Subgroups — C₂²×#15689532 #27085888

Order 12032.

2142 conjugacy classes of subgroups.

The subgroup lattice has 236 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×#13094016	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689532	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093958	C₂	1	2	yes	maximal
5	C₂²×#13093956	C₂	1	2	yes	maximal
6	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	—	2	1	no	—
8	C₂×#13094024	—	2	6	no	—
9	C₂×#13094024	C₂²	1	12	yes	—
10	C₂×#13094016	—	2	6	no	—
11	C₂×#13094016	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
17	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
18	#15689532	C₂²	1	64	yes	direct product factor
19	C₂×#13093958	C₂²	1	24	yes	—
20	C₂×#13093956	C₂²	1	24	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₇	C₂²	1	12	yes	—
24	C₂²×Dic₉₄	—	2	12	no	—
25	C₂²×Dic₉₄	—	4	8	no	—
26	C₂³×Dic₄₇	—	2	1	no	—
27	C₂³×Dic₄₇	—	2	1	no	—
28	C₂³×Dic₄₇	—	2	2	no	—
29	#13094024	—	2	24	no	—
30	#13094024	C₂³	1	16	yes	—
31	#13094016	—	2	24	no	—
32	#13094016	C₂³	1	16	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	12	no	—
34	C₂×C₄⋊Dic₄₇	—	2	3	no	—
35	C₂×C₄⋊Dic₄₇	—	2	3	no	—
36	C₂×C₄⋊Dic₄₇	—	2	6	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
40	C₂×Dic₄₇⋊C₄	—	2	12	no	—
41	C₂×Dic₄₇⋊C₄	—	2	12	no	—
42	#13093958	C₂³	1	32	yes	—
43	#13093956	C₂³	1	32	yes	—
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	C₂²×C₈×C₄₇	Q₈	1	16	yes	complemented
48	C₂²×C₈×C₄₇	C₂³	1	12	yes	—
49	C₂×C₄²×C₄₇	D₄	1	6	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
51	#97×C₄₇	C₂³	1	16	yes	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
60	C₂²×Dic₄₇	—	2	6	no	—
61	C₂²×Dic₄₇	—	2	6	no	—
62	C₂²×Dic₄₇	—	2	12	no	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	C₂²×Dic₄₇	—	4	8	no	—
65	C₂²×Dic₄₇	—	4	4	no	—
66	C₂×Dic₉₄	—	2	16	no	—
67	C₂×Dic₉₄	—	4	48	no	—
68	C₂×C₈×C₄₇	C₂×Q₈	1	48	yes	complemented
69	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
70	C₄²×C₄₇	C₂×D₄	1	12	yes	—
71	C₄²×C₄₇	C₂⁴	1	4	yes	—
72	C₄⋊Dic₄₇	—	2	16	no	—
73	C₄⋊Dic₄₇	—	2	6	no	—
74	C₄⋊Dic₄₇	—	2	6	no	—
75	C₄⋊Dic₄₇	—	2	12	no	—
76	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
77	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
78	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
79	Dic₄₇⋊C₄	—	2	16	no	—
80	Dic₄₇⋊C₄	—	2	16	no	—
81	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
82	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
83	C₂³×C₄₇	#137	1	4	yes	—
84	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
85	C₂×C₄×C₄₇	—	2	12	no	—
86	C₂×C₄×C₄₇	—	2	12	no	—
87	C₂×C₄×C₄₇	C₄⋊Q₈	1	12	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
90	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
91	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
92	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₄₇	—	2	4	no	—
94	C₂×Dic₄₇	—	2	4	no	—
95	C₂×Dic₄₇	—	2	8	no	—
96	C₂×Dic₄₇	—	4	24	no	—
97	C₂×Dic₄₇	—	4	12	no	—
98	C₂×Dic₄₇	—	4	12	no	—
99	Dic₉₄	—	4	32	no	—
100	C₈×C₄₇	C₂²×Q₈	1	16	yes	complemented
101	C₂²×#501	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	#501	1	16	yes	complemented
103	C₂²×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
104	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
105	C₂²×C₄₇	C₂×#137	1	6	yes	—
106	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
107	C₄×C₄₇	—	2	4	no	—
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
110	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
111	Dic₄₇	—	4	8	no	—
112	Dic₄₇	—	4	4	no	—
113	Dic₄₇	—	4	4	no	—
114	C₂²×C₄⋊Q₈	—	47	1	no	—
115	C₂²×#125	—	47	1	no	—
116	C₂×#501	—	47	24	no	—
117	C₂²×#107	—	47	2	no	—
118	C₂²×#106	—	47	2	no	—
119	C₂²×#97	D₄₇	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#501	1	12	yes	complemented
121	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄₇	C₂²×#137	1	1	yes	characteristic
124	C₂³×Q₈	—	94	1	no	—
125	C₂³×C₈	C₂×D₄₇	1	2	yes	—
126	C₂×C₄⋊Q₈	—	47	12	no	—
127	C₂×C₄⋊Q₈	—	94	6	no	—
128	C₂×#125	—	47	12	no	—
129	C₂×#125	—	94	6	no	—
130	C₂²×C₄⋊C₄	—	47	1	no	—
