Subgroups — C₂×#20243544 #26216675

Order 12032.

1358 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093955	C₂	1	4	yes	maximal
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
4	#20243544	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×#13093955	—	2	16	no	—
10	C₂×#13093955	C₂²	1	16	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
13	#15688993	—	2	2	no	—
14	#15688993	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#384	—	2	2	no	—
19	C₄₇×#384	C₂²	1	4	yes	—
20	C₂³×Dic₄₇	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
22	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
23	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	8	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	4	8	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	#13093955	—	2	32	no	—
31	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
32	C₂³×C₄×C₄₇	—	2	2	no	—
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	8	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
44	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	—	4	8	no	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
49	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
50	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	—	4	4	no	—
60	C₂²×Dic₄₇	—	4	8	no	—
61	C₄⋊C₄×C₄₇	—	2	8	no	—
62	C₄⋊C₄×C₄₇	—	4	4	no	—
63	C₄⋊C₄×C₄₇	—	4	16	no	—
64	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
66	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
67	C₄⋊Dic₄₇	—	2	8	no	—
68	C₄⋊Dic₄₇	—	4	4	no	—
69	C₄⋊Dic₄₇	—	4	16	no	—
70	C₂×C₄₇⋊C₈	C₄⋊C₄	1	32	yes	complemented
71	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
72	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
73	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
74	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₄₇	#127	1	2	yes	—
76	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
77	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
78	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
79	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	8	no	—
82	C₂×C₄×C₄₇	—	8	8	no	—
83	C₂×C₄×C₄₇	#106	1	32	yes	complemented
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₄₇	#127	1	2	yes	—
88	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
90	C₂×C₄×C₄₇	C₂×SD₁₆	1	8	yes	—
91	C₂×C₄×C₄₇	C₂×SD₁₆	1	8	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×Dic₄₇	—	4	4	no	—
97	C₂×Dic₄₇	—	4	8	no	—
98	C₂×Dic₄₇	—	8	8	no	—
99	C₄₇⋊C₈	C₂×C₄⋊C₄	1	16	yes	complemented
100	C₂×#1844	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	#384	1	4	yes	—
102	C₂²×C₄₇	C₂×#106	1	8	yes	—
103	C₂²×C₄₇	#492	1	4	yes	—
104	C₂²×C₄₇	#492	1	4	yes	—
105	C₂²×C₄₇	#498	1	8	yes	—
106	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
110	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
111	C₄×C₄₇	—	8	4	no	—
112	C₄×C₄₇	#384	1	4	yes	—
113	C₄×C₄₇	C₂×#106	1	16	yes	complemented
114	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
115	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
116	Dic₄₇	—	8	4	no	—
117	C₂²×#106	—	47	4	no	—
118	C₂²×C₄×C₈	—	47	1	no	—
119	C₂×#384	—	47	1	no	—
120	C₂×#384	D₄₇	1	1	yes	characteristic
121	#1844	—	47	8	no	—
122	C₂×C₄₇	#1844	1	8	yes	complemented
123	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
124	C₂×C₄₇	C₂²×#106	1	2	yes	—
125	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#498	1	2	yes	—
128	C₂³×C₈	—	47	2	no	—
129	C₂²×C₄⋊C₄	—	2	2	no	—
130	C₂²×C₄⋊C₄	—	47	2	no	—
131	C₂²×C₄⋊C₄	—	94	2	no	—
132	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
133	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	C₂×#106	—	47	16	no	—
135	C₂×#106	—	94	16	no	—
136	C₂×C₄×C₈	—	47	8	no	—
137	C₂×C₄×C₈	—	47	4	no	—
138	#384	—	2	2	no	—
139	#384	—	47	4	no	—
140	#384	—	94	2	no	—
141	#384	C₂×D₄₇	1	4	yes	—
