Subgroups — C₂×#20245181 #26369332

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 250 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×#108×D₄₇	C₂	1	2	yes	maximal
3	#20245181	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688953	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688952	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₄₇	—	2	2	no	—
8	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×#13093963	C₄	1	4	yes	—
10	C₂×#13093963	C₄	1	4	yes	—
11	C₂×#13093963	C₂²	1	4	yes	—
12	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#13093960	C₂²	1	2	yes	—
14	C₂×#13093959	—	2	2	no	—
15	#108×D₄₇	—	2	4	no	—
16	#108×D₄₇	C₂²	1	8	yes	—
17	#15688953	C₂²	1	4	yes	complemented
18	#15688952	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#108×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#368	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625324	—	4	2	no	—
25	M₁₆×D₄₇	—	2	8	no	—
26	M₁₆×D₄₇	—	4	4	no	—
27	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₈×D₄₇	—	4	2	no	—
29	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₄₇	D₄	1	2	yes	—
32	C₂×C₈×D₄₇	Q₈	1	2	yes	—
33	C₂×C₈×D₄₇	Q₈	1	2	yes	—
34	C₂×C₈×D₄₇	C₂³	1	2	yes	—
35	#13093963	C₂×C₄	1	16	yes	—
36	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
39	#13093960	—	2	2	no	—
40	#13093960	C₂³	1	4	yes	—
41	#13093959	—	2	8	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
47	#108×C₄₇	—	2	2	no	—
48	#108×C₄₇	C₂³	1	4	yes	—
49	C₂³×D₄₇	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	Q₁₆	1	1	yes	characteristic
52	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₄₇	D₈	1	2	yes	—
55	C₂×C₄×D₄₇	D₈	1	2	yes	—
56	C₂×C₄×D₄₇	Q₁₆	1	2	yes	—
57	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
58	M₁₆×C₄₇	—	2	4	no	—
59	M₁₆×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
67	#10625329	—	2	4	no	—
68	#10625329	—	4	2	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
73	#10625324	—	4	8	no	—
74	C₈×D₄₇	—	4	8	no	—
75	C₈×D₄₇	C₄⋊C₄	1	4	yes	—
76	C₈×D₄₇	C₄⋊C₄	1	4	yes	—
77	C₈×D₄₇	C₂²×C₄	1	4	yes	—
78	C₈×D₄₇	C₂×Q₈	1	4	yes	—
79	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
80	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	—
81	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
82	C₂²×D₄₇	—	4	2	no	—
83	C₂²×D₄₇	#107	1	2	yes	—
84	C₂²×D₄₇	#107	1	2	yes	—
85	C₂²×D₄₇	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₄₇	#107	1	2	yes	—
93	C₂×Dic₄₇	#107	1	2	yes	—
94	C₂×Dic₄₇	C₂×Q₁₆	1	2	yes	—
95	C₄×D₄₇	#107	1	4	yes	—
96	C₄×D₄₇	#107	1	4	yes	—
97	C₄×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₄×D₄₇	C₂×D₈	1	4	yes	—
99	C₈×C₄₇	—	4	4	no	—
100	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
104	C₄₇⋊C₈	—	4	4	no	—
105	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×#2161	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	2	2	no	—
108	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
110	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
111	C₂×D₄₇	—	2	4	no	—
112	C₂×D₄₇	—	4	2	no	—
113	C₂×D₄₇	—	4	2	no	—
114	C₂×D₄₇	C₂×#107	1	4	yes	—
115	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
116	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
117	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
119	Dic₄₇	C₂×#107	1	4	yes	—
120	C₂²×M₃₂	—	47	1	no	—
121	C₂²×#108	—	47	2	no	—
122	#2161	—	47	8	no	—
123	C₂×#368	—	47	1	no	—
124	C₂×#368	—	47	2	no	—
125	C₂×#368	D₄₇	1	1	yes	characteristic, complemented
126	C₂×C₄₇	—	2	1	no	—
127	C₂×C₄₇	—	2	1	no	—
128	C₂×C₄₇	#2161	1	2	yes	complemented
129	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
130	D₄₇	—	2	4	no	—
131	C₂²×M₁₆	—	94	2	no	—
132	C₂³×C₈	—	47	1	no	—
133	C₂×M₃₂	—	47	4	no	—
134	C₂×M₃₂	—	47	4	no	—
135	C₂×M₃₂	—	47	4	no	—
136	C₂²×C₁₆	—	47	1	no	—
137	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
138	C₂×#108	—	47	2	no	—
139	C₂×#108	—	47	8	no	—
140	C₂×#108	—	94	4	no	—
