Subgroups — C₂×#20241527 #26136478

Order 12032.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093963	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688957	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241527	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₄₇	C₄	1	1	yes	characteristic
6	C₂×#13093963	—	2	1	no	—
7	C₂×#13093963	—	2	1	no	—
8	C₂×#13093963	—	2	1	no	—
9	C₂×#13093963	—	2	1	no	—
10	C₂×#13093963	C₄	1	2	yes	—
11	C₂×#13093963	C₂²	1	2	yes	—
12	C₂²×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
13	#15688957	—	2	1	no	—
14	#15688957	C₂²	1	2	yes	complemented
15	C₂×C₄₇⋊C₃₂	—	2	1	no	—
16	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×M₆₄	—	2	1	no	—
18	C₄₇×M₆₄	C₂²	1	2	yes	complemented
19	C₂×C₃₂×C₄₇	—	2	1	no	—
20	C₂²×C₄×D₄₇	C₈	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	C₈	1	1	yes	characteristic
22	C₂×C₈×D₄₇	—	2	1	no	—
23	C₂×C₈×D₄₇	—	2	1	no	—
24	C₂×C₈×D₄₇	—	2	1	no	—
25	C₂×C₈×D₄₇	—	2	1	no	—
26	C₂×C₈×D₄₇	C₈	1	2	yes	—
27	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
28	#13093963	—	2	2	no	—
29	#13093963	—	2	2	no	—
30	#13093963	—	2	2	no	—
31	#13093963	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	—	2	1	no	—
33	C₂×C₄₇⋊C₁₆	—	2	1	no	—
34	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₄₇⋊C₃₂	—	2	2	no	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
43	C₃₂×C₄₇	—	2	2	no	—
44	C₂³×D₄₇	C₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×Dic₄₇	C₁₆	1	1	yes	characteristic
47	C₂×C₄×D₄₇	—	2	1	no	—
48	C₂×C₄×D₄₇	—	2	1	no	—
49	C₂×C₄×D₄₇	—	2	1	no	—
50	C₂×C₄×D₄₇	—	2	1	no	—
51	C₂×C₄×D₄₇	C₁₆	1	2	yes	—
52	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄₇⋊C₈	—	2	1	no	—
60	C₂×C₄₇⋊C₈	—	2	1	no	—
61	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
62	C₈×D₄₇	—	2	2	no	—
63	C₈×D₄₇	—	2	2	no	—
64	C₈×D₄₇	—	2	2	no	—
65	C₈×D₄₇	—	2	2	no	—
66	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄₇⋊C₁₆	—	2	2	no	—
71	C₂³×C₄₇	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₄₇	—	2	1	no	—
73	C₂²×D₄₇	—	2	1	no	—
74	C₂²×D₄₇	—	4	2	no	—
75	C₂²×D₄₇	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
80	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×Dic₄₇	—	2	1	no	—
83	C₂×Dic₄₇	—	2	1	no	—
84	C₂×Dic₄₇	C₂×C₁₆	1	2	yes	—
85	C₄×D₄₇	—	2	2	no	—
86	C₄×D₄₇	—	2	2	no	—
87	C₄×D₄₇	—	2	2	no	—
88	C₄×D₄₇	—	2	2	no	—
89	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
93	C₄₇⋊C₈	—	2	2	no	—
94	C₂×#1400	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
99	C₂×D₄₇	—	2	2	no	—
100	C₂×D₄₇	—	4	2	no	—
101	C₂×D₄₇	—	4	2	no	—
102	C₂×D₄₇	—	4	2	no	—
103	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
104	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
105	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
107	Dic₄₇	—	2	2	no	—
108	C₂²×M₃₂	—	47	1	no	—
109	C₂×M₆₄	—	47	1	no	—
110	C₂×M₆₄	D₄₇	1	1	yes	characteristic, complemented
111	#1400	—	47	4	no	—
112	C₂×C₄₇	—	2	1	no	—
113	C₂×C₄₇	—	2	1	no	—
114	C₂×C₄₇	#1400	1	2	yes	complemented
115	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
116	D₄₇	—	4	2	no	—
117	C₂³×C₈	—	47	1	no	—
118	C₂×M₃₂	—	47	2	no	—
119	C₂×M₃₂	—	47	2	no	—
120	C₂×M₃₂	—	94	1	no	—
121	C₂×M₃₂	—	94	1	no	—
122	C₂×M₃₂	—	94	1	no	—
123	C₂×M₃₂	—	94	1	no	—
124	C₂²×C₁₆	—	47	1	no	—
125	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
126	M₆₄	—	2	1	no	—
127	M₆₄	—	47	2	no	—
128	M₆₄	—	94	1	no	—
129	M₆₄	C₂×D₄₇	1	2	yes	complemented
130	C₂×C₃₂	—	2	1	no	—
131	C₂×C₃₂	—	94	1	no	—
132	C₄₇	C₂×#1400	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	—	47	1	no	—
134	C₂²×C₈	—	47	1	no	—
135	C₂²×C₈	—	47	2	no	—
136	C₂²×C₈	—	47	2	no	—
137	C₂²×C₈	—	94	1	no	—
138	C₂²×C₈	—	94	1	no	—
