Subgroups — C₂×#20241686 #26181899

Order 12032.

866 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689174	C₂	1	2	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20241686	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688844	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625324	C₄	1	2	yes	—
8	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
9	#15689174	C₄	1	8	yes	—
10	#15689174	C₂²	1	8	yes	—
11	C₂×Dic₄₇⋊C₈	C₄	1	4	yes	—
12	C₂×(C₂×D₄₇)⋊C₈	C₄	1	4	yes	—
13	C₄×#10625324	C₄	1	4	yes	—
14	C₄×#10625324	C₂²	1	4	yes	—
15	C₂×#13093934	C₄	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	#15688844	C₂²	1	8	yes	complemented
19	C₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
20	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#322	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×#10625324	—	2	8	no	—
26	C₂×#10625324	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
31	C₄²×D₄₇	C₂×C₄	1	2	yes	—
32	C₄²×D₄₇	C₂×C₄	1	4	yes	—
33	C₄²×D₄₇	C₂³	1	2	yes	—
34	Dic₄₇⋊C₈	C₂×C₄	1	16	yes	—
35	(C₂×D₄₇)⋊C₈	C₂×C₄	1	16	yes	—
36	#13093934	C₂×C₄	1	8	yes	—
37	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
41	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₄₇	C₂³	1	2	yes	—
47	C₄×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂³×D₄₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
52	C₂²×Dic₄₇	C₄²	1	2	yes	—
53	C₂×C₄×D₄₇	—	2	4	no	—
54	C₂×C₄×D₄₇	—	2	8	no	—
55	C₂×C₄×D₄₇	C₄²	1	2	yes	—
56	C₂×C₄×D₄₇	C₄²	1	4	yes	—
57	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
58	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₄₇	—	2	8	no	—
60	C₂×C₈×C₄₇	M₁₆	1	8	yes	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₄×Dic₄₇	C₄²	1	2	yes	—
67	C₄×Dic₄₇	C₄²	1	4	yes	—
68	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
69	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	8	no	—
72	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
74	#10625324	—	2	16	no	—
75	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
76	C₂²×D₄₇	—	2	2	no	—
77	C₂²×D₄₇	—	2	4	no	—
78	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₄₇	#131	1	8	yes	—
83	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
85	C₂×Dic₄₇	—	2	2	no	—
86	C₂×Dic₄₇	—	2	4	no	—
87	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
88	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
89	C₄×D₄₇	—	2	4	no	—
90	C₄×D₄₇	—	2	8	no	—
91	C₄×D₄₇	—	2	4	no	—
92	C₄×D₄₇	—	2	8	no	—
93	C₈×C₄₇	—	2	8	no	—
94	C₈×C₄₇	C₂×M₁₆	1	8	yes	—
95	C₄₇⋊C₈	—	2	4	no	—
96	C₄₇⋊C₈	—	2	8	no	—
97	C₂×#1453	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
99	C₂²×C₄₇	#405	1	4	yes	—
100	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
101	C₂×D₄₇	—	2	2	no	—
102	C₂×D₄₇	—	2	4	no	—
103	C₂×D₄₇	—	4	4	no	—
104	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
105	C₄×C₄₇	#405	1	4	yes	—
106	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
107	C₄×C₄₇	C₂×#131	1	4	yes	—
108	Dic₄₇	—	2	2	no	—
109	Dic₄₇	—	2	4	no	—
110	C₂×#431	—	47	2	no	—
111	C₂×C₄×M₁₆	—	47	1	no	—
112	#1453	—	47	8	no	—
113	C₂×#322	—	47	2	no	—
114	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
115	C₂×C₈²	—	47	1	no	—
116	C₂×C₄₇	#1453	1	4	yes	complemented
117	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#405	1	2	yes	—
119	D₄₇	—	4	2	no	—
120	C₂²×M₁₆	—	47	2	no	—
121	C₂²×C₄²	—	47	1	no	—
122	#431	—	47	8	no	—
123	#431	—	47	8	no	—
124	C₂×C₄⋊C₈	—	47	4	no	—
125	C₂×C₂²⋊C₈	—	47	4	no	—
126	C₄×M₁₆	—	47	4	no	—
127	C₄×M₁₆	—	47	4	no	—
128	C₂×#97	—	47	2	no	—
129	C₂×C₄×C₈	—	47	1	no	—