141	C₂×#108	—	94	2	no	—
142	C₂×#108	C₂×D₄₇	1	2	yes	—
143	#368	—	47	4	no	—
144	#368	—	47	8	no	—
145	#368	C₂×D₄₇	1	4	yes	complemented
146	C₄₇	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	47	1	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	94	8	no	—
150	C₂×M₁₆	—	94	2	no	—
151	C₂×M₁₆	—	188	2	no	—
152	C₂×M₁₆	—	188	4	no	—
153	C₂²×C₈	—	47	1	no	—
154	C₂²×C₈	—	47	2	no	—
155	C₂²×C₈	—	47	2	no	—
156	C₂²×C₈	—	47	2	no	—
157	C₂²×C₈	—	47	2	no	—
158	C₂²×C₈	—	47	2	no	—
159	C₂²×C₈	—	47	2	no	—
160	C₂²×C₈	—	188	2	no	—
161	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
162	M₃₂	—	47	16	no	—
163	C₂×C₁₆	—	47	2	no	—
164	C₂×C₁₆	—	47	2	no	—
165	C₂×C₁₆	—	47	2	no	—
166	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
167	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
168	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
169	#108	—	2	2	no	—
170	#108	—	47	4	no	—
171	#108	—	94	8	no	—
172	#108	—	94	2	no	—
173	#108	C₂²×D₄₇	1	4	yes	—
174	C₂⁴	—	47	1	no	—
175	C₂²×C₄	—	47	1	no	—
176	C₂²×C₄	—	47	2	no	—
177	C₂²×C₄	—	47	2	no	—
178	C₂²×C₄	—	47	2	no	—
179	C₂²×C₄	—	47	2	no	—
180	C₂²×C₄	—	47	2	no	—
181	C₂²×C₄	—	47	2	no	—
182	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
183	M₁₆	—	2	4	no	—
184	M₁₆	—	4	2	no	—
185	M₁₆	—	94	4	no	—
186	M₁₆	—	188	8	no	—
187	M₁₆	—	188	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	47	2	no	—
190	C₂×C₈	—	47	2	no	—
191	C₂×C₈	—	47	2	no	—
192	C₂×C₈	—	47	4	no	—
193	C₂×C₈	—	47	4	no	—
194	C₂×C₈	—	47	4	no	—
195	C₂×C₈	—	47	4	no	—
196	C₂×C₈	—	188	8	no	—
197	C₂×C₈	—	188	2	no	—
198	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
202	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
204	C₁₆	—	47	4	no	—
205	C₁₆	C₂×C₄×D₄₇	1	4	yes	—
206	C₂³	—	47	2	no	—
207	C₂³	—	47	2	no	—
208	C₂³	—	47	2	no	—
209	C₂³	—	188	2	no	—
210	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
211	C₂×C₄	—	47	2	no	—
212	C₂×C₄	—	47	2	no	—
213	C₂×C₄	—	47	2	no	—
214	C₂×C₄	—	47	4	no	—
215	C₂×C₄	—	47	4	no	—
216	C₂×C₄	—	47	4	no	—
217	C₂×C₄	—	47	4	no	—
218	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
219	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
221	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
224	C₈	—	4	4	no	—
225	C₈	—	47	4	no	—
226	C₈	—	188	4	no	—
227	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
228	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
229	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
230	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	47	4	no	—
233	C₂²	—	94	4	no	—
234	C₂²	—	188	2	no	—
235	C₂²	—	188	2	no	—
236	C₂²	#107×D₄₇	1	1	yes	characteristic
237	C₂²	#107×D₄₇	1	1	yes	characteristic
238	C₂²	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
239	C₄	—	47	4	no	—
240	C₄	#107×D₄₇	1	1	yes	characteristic
241	C₄	#107×D₄₇	1	1	yes	characteristic
242	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
243	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	94	4	no	—
247	C₂	#20245181	1	2	yes	direct product factor
248	C₂	C₂×#107×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#368	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×D₄₇	Q₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#368	D₄₇
hypocenter	C₄₇	C₂×#2161
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. #20245181	C₂	direct (N × H)
4. C₂×#15688953	C₂	semidirect
5. C₂×#15688952	C₂	semidirect
6. C₂×C₄₇×#368	C₂	semidirect
17. #15688953	C₂²	semidirect
18. #15688952	C₂²	semidirect
21. C₄₇×#368	C₂²	semidirect
125. C₂×#368	D₄₇	semidirect
128. C₂×C₄₇	#2161	semidirect
145. #368	C₂×D₄₇	semidirect
146. C₄₇	C₂×#2161	semidirect
247. C₂	#20245181	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×#108×D₄₇	2
#20245181	2
C₂×#15688953	2
C₂×#15688952	2
C₂×C₄₇×#368	2
C₂×#2161	47

Subgroups — C2×#20245181 #26369332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245181 #26369332