139	C₂²×C₈	—	94	1	no	—
140	C₂²×C₈	—	94	1	no	—
141	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
142	M₃₂	—	94	2	no	—
143	M₃₂	—	94	2	no	—
144	M₃₂	—	94	2	no	—
145	M₃₂	—	94	2	no	—
146	C₂×C₁₆	—	47	2	no	—
147	C₂×C₁₆	—	94	1	no	—
148	C₂×C₁₆	—	94	1	no	—
149	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
150	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
151	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
152	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
154	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
155	C₃₂	—	2	2	no	—
156	C₃₂	—	94	2	no	—
157	C₂⁴	—	47	1	no	—
158	C₂²×C₄	—	47	1	no	—
159	C₂²×C₄	—	47	2	no	—
160	C₂²×C₄	—	47	2	no	—
161	C₂²×C₄	—	94	1	no	—
162	C₂²×C₄	—	94	1	no	—
163	C₂²×C₄	—	94	1	no	—
164	C₂²×C₄	—	94	1	no	—
165	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
166	C₂×C₈	—	47	2	no	—
167	C₂×C₈	—	94	1	no	—
168	C₂×C₈	—	94	1	no	—
169	C₂×C₈	—	94	2	no	—
170	C₂×C₈	—	94	2	no	—
171	C₂×C₈	—	94	2	no	—
172	C₂×C₈	—	94	2	no	—
173	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
174	C₂×C₈	#10625324	1	1	yes	characteristic
175	C₂×C₈	#10625324	1	1	yes	characteristic
176	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
177	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
179	C₁₆	—	94	2	no	—
180	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
181	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
182	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
183	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂³	—	47	2	no	—
185	C₂³	—	94	1	no	—
186	C₂³	—	94	1	no	—
187	C₂³	—	188	2	no	—
188	C₂³	C₁₆×D₄₇	1	1	yes	characteristic
189	C₂×C₄	—	47	2	no	—
190	C₂×C₄	—	94	1	no	—
191	C₂×C₄	—	94	1	no	—
192	C₂×C₄	—	94	2	no	—
193	C₂×C₄	—	94	2	no	—
194	C₂×C₄	—	94	2	no	—
195	C₂×C₄	—	94	2	no	—
196	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
197	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
199	C₂×C₄	#13093963	1	1	yes	characteristic
200	C₂×C₄	#13093963	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
202	C₈	—	94	2	no	—
203	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
204	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
205	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
206	C₈	C₂×#10625324	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	94	2	no	—
209	C₂²	—	188	2	no	—
210	C₂²	—	188	2	no	—
211	C₂²	—	188	2	no	—
212	C₂²	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
213	C₂²	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
214	C₂²	C₂×C₁₆×D₄₇	1	1	yes	characteristic
215	C₄	—	94	2	no	—
216	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
217	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
218	C₄	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
219	C₄	C₂×#13093963	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	—	188	2	no	—
223	C₂	#20241527	1	2	yes	direct product factor
224	C₂	C₂×(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₂×C₄₇×M₆₄	C₂
socle	C₂²×C₄₇	C₂²⋊C₁₆
prime radical	C₂³×D₄₇	C₁₆
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	C₂×M₆₄	D₄₇
hypocenter	C₄₇	C₂×#1400
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#15688957	C₂	semidirect
3. #20241527	C₂	direct (N × H)
4. C₂×C₄₇×M₆₄	C₂	semidirect
14. #15688957	C₂²	semidirect
18. C₄₇×M₆₄	C₂²	semidirect
110. C₂×M₆₄	D₄₇	semidirect
114. C₂×C₄₇	#1400	semidirect
129. M₆₄	C₂×D₄₇	semidirect
132. C₄₇	C₂×#1400	semidirect
223. C₂	#20241527	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×#13093963	2
C₂×#15688957	2
#20241527	2
C₂×C₄₇×M₆₄	2
C₂×#1400	47

Subgroups — C2×#20241527 #26136478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241527 #26136478