Subgroups — C₂×#20241219 #26105694

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20241219	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	2	no	—
5	#15689176	—	2	2	no	—
6	#15689176	C₄	1	4	yes	—
7	#15689176	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
11	C₂³×C₈×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#97×C₄₇	—	2	2	no	—
14	C₂×#97×C₄₇	—	2	2	no	—
15	C₂×#97×C₄₇	C₄	1	2	yes	—
16	C₂×#97×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₄×C₄₇⋊C₈	—	2	4	no	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	—	2	4	no	—
30	C₂²×C₈×C₄₇	C₈	1	8	yes	complemented
31	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	#97×C₄₇	—	2	4	no	—
41	#97×C₄₇	—	2	4	no	—
42	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
57	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₄₇	—	2	8	no	—
62	C₂×C₈×C₄₇	—	2	8	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	C₂×C₈	1	8	yes	complemented
66	C₄²×C₄₇	C₄²	1	2	yes	—
67	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₇	C₂×D₄	1	2	yes	—
74	C₂×C₄₇⋊C₈	—	2	8	no	—
75	C₂×C₄₇⋊C₈	—	4	8	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	—	2	2	no	—
78	C₂³×C₄₇	#95	1	1	yes	characteristic
79	C₂³×C₄₇	#95	1	1	yes	characteristic
80	C₂³×C₄₇	C₄⋊C₈	1	4	yes	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	4	2	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	4	1	no	—
87	C₂×C₄×C₄₇	—	4	1	no	—
88	C₂×C₄×C₄₇	—	4	1	no	—
89	C₂×C₄×C₄₇	—	4	1	no	—
90	C₂×C₄×C₄₇	#95	1	2	yes	—
91	C₂×C₄×C₄₇	#95	1	2	yes	—
92	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₄₇	#97	1	2	yes	—
97	C₂×C₄×C₄₇	#97	1	2	yes	—
98	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
107	C₈×C₄₇	—	2	8	no	—
108	C₄₇⋊C₈	—	4	8	no	—
109	C₂×#1275	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	1	no	—
111	C₂²×C₄₇	—	2	1	no	—
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	2	8	no	—
115	C₂²×C₄₇	#336	1	2	yes	—
116	C₂²×C₄₇	#336	1	2	yes	—
117	C₂²×C₄₇	H(3,ℤ/4)	1	2	yes	—
118	C₂²×C₄₇	#338	1	2	yes	—
119	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	#336	1	2	yes	—
126	C₄×C₄₇	#336	1	2	yes	—
127	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
128	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
129	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
130	C₂×#431	—	47	2	no	—
131	C₂²×#97	D₄₇	1	1	yes	characteristic
132	#1275	—	47	4	no	—
133	C₂×C₄₇	—	2	2	no	—
134	C₂×C₄₇	—	2	2	no	—
135	C₂×C₄₇	#1275	1	4	yes	complemented
136	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
137	C₂×C₄₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#338	1	1	yes	characteristic
139	C₂³×C₈	Dic₄₇	1	2	yes	—
140	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
141	#431	—	47	4	no	—
142	#431	—	47	4	no	—
143	#431	—	94	2	no	—
144	#431	—	94	2	no	—
145	C₂×C₄⋊C₈	—	94	2	no	—
146	C₂×C₂²⋊C₈	—	94	2	no	—
147	C₂×#97	—	2	2	no	—
148	C₂×#97	—	2	2	no	—
149	C₂×#97	Dic₄₇	1	2	yes	—
150	C₂×#97	C₂×D₄₇	1	2	yes	—
151	C₂×C₄×C₈	—	47	4	no	—
152	C₄₇	C₂×#1275	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
154	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	94	4	no	—
160	C₂²×C₈	C₄₇⋊C₈	1	8	yes	complemented
161	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
162	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₄₇	1	2	yes	—
164	C₂×C₄²	C₄×D₄₇	1	2	yes	—
165	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₄⋊C₈	—	94	4	no	—
171	C₄⋊C₈	—	94	4	no	—
172	C₂²⋊C₈	—	94	4	no	—
173	C₂²⋊C₈	—	94	4	no	—
174	#97	—	2	4	no	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	47	8	no	—
177	C₄×C₈	—	94	4	no	—
178	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
182	C₂²×C₄	#10625329	1	4	yes	—
183	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
184	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
186	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
191	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
192	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	94	8	no	—
202	C₂×C₈	—	188	8	no	—
203	C₂×C₈	C₂×C₄₇⋊C₈	1	8	yes	complemented
204	C₄²	C₄×Dic₄₇	1	2	yes	—
205	C₄²	Dic₄₇⋊C₄	1	2	yes	—
206	C₄²	Dic₄₇⋊C₄	1	2	yes	—
207	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
208	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
209	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
210	C₄²	C₂×C₄×D₄₇	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₄₇⋊#95	1	1	yes	characteristic
215	C₂³	C₄₇⋊#95	1	1	yes	characteristic
216	C₂³	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
217	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	C₄₇⋊#95	1	2	yes	—
227	C₂×C₄	C₄₇⋊#95	1	2	yes	—
228	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
229	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
230	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
231	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
232	C₂×C₄	#13093935	1	2	yes	—
233	C₂×C₄	#13093935	1	2	yes	—
234	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
235	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
236	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
237	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
241	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
242	C₂×C₄	C₂×#10625329	1	2	yes	—
243	C₈	—	2	8	no	—
244	C₈	—	188	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	#15688878	1	2	yes	—
251	C₂²	#15688878	1	2	yes	—
252	C₂²	#15688880	1	2	yes	—
253	C₂²	C₄₇⋊#338	1	2	yes	—
254	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
255	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	#15688878	1	2	yes	—
261	C₄	#15688878	1	2	yes	—
262	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
263	C₄	C₂×#13093935	1	1	yes	characteristic
264	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	—	2	2	no	—
267	C₂	#20241219	1	4	yes	direct product factor
268	C₂	C₂×#15688878	1	1	yes	characteristic
269	C₂	C₂×#15688880	1	1	yes	characteristic
270	C₂	C₂×C₄₇⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂×#1275
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #20241219	C₂	direct (N × H)
30. C₂²×C₈×C₄₇	C₈	semidirect
65. C₂×C₈×C₄₇	C₂×C₈	semidirect
135. C₂×C₄₇	#1275	semidirect
152. C₄₇	C₂×#1275	semidirect
160. C₂²×C₈	C₄₇⋊C₈	semidirect
203. C₂×C₈	C₂×C₄₇⋊C₈	semidirect
267. C₂	#20241219	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×#15689176	2
#20241219	2
C₂²×#97×C₄₇	2
C₂×#1275	47

Subgroups — C2×#20241219 #26105694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241219 #26105694