Subgroups — C₂×#20252494 #27081247

Order 12032.

1706 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252494	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689254	C₂	1	8	yes	maximal, complemented
3	C₂×#15689184	C₂	1	4	yes	maximal
4	C₂×#15689180	C₂	1	2	yes	maximal
5	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
6	#15689254	C₂²	1	64	yes	complemented
7	#15689184	C₂²	1	32	yes	—
8	#15689180	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	4	yes	—
13	C₂×#13093935	C₂²	1	4	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
15	C₄₇×#533	C₄	1	8	yes	—
16	C₄₇×#533	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
18	C₂×#125×C₄₇	C₄	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
21	C₂×#117×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
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	16	yes	—
28	#13093935	C₂³	1	16	yes	—
29	C₄×C₄₇⋊C₈	C₂³	1	16	yes	—
30	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
34	#125×C₄₇	C₂×C₄	1	16	yes	—
35	C₂²⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
36	C₄×Q₈×C₄₇	C₂×C₄	1	16	yes	—
37	C₄×Q₈×C₄₇	C₂³	1	16	yes	—
38	#117×C₄₇	C₂×C₄	1	8	yes	—
39	#117×C₄₇	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	4	8	no	—
52	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
64	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
65	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
66	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
71	C₄²×C₄₇	C₂⁴	1	4	yes	—
72	C₄²×C₄₇	C₂⁴	1	8	yes	—
73	C₂×C₄₇⋊C₈	—	2	16	no	—
74	C₂×C₄₇⋊C₈	C₂⁴	1	16	yes	—
75	C₂³×C₄₇	—	2	2	no	—
76	C₂³×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
78	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₄₇	—	4	16	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	16	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	4	8	no	—
87	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
93	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
94	C₄₇⋊C₈	—	2	16	no	—
95	C₂×#2987	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	2	4	no	—
98	C₂²×C₄₇	—	4	4	no	—
99	C₂²×C₄₇	#519	1	2	yes	—
100	C₂²×C₄₇	#568	1	4	yes	—
101	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	2	4	no	—
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	—	2	16	no	—
106	C₄×C₄₇	#519	1	2	yes	—
107	C₄×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
108	C₂×#533	D₄₇	1	1	yes	characteristic
109	#2987	—	47	16	no	—
110	C₂×#446	—	47	8	no	—
111	C₂×#433	—	47	4	no	—
112	C₂×#432	—	47	2	no	—
113	C₂×C₄₇	—	4	2	no	—
114	C₂×C₄₇	#2987	1	4	yes	complemented
115	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
116	C₂×C₄₇	C₂×#568	1	2	yes	—
117	#533	Dic₄₇	1	8	yes	—
118	#533	C₂×D₄₇	1	8	yes	—
119	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
120	C₂×#125	Dic₄₇	1	2	yes	—
121	C₂×C₂²⋊Q₈	Dic₄₇	1	4	yes	—
122	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
123	C₂×#117	C₂×D₄₇	1	2	yes	—
124	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
125	#446	—	47	64	no	—
126	#433	—	47	32	no	—
127	#432	—	47	16	no	—
128	C₂×C₄⋊C₈	—	47	4	no	—
129	C₂×C₄⋊C₈	—	47	8	no	—
130	C₂×C₂²⋊C₈	—	47	4	no	—
131	C₂×#97	—	47	4	no	—
132	C₂×#97	—	47	4	no	—
133	C₂×C₄×C₈	—	47	4	no	—
134	C₄₇	C₂×#2987	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
136	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
137	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂²×C₈	—	47	8	no	—
139	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
140	#125	C₂×Dic₄₇	1	16	yes	—
141	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
142	C₄×Q₈	C₂×Dic₄₇	1	16	yes	—
143	C₄×Q₈	C₂²×D₄₇	1	16	yes	—
144	#117	C₂×Dic₄₇	1	8	yes	—
145	#117	C₂²×D₄₇	1	8	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
147	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
149	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
150	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
153	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
156	C₄⋊C₈	—	47	16	no	—
157	C₄⋊C₈	—	47	32	no	—
158	C₂²⋊C₈	—	47	16	no	—
159	#97	—	47	16	no	—
160	#97	—	47	16	no	—
161	C₄×C₈	—	47	16	no	—
162	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
163	C₂×Q₈	—	4	8	no	—
164	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
172	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
173	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	C₂×C₈	—	47	16	no	—
176	C₂×C₈	—	94	16	no	—
177	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
178	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
179	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
180	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
181	C₄²	C₂²×Dic₄₇	1	4	yes	—
182	C₄²	C₂²×Dic₄₇	1	4	yes	—
183	C₄²	C₂²×Dic₄₇	1	4	yes	—
184	C₄²	C₂²×Dic₄₇	1	8	yes	—
185	C₄²	C₂³×D₄₇	1	8	yes	—
186	C₄²	C₂³×D₄₇	1	4	yes	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₂³×Dic₄₇	1	2	yes	—
190	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
191	Q₈	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
200	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
201	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
202	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
203	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
204	C₂×C₄	#13094072	1	4	yes	—
205	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
206	C₈	—	94	16	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	#15689601	1	2	yes	—
211	C₂²	#15689890	1	4	yes	—
212	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	2	16	no	—
216	C₄	—	2	2	no	—
217	C₄	#15689601	1	2	yes	—
218	C₄	C₂×#13094072	1	2	yes	—
219	C₂	—	4	2	no	—
220	C₂	#20252494	1	4	yes	direct product factor
221	C₂	C₂×#15689601	1	1	yes	characteristic
222	C₂	C₂×#15689890	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#2987
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20252494	C₂	direct (N × H)
2. C₂×#15689254	C₂	semidirect
6. #15689254	C₂²	semidirect
114. C₂×C₄₇	#2987	semidirect
134. C₄₇	C₂×#2987	semidirect
220. C₂	#20252494	direct (N × H)

Maximal subgroups

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

subgroup	index
#20252494	2
C₂×#15689254	2
C₂×#15689184	2
C₂×#15689180	2
C₂×C₄₇×#533	2
C₂×#2987	47

Subgroups — C2×#20252494 #27081247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20252494 #27081247