	—	47	6	no	—
157	C₂²×Q₈	—	94	4	no	—
158	C₂²×Q₈	—	94	3	no	—
159	C₂³×C₄	—	2	2	no	—
160	C₂³×C₄	—	94	1	no	—
161	C₂³×C₄	—	94	1	no	—
162	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
163	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂²×C₈	—	47	12	no	—
166	C₂²×C₈	—	94	8	no	—
167	C₄⋊Q₈	—	2	24	no	—
168	C₄⋊Q₈	C₂²×D₄₇	1	16	yes	—
169	C₄×Q₈	—	47	16	no	—
170	C₄×Q₈	—	94	24	no	—
171	C₂×C₄⋊C₄	—	2	12	no	—
172	C₂×C₄⋊C₄	—	2	6	no	—
173	C₂×C₄⋊C₄	—	47	6	no	—
174	C₂×C₄⋊C₄	—	94	3	no	—
175	C₂×C₄⋊C₄	—	94	12	no	—
176	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
177	C₂×C₄²	—	94	12	no	—
178	C₂×C₄²	C₂²⋊D₄₇	1	6	yes	—
179	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
180	#106	—	47	32	no	—
181	C₄⋊C₈	—	47	16	no	—
182	#103	—	47	32	no	—
183	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	16	no	—
185	C₂×Q₈	—	4	48	no	—
186	C₂×Q₈	—	47	24	no	—
187	C₂×Q₈	—	47	4	no	—
188	C₂×Q₈	—	94	24	no	—
189	C₂×Q₈	—	94	6	no	—
190	C₂×Q₈	—	94	12	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	12	no	—
193	C₂²×C₄	—	4	8	no	—
194	C₂²×C₄	—	94	8	no	—
195	C₂²×C₄	—	94	6	no	—
196	C₂²×C₄	—	94	6	no	—
197	C₂²×C₄	—	188	4	no	—
198	C₂²×C₄	#10625335	1	8	yes	—
199	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
200	C₂²×C₄	#10625354	1	4	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
204	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
205	C₂×C₈	—	47	8	no	—
206	C₂×C₈	—	94	24	no	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	12	no	—
209	C₄⋊C₄	—	47	4	no	—
210	C₄⋊C₄	—	94	16	no	—
211	C₄⋊C₄	—	94	6	no	—
212	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
213	C₄²	—	94	16	no	—
214	C₄²	C₂×C₂²⋊D₄₇	1	12	yes	—
215	C₄²	C₂³×D₄₇	1	4	yes	—
216	C₂³	#13094000	1	4	yes	—
217	C₂³	C₂×#10625335	1	4	yes	—
218	C₂³	#13094055	1	4	yes	—
219	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
220	Q₈	—	4	32	no	—
221	Q₈	—	47	8	no	—
222	Q₈	—	94	24	no	—
223	Q₈	—	94	4	no	—
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	24	no	—
227	C₂×C₄	—	94	24	no	—
228	C₂×C₄	—	94	4	no	—
229	C₂×C₄	—	94	4	no	—
230	C₂×C₄	—	188	12	no	—
231	C₂×C₄	#13094000	1	12	yes	—
232	C₂×C₄	C₂×#10625335	1	24	yes	—
233	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
236	C₂×C₄	C₂×Q₈×D₄₇	1	6	yes	—
237	C₂×C₄	C₂×#10625354	1	6	yes	—
238	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
239	C₈	—	94	8	no	—
240	C₂²	#15689405	1	16	yes	direct product factor
241	C₂²	C₂×#13094000	1	6	yes	—
242	C₂²	C₂²×#10625335	1	6	yes	—
243	C₂²	C₂×#13094055	1	6	yes	—
244	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	4	8	no	—
247	C₄	—	94	8	no	—
248	C₄	—	188	4	no	—
249	C₄	C₂×#13094000	1	6	yes	—
250	C₄	C₂²×#10625335	1	8	yes	—
251	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
252	C₄	C₂²×#10625354	1	1	yes	characteristic
253	C₂	C₂×#15689405	1	12	yes	direct product factor
254	C₂	C₂²×#13094000	1	1	yes	characteristic
255	C₂	C₂³×#10625335	1	1	yes	characteristic
256	C₂	C₂²×#13094055	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#475
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15689405	C₂	direct (N × H)
14. #15689405	C₂²	direct (N × H)
26. C₂²×Dic₉₄	Q₈	semidirect
74. C₂×Dic₉₄	C₂×Q₈	semidirect
109. Dic₉₄	C₂²×Q₈	semidirect
112. C₂²×C₄₇	#475	semidirect
131. C₂×C₄₇	C₂×#475	semidirect
152. C₄₇	C₂²×#475	semidirect
240. C₂²	#15689405	direct (N × H)
253. C₂	C₂×#15689405	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×C₄×Dic₉₄	2
C₂×#15689405	2
C₂²×#13093955	2
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₂²×#13093947	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×#475	47

Subgroups — C22×#15689405 #27083752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689405 #27083752