Subgroups — C₂×#20241435 #26127515

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241435	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₇₆	—	2	1	no	—
6	C₂²×D₃₇₆	C₄	1	2	yes	—
7	C₂×#13094021	—	2	1	no	—
8	#15689500	—	2	1	no	—
9	#15689500	—	2	1	no	—
10	#15689500	C₄	1	2	yes	—
11	#15689500	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
13	#15688876	—	2	1	no	—
14	#15688876	C₂²	1	2	yes	complemented
15	C₂²×C₁₆×C₄₇	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×C₄⋊C₁₆	—	2	1	no	—
18	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	complemented
19	C₂²×D₁₈₈	—	2	2	no	—
20	C₂²×D₁₈₈	—	4	1	no	—
21	C₂²×D₁₈₈	—	4	1	no	—
22	C₂×D₃₇₆	—	2	2	no	—
23	C₂×D₃₇₆	—	2	2	no	—
24	C₂×D₃₇₆	—	2	2	no	—
25	C₂×D₃₇₆	—	4	4	no	—
26	C₂×D₃₇₆	—	4	2	no	—
27	C₂×D₃₇₆	—	4	2	no	—
28	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	4	1	no	—
30	#13094021	—	2	2	no	—
31	#13094021	—	2	2	no	—
32	#13094021	—	4	1	no	—
33	#13094021	—	4	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₄₇	—	2	2	no	—
40	C₂×C₁₆×C₄₇	—	4	2	no	—
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂³×D₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂×D₁₈₈	—	2	4	no	—
50	C₂×D₁₈₈	—	4	4	no	—
51	C₂×D₁₈₈	—	4	4	no	—
52	C₂×D₁₈₈	—	4	2	no	—
53	C₂×D₁₈₈	—	4	2	no	—
54	C₂×D₁₈₈	—	4	2	no	—
55	C₂×D₁₈₈	—	4	2	no	—
56	C₂×D₁₈₈	—	4	2	no	—
57	C₂×D₁₈₈	—	8	4	no	—
58	C₂×D₁₈₈	—	8	2	no	—
59	C₂×D₁₈₈	—	8	2	no	—
60	C₂×C₈×C₄₇	—	2	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	D₈	1	2	yes	—
64	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₄₇	D₈	1	1	yes	characteristic
69	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂×C₄₇⋊C₈	—	8	2	no	—
73	D₃₇₆	—	4	4	no	—
74	D₃₇₆	—	4	4	no	—
75	D₃₇₆	—	4	4	no	—
76	D₃₇₆	—	4	2	no	—
77	D₃₇₆	—	4	2	no	—
78	C₁₆×C₄₇	—	4	2	no	—
79	C₂³×C₄₇	#102	1	1	yes	characteristic
80	C₂²×D₄₇	—	4	4	no	—
81	C₂²×D₄₇	—	8	4	no	—
82	C₂²×D₄₇	—	8	4	no	—
83	C₂²×D₄₇	—	8	4	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	#100	1	2	yes	—
86	C₂×C₄×C₄₇	#102	1	2	yes	—
87	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
90	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
95	D₁₈₈	—	4	4	no	—
96	D₁₈₈	—	4	4	no	—
97	D₁₈₈	—	8	4	no	—
98	D₁₈₈	—	8	4	no	—
99	D₁₈₈	—	8	4	no	—
100	D₁₈₈	—	8	4	no	—
101	D₁₈₈	—	8	2	no	—
102	D₁₈₈	—	8	2	no	—
103	C₈×C₄₇	—	2	1	no	—
104	C₈×C₄₇	—	2	1	no	—
105	C₈×C₄₇	#102	1	2	yes	—
106	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
107	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₄₇⋊C₈	—	8	2	no	—
109	C₂×#1361	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	#331	1	2	yes	—
111	C₂²×C₄₇	#357	1	2	yes	—
112	C₂²×C₄₇	#361	1	2	yes	—
113	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
114	C₂×D₄₇	—	8	4	no	—
115	C₂×D₄₇	—	8	4	no	—
116	C₂×D₄₇	—	8	4	no	—
117	C₂×D₄₇	—	16	8	no	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	#331	1	2	yes	—
121	C₄×C₄₇	#357	1	2	yes	—
122	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
123	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
124	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
125	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
126	C₂×#493	—	47	1	no	—
127	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
128	C₂×#335	—	47	1	no	—
129	#1361	—	47	4	no	—
130	C₂×C₄₇	#1361	1	4	yes	complemented
131	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
132	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#361	1	1	yes	characteristic
134	D₄₇	—	16	4	no	—
135	C₂²×D₈	—	47	2	no	—
136	C₂²×D₈	—	94	1	no	—
137	C₂×#127	—	94	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	#493	—	47	2	no	—
140	#493	—	47	2	no	—
141	#493	—	94	1	no	—
142	#493	—	94	1	no	—
143	C₂×C₄⋊C₈	—	94	1	no	—
144	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
145	C₄⋊C₁₆	—	2	1	no	—
146	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	complemented
147	#335	—	47	2	no	—
148	#335	—	94	1	no	—
