Subgroups — C₂×#20252473 #27081222

Order 12032.

1978 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252473	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689254	C₂	1	8	yes	maximal, complemented
3	C₂×#15689180	C₂	1	4	yes	maximal
4	C₂×#15689176	C₂	1	2	yes	maximal
5	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689254	C₂²	1	64	yes	complemented
7	#15689180	C₂²	1	32	yes	—
8	#15689176	C₂²	1	16	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	8	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
14	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₄₇	C₄	1	4	yes	—
16	C₂×C₄×Q₈×C₄₇	C₄	1	8	yes	—
17	C₂×C₄×Q₈×C₄₇	C₂²	1	8	yes	—
18	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
24	C₄⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
25	C₄⋊(C₄₇⋊C₈)	C₂³	1	32	yes	—
26	C₂²⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
27	#13093935	C₂³	1	32	yes	—
28	C₄×C₄₇⋊C₈	C₂³	1	16	yes	—
29	C₂²×Q₈×C₄₇	—	2	8	no	—
30	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
31	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₄₇	C₂×C₄	1	32	yes	—
38	C₄×Q₈×C₄₇	C₂×C₄	1	16	yes	—
39	C₄×Q₈×C₄₇	C₂³	1	16	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Q₈×C₄₇	—	2	32	no	—
54	C₂×Q₈×C₄₇	—	2	16	no	—
55	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
74	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂⁴	1	4	yes	—
80	C₄²×C₄₇	C₂⁴	1	8	yes	—
81	C₂×C₄₇⋊C₈	—	2	16	no	—
82	C₂×C₄₇⋊C₈	C₂⁴	1	16	yes	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
86	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
87	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₄₇	—	2	32	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	16	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	4	8	no	—
98	C₂×C₄×C₄₇	C₂×M₁₆	1	16	yes	—
99	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
107	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
108	C₄₇⋊C₈	—	2	16	no	—
109	C₂×#2983	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	—	2	8	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	#519	1	2	yes	—
114	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
115	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
116	C₂²×C₄₇	#568	1	2	yes	—
117	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	8	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	—	2	8	no	—
122	C₄×C₄₇	#519	1	2	yes	—
123	C₄×C₄₇	C₂²×M₁₆	1	8	yes	—
124	C₄×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
125	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
126	#2983	—	47	16	no	—
127	C₂×#446	—	47	8	no	—
128	C₂×#432	—	47	4	no	—
129	C₂×#431	—	47	2	no	—
130	C₂×C₄₇	—	2	4	no	—
131	C₂×C₄₇	#2983	1	4	yes	complemented
132	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
133	C₂×C₄₇	C₂³×M₁₆	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#568	1	1	yes	characteristic
135	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
136	C₂×C₄×Q₈	Dic₄₇	1	4	yes	—
137	C₂×C₄×Q₈	Dic₄₇	1	8	yes	—
138	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
139	C₂×C₄×Q₈	C₂×D₄₇	1	8	yes	—
140	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
141	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
142	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
143	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
144	#446	—	47	64	no	—
145	#432	—	47	32	no	—
146	#431	—	47	16	no	—
147	C₂×C₄⋊C₈	—	47	4	no	—
148	C₂×C₄⋊C₈	—	47	8	no	—
149	C₂×C₂²⋊C₈	—	47	4	no	—
150	C₂×#97	—	47	8	no	—
151	C₂×C₄×C₈	—	47	4	no	—
152	C₄₇	C₂×#2983	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	2	8	no	—
154	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
155	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
156	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
157	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
158	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
159	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
160	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
161	C₂²×C₈	—	47	8	no	—
162	C₄×Q₈	C₂×Dic₄₇	1	16	yes	—
163	C₄×Q₈	C₂×Dic₄₇	1	32	yes	—
164	C₄×Q₈	C₂²×D₄₇	1	16	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
168	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
169	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
170	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
171	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
172	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
173	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
174	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
175	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
176	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
177	C₄⋊C₈	—	47	16	no	—
178	C₄⋊C₈	—	47	32	no	—
179	C₂²⋊C₈	—	47	16	no	—
180	#97	—	47	32	no	—
181	C₄×C₈	—	47	16	no	—
182	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
183	C₂×Q₈	—	2	32	no	—
184	C₂×Q₈	—	2	16	no	—
185	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	#10625329	1	8	yes	—
191	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
195	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
196	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	C₂×C₈	—	47	16	no	—
204	C₂×C₈	—	94	16	no	—
205	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
206	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
207	C₄²	C₂²×Dic₄₇	1	16	yes	—
208	C₄²	C₂²×Dic₄₇	1	8	yes	—
209	C₄²	C₂²×Dic₄₇	1	8	yes	—
210	C₄²	C₂²×Dic₄₇	1	4	yes	—
211	C₄²	C₂³×D₄₇	1	4	yes	—
212	C₄²	C₂³×D₄₇	1	8	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×#10625329	1	4	yes	—
216	C₂³	C₂³×Dic₄₇	1	2	yes	—
217	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
218	Q₈	—	2	32	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	C₂×#10625329	1	16	yes	—
229	C₂×C₄	C₂×#10625329	1	8	yes	—
230	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
231	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
232	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
233	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
234	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
236	C₂×C₄	#13094072	1	4	yes	—
237	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
238	C₈	—	94	16	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	#15689601	1	2	yes	—
243	C₂²	C₂²×#10625329	1	4	yes	—
244	C₂²	C₂²×#10625329	1	2	yes	—
245	C₂²	#15689890	1	2	yes	—
246	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
247	C₄	—	2	8	no	—
248	C₄	—	2	4	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	2	no	—
251	C₄	#15689601	1	2	yes	—
252	C₄	C₂²×#10625329	1	8	yes	—
253	C₄	C₂×#13094072	1	2	yes	—
254	C₂	—	2	4	no	—
255	C₂	#20252473	1	4	yes	direct product factor
256	C₂	C₂×#15689601	1	1	yes	characteristic
257	C₂	C₂³×#10625329	1	1	yes	characteristic
258	C₂	C₂×#15689890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#2983
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20252473	C₂	direct (N × H)
2. C₂×#15689254	C₂	semidirect
6. #15689254	C₂²	semidirect
131. C₂×C₄₇	#2983	semidirect
152. C₄₇	C₂×#2983	semidirect
255. C₂	#20252473	direct (N × H)

Maximal subgroups

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

subgroup	index
#20252473	2
C₂×#15689254	2
C₂×#15689180	2
C₂×#15689176	2
C₂²×C₄×Q₈×C₄₇	2
C₂×#2983	47

Subgroups — C2×#20252473 #27081222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20252473 #27081222