130	C₂×C₄×C₈	—	47	2	no	—
131	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
133	#322	—	47	8	no	—
134	#322	C₂×D₄₇	1	4	yes	complemented
135	C₈²	—	47	4	no	—
136	C₄₇	C₂×#1453	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	47	1	no	—
138	C₂³×C₄	—	47	2	no	—
139	C₂×M₁₆	—	47	8	no	—
140	C₂×M₁₆	—	94	8	no	—
141	C₂²×C₈	—	47	2	no	—
142	C₂²×C₈	—	47	4	no	—
143	C₂²×C₈	C₄×D₄₇	1	4	yes	—
144	C₂²×C₈	C₄×D₄₇	1	2	yes	—
145	C₂×C₄²	—	47	1	no	—
146	C₂×C₄²	—	47	2	no	—
147	C₂×C₄²	—	47	2	no	—
148	C₂×C₄²	—	47	4	no	—
149	C₂×C₄²	—	47	2	no	—
150	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
151	C₄⋊C₈	—	47	16	no	—
152	C₂²⋊C₈	—	47	16	no	—
153	#97	—	47	8	no	—
154	C₄×C₈	—	47	2	no	—
155	C₄×C₈	—	47	4	no	—
156	C₄×C₈	—	47	4	no	—
157	C₄×C₈	—	47	2	no	—
158	C₄×C₈	C₄×D₄₇	1	2	yes	—
159	C₄×C₈	C₄×D₄₇	1	4	yes	—
160	C₄×C₈	C₂²×D₄₇	1	2	yes	—
161	C₄×C₈	C₂²×D₄₇	1	4	yes	—
162	C₂⁴	—	47	1	no	—
163	C₂²×C₄	—	47	1	no	—
164	C₂²×C₄	—	47	2	no	—
165	C₂²×C₄	—	47	2	no	—
166	C₂²×C₄	—	47	4	no	—
167	C₂²×C₄	—	47	4	no	—
168	C₂²×C₄	—	47	2	no	—
169	C₂²×C₄	—	94	4	no	—
170	C₂²×C₄	—	94	8	no	—
171	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
173	M₁₆	—	94	16	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	47	4	no	—
176	C₂×C₈	—	47	8	no	—
177	C₂×C₈	—	94	4	no	—
178	C₂×C₈	—	94	8	no	—
179	C₂×C₈	#10625324	1	8	yes	—
180	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
181	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
182	C₄²	—	47	2	no	—
183	C₄²	—	47	4	no	—
184	C₄²	—	47	2	no	—
185	C₄²	—	47	4	no	—
186	C₄²	C₂×C₄×D₄₇	1	2	yes	—
187	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
188	C₄²	C₂³×D₄₇	1	1	yes	characteristic
189	C₂³	—	47	2	no	—
190	C₂³	—	94	2	no	—
191	C₂³	—	94	4	no	—
192	C₂³	C₄²×D₄₇	1	1	yes	characteristic
193	C₂×C₄	—	47	2	no	—
194	C₂×C₄	—	47	4	no	—
195	C₂×C₄	—	94	2	no	—
196	C₂×C₄	—	94	4	no	—
197	C₂×C₄	—	94	4	no	—
198	C₂×C₄	—	94	8	no	—
199	C₂×C₄	—	94	4	no	—
200	C₂×C₄	—	94	8	no	—
201	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
202	C₂×C₄	C₄²×D₄₇	1	2	yes	—
203	C₂×C₄	C₂×#10625324	1	4	yes	—
204	C₂×C₄	#13094032	1	8	yes	—
205	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
207	C₈	—	2	8	no	—
208	C₈	—	94	4	no	—
209	C₈	—	94	8	no	—
210	C₈	C₂×#10625324	1	8	yes	—
211	C₂²	—	94	2	no	—
212	C₂²	—	94	4	no	—
213	C₂²	—	188	4	no	—
214	C₂²	C₄×#10625324	1	2	yes	—
215	C₂²	#15689080	1	4	yes	—
216	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
217	C₄	—	94	2	no	—
218	C₄	—	94	4	no	—
219	C₄	C₄×#10625324	1	2	yes	—
220	C₄	#15689080	1	4	yes	—
221	C₄	C₂²×#10625324	1	2	yes	—
222	C₄	C₂×#13094032	1	4	yes	—
223	C₂	—	188	2	no	—
224	C₂	#20241686	1	4	yes	direct product factor
225	C₂	C₂×C₄×#10625324	1	1	yes	characteristic
226	C₂	C₂×#15689080	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	C₂×C₄²
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	C₂×#1453
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #20241686	C₂	direct (N × H)
4. C₂×#15688844	C₂	semidirect
5. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
6. C₂×C₄₇×#322	C₂	semidirect
18. #15688844	C₂²	semidirect
19. C₈×C₄₇⋊C₈	C₂²	semidirect
22. C₄₇×#322	C₂²	semidirect
114. C₂×#322	D₄₇	semidirect
116. C₂×C₄₇	#1453	semidirect
134. #322	C₂×D₄₇	semidirect
136. C₄₇	C₂×#1453	semidirect
224. C₂	#20241686	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689174	2
C₂×C₄×#10625324	2
#20241686	2
C₂×#15688844	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄₇×#322	2
C₂×#1453	47

Subgroups — C2×#20241686 #26181899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241686 #26181899