149	C₄₇	C₂×#1361	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	94	2	no	—
151	C₂²×D₄	—	188	1	no	—
152	C₂²×D₄	—	188	1	no	—
153	C₂×D₈	—	47	4	no	—
154	C₂×D₈	—	94	2	no	—
155	C₂×D₈	—	94	2	no	—
156	C₂×D₈	—	94	2	no	—
157	C₂×D₈	—	188	4	no	—
158	C₂×D₈	—	188	2	no	—
159	C₂×D₈	—	188	2	no	—
160	C₂²×C₈	—	188	1	no	—
161	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
163	#127	—	94	2	no	—
164	#127	—	94	2	no	—
165	#127	—	188	1	no	—
166	#127	—	188	1	no	—
167	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	4	2	no	—
170	C₄⋊C₈	—	94	2	no	—
171	C₄⋊C₈	—	188	1	no	—
172	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
173	C₄×C₈	D₁₈₈	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
176	C₂⁴	—	188	2	no	—
177	C₂×D₄	—	94	4	no	—
178	C₂×D₄	—	376	4	no	—
179	C₂×D₄	—	376	2	no	—
180	C₂×D₄	—	376	2	no	—
181	C₂×D₄	—	188	4	no	—
182	C₂×D₄	—	188	4	no	—
183	C₂×D₄	—	188	2	no	—
184	C₂×D₄	—	188	2	no	—
185	C₂×D₄	—	188	2	no	—
186	C₂×D₄	—	188	2	no	—
187	C₂×D₄	—	188	2	no	—
188	C₂²×C₄	#10625325	1	1	yes	characteristic
189	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
191	D₈	—	188	4	no	—
192	D₈	—	188	4	no	—
193	D₈	—	188	4	no	—
194	D₈	—	188	2	no	—
195	D₈	—	188	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	376	2	no	—
198	C₂×C₈	—	188	2	no	—
199	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
201	C₂×C₈	#10625334	1	2	yes	—
202	C₂×C₈	#10625336	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
205	C₄²	#10625325	1	1	yes	characteristic
206	C₄²	D₃₇₆	1	1	yes	characteristic
207	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₂³	—	376	4	no	—
211	C₂³	—	376	4	no	—
212	C₂³	—	376	4	no	—
213	C₂³	—	188	4	no	—
214	C₂³	#13093946	1	1	yes	characteristic
215	D₄	—	376	4	no	—
216	D₄	—	376	4	no	—
217	D₄	—	376	2	no	—
218	D₄	—	376	2	no	—
219	D₄	—	376	4	no	—
220	D₄	—	376	4	no	—
221	D₄	—	188	4	no	—
222	D₄	—	188	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	#13093940	1	2	yes	—
225	C₂×C₄	#13093944	1	2	yes	—
226	C₂×C₄	#13093946	1	1	yes	characteristic
227	C₂×C₄	#13093946	1	1	yes	characteristic
228	C₂×C₄	#13093946	1	1	yes	characteristic
229	C₂×C₄	D₇₅₂	1	2	yes	—
230	C₂×C₄	#13093969	1	2	yes	—
231	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
233	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	376	2	no	—
237	C₈	#13093944	1	2	yes	—
238	C₈	C₂×#10625334	1	1	yes	characteristic
239	C₈	C₂×#10625336	1	1	yes	characteristic
240	C₂²	—	376	4	no	—
241	C₂²	—	376	4	no	—
242	C₂²	—	376	4	no	—
243	C₂²	—	752	8	no	—
244	C₂²	#15688866	1	2	yes	—
245	C₂²	#15688925	1	2	yes	—
246	C₂²	#15688937	1	2	yes	—
247	C₂²	C₂×#13093946	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	#15688866	1	2	yes	—
251	C₄	#15688925	1	2	yes	—
252	C₄	C₂×#13093940	1	1	yes	characteristic
253	C₄	C₂×#13093944	1	1	yes	characteristic
254	C₄	C₂×D₇₅₂	1	1	yes	characteristic
255	C₄	C₂×#13093969	1	1	yes	characteristic
256	C₂	—	752	4	no	—
257	C₂	#20241435	1	4	yes	direct product factor
258	C₂	C₂×#15688866	1	1	yes	characteristic
259	C₂	C₂×#15688925	1	1	yes	characteristic
260	C₂	C₂×#15688937	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093946
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle	C₂³×C₄₇	#102
prime radical	C₂×#15689500	C₂
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#1361
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#15688876	C₂	semidirect
3. #20241435	C₂	direct (N × H)
4. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
14. #15688876	C₂²	semidirect
18. C₄₇×C₄⋊C₁₆	C₂²	semidirect
127. C₂×C₄⋊C₁₆	D₄₇	semidirect
130. C₂×C₄₇	#1361	semidirect
146. C₄⋊C₁₆	C₂×D₄₇	semidirect
149. C₄₇	C₂×#1361	semidirect
257. C₂	#20241435	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×#15689500	2
C₂×#15688876	2
#20241435	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×#1361	47

Subgroups — C2×#20241435 #26127515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241435 #26127515