131	C₂²×C₄⋊C₄	—	47	2	no	—
132	C₂²×C₄⋊C₄	—	47	1	no	—
133	C₂²×C₄⋊C₄	—	94	1	no	—
134	C₂²×C₄⋊C₄	—	94	1	no	—
135	C₂²×C₄⋊C₄	—	94	1	no	—
136	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
137	#501	—	47	64	no	—
138	C₂×#107	—	47	24	no	—
139	C₂×#106	—	47	24	no	—
140	C₂×#97	C₂×D₄₇	1	12	yes	—
141	C₄₇	C₂²×#501	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	94	12	no	—
143	C₂²×Q₈	—	188	8	no	—
144	C₂³×C₄	—	94	1	no	—
145	C₂³×C₄	—	94	1	no	—
146	C₂³×C₄	—	94	2	no	—
147	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
148	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
149	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂²×C₈	Dic₉₄	1	16	yes	complemented
151	C₂²×C₈	C₂²×D₄₇	1	12	yes	—
152	C₄⋊Q₈	—	47	16	no	—
153	C₄⋊Q₈	—	94	24	no	—
154	#125	—	47	16	no	—
155	#125	—	94	24	no	—
156	C₂×C₄⋊C₄	—	47	6	no	—
157	C₂×C₄⋊C₄	—	47	6	no	—
158	C₂×C₄⋊C₄	—	47	12	no	—
159	C₂×C₄⋊C₄	—	94	3	no	—
160	C₂×C₄⋊C₄	—	94	3	no	—
161	C₂×C₄⋊C₄	—	94	6	no	—
162	C₂×C₄⋊C₄	—	94	12	no	—
163	C₂×C₄⋊C₄	—	94	12	no	—
164	C₂×C₄⋊C₄	—	94	12	no	—
165	C₂×C₄²	D₁₈₈	1	6	yes	—
166	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
167	#107	—	47	32	no	—
168	#106	—	47	32	no	—
169	#97	C₂²×D₄₇	1	16	yes	—
170	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
171	C₂×Q₈	—	94	16	no	—
172	C₂×Q₈	—	188	48	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	94	6	no	—
176	C₂²×C₄	—	94	6	no	—
177	C₂²×C₄	—	94	12	no	—
178	C₂²×C₄	—	188	4	no	—
179	C₂²×C₄	—	188	8	no	—
180	C₂²×C₄	—	188	4	no	—
181	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
182	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
183	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
184	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
185	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
186	C₂×C₈	C₂×Dic₉₄	1	48	yes	complemented
187	C₂×C₈	C₂³×D₄₇	1	8	yes	—
188	C₄⋊C₄	—	47	4	no	—
189	C₄⋊C₄	—	47	8	no	—
190	C₄⋊C₄	—	47	4	no	—
191	C₄⋊C₄	—	94	16	no	—
192	C₄⋊C₄	—	94	16	no	—
193	C₄⋊C₄	—	94	6	no	—
194	C₄⋊C₄	—	94	6	no	—
195	C₄⋊C₄	—	94	12	no	—
196	C₄⋊C₄	—	94	16	no	—
197	C₄²	C₂×D₁₈₈	1	12	yes	—
198	C₄²	C₂³×D₄₇	1	4	yes	—
199	C₂³	#13094024	1	4	yes	—
200	C₂³	#13094052	1	4	yes	—
201	C₂³	#13094057	1	4	yes	—
202	C₂³	C₂²×D₁₈₈	1	3	yes	—
203	Q₈	—	188	32	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	94	4	no	—
207	C₂×C₄	—	94	4	no	—
208	C₂×C₄	—	94	8	no	—
209	C₂×C₄	—	188	12	no	—
210	C₂×C₄	—	188	24	no	—
211	C₂×C₄	—	188	12	no	—
212	C₂×C₄	#13094024	1	12	yes	—
213	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
214	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
215	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
216	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
217	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
218	C₈	C₂²×Dic₉₄	1	16	yes	complemented
219	C₂²	#15689532	1	16	yes	direct product factor
220	C₂²	C₂×#13094024	1	6	yes	—
221	C₂²	C₂×#13094052	1	6	yes	—
222	C₂²	C₂×#13094057	1	6	yes	—
223	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	188	8	no	—
227	C₄	—	188	4	no	—
228	C₄	—	188	4	no	—
229	C₄	C₂×#13094024	1	6	yes	—
230	C₄	C₂³×Dic₉₄	1	2	yes	—
231	C₂	C₂×#15689532	1	12	yes	direct product factor
232	C₂	C₂²×#13094024	1	1	yes	characteristic
233	C₂	C₂²×#13094052	1	1	yes	characteristic
234	C₂	C₂²×#13094057	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂²×#501
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. C₂×#15689532	C₂	direct (N × H)
18. #15689532	C₂²	direct (N × H)
47. C₂²×C₈×C₄₇	Q₈	semidirect
68. C₂×C₈×C₄₇	C₂×Q₈	semidirect
100. C₈×C₄₇	C₂²×Q₈	semidirect
102. C₂²×C₄₇	#501	semidirect
120. C₂×C₄₇	C₂×#501	semidirect
141. C₄₇	C₂²×#501	semidirect
150. C₂²×C₈	Dic₉₄	semidirect
186. C₂×C₈	C₂×Dic₉₄	semidirect
218. C₈	C₂²×Dic₉₄	semidirect
219. C₂²	#15689532	direct (N × H)
231. C₂	C₂×#15689532	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×#13094016	2
C₂×#15689532	2
C₂²×#13093958	2
C₂²×#13093956	2
C₂²×#97×C₄₇	2
C₂²×#501	47

Subgroups — C22×#15689532 #27085888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689532 #27085888