142	C₄₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₄	—	2	2	no	—
144	C₂³×C₄	—	94	2	no	—
145	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
146	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	C₂²×C₈	—	47	8	no	—
148	C₂²×C₈	—	47	8	no	—
149	C₂²×C₈	—	47	8	no	—
150	C₂²×C₈	—	47	4	no	—
151	C₂×C₄⋊C₄	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	8	no	—
155	C₂×C₄⋊C₄	—	47	4	no	—
156	C₂×C₄⋊C₄	—	94	8	no	—
157	C₂×C₄⋊C₄	—	94	2	no	—
158	C₂×C₄⋊C₄	—	94	8	no	—
159	C₂×C₄⋊C₄	—	188	8	no	—
160	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
161	C₂×C₄²	Dic₉₄	1	4	yes	—
162	C₂×C₄²	C₄×D₄₇	1	4	yes	—
163	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
165	#106	—	94	32	no	—
166	C₄×C₈	—	47	16	no	—
167	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	4	8	no	—
171	C₂²×C₄	—	94	4	no	—
172	C₂²×C₄	—	188	4	no	—
173	C₂²×C₄	—	188	8	no	—
174	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
175	C₂²×C₄	#10625335	1	8	yes	—
176	C₂²×C₄	#10625336	1	8	yes	—
177	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
179	C₂²×C₄	D₄×D₄₇	1	4	yes	—
180	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
183	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
184	C₂×C₈	—	47	32	no	—
185	C₂×C₈	—	47	8	no	—
186	C₂×C₈	—	47	8	no	—
187	C₂×C₈	—	47	8	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	16	no	—
191	C₄⋊C₄	—	94	8	no	—
192	C₄⋊C₄	—	188	4	no	—
193	C₄⋊C₄	—	188	16	no	—
194	C₄²	Dic₄₇⋊C₄	1	8	yes	—
195	C₄²	C₂×Dic₉₄	1	4	yes	—
196	C₄²	C₂×C₄×D₄₇	1	4	yes	—
197	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
198	C₂³	#13094020	1	2	yes	—
199	C₂³	#13094022	1	2	yes	—
200	C₂³	C₂×#10625336	1	4	yes	—
201	C₂³	C₂×#10625335	1	4	yes	—
202	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	8	no	—
206	C₂×C₄	—	376	8	no	—
207	C₂×C₄	—	188	8	no	—
208	C₂×C₄	—	188	4	no	—
209	C₂×C₄	#13093955	1	32	yes	complemented
210	C₂×C₄	C₄⋊C₄×D₄₇	1	8	yes	—
211	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
213	C₂×C₄	#13094020	1	2	yes	—
214	C₂×C₄	#13094022	1	2	yes	—
215	C₂×C₄	#13094023	1	8	yes	—
216	C₂×C₄	C₂×#10625336	1	8	yes	—
217	C₂×C₄	C₂×#10625335	1	8	yes	—
218	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
221	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
222	C₈	—	47	16	no	—
223	C₂²	#15688992	1	4	yes	—
224	C₂²	C₂×#13093955	1	8	yes	—
225	C₂²	#15689495	1	4	yes	—
226	C₂²	#15689497	1	4	yes	—
227	C₂²	#15689519	1	8	yes	—
228	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
229	C₂²	C₂×#13094020	1	1	yes	characteristic
230	C₂²	C₂×#13094022	1	1	yes	characteristic
231	C₂²	C₂²×#10625336	1	2	yes	—
232	C₂²	C₂²×#10625335	1	2	yes	—
233	C₄	—	8	4	no	—
234	C₄	—	376	4	no	—
235	C₄	#15688992	1	4	yes	—
236	C₄	C₂×#13093955	1	16	yes	complemented
237	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
238	C₄	C₂×#13094023	1	2	yes	—
239	C₂	#20243544	1	8	yes	direct product factor
240	C₂	C₂×#15688992	1	1	yes	characteristic
241	C₂	C₂²×#13093955	1	2	yes	—
242	C₂	C₂×#15689495	1	1	yes	characteristic
243	C₂	C₂×#15689497	1	1	yes	characteristic
244	C₂	C₂×#15689519	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
4. #20243544	C₂	direct (N × H)
11. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
31. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
70. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
83. C₂×C₄×C₄₇	#106	semidirect
99. C₄₇⋊C₈	C₂×C₄⋊C₄	semidirect
113. C₄×C₄₇	C₂×#106	semidirect
122. C₂×C₄₇	#1844	semidirect
142. C₄₇	C₂×#1844	semidirect
209. C₂×C₄	#13093955	semidirect
236. C₄	C₂×#13093955	semidirect
239. C₂	#20243544	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×#13093955	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688993	2
#20243544	2
C₂×C₄₇×#384	2
C₂×#1844	47

Subgroups — C2×#20243544 #26216675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243544 #26216675