Subgroups — C₂×#20243551 #26216684

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093955	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688988	C₂	1	2	yes	maximal
5	#20243551	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093958	—	2	4	no	—
11	C₂×#13093958	C₂²	1	4	yes	—
12	C₂×#13093955	—	2	8	no	—
13	C₂×#13093955	C₂²	1	8	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
15	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
16	#15688988	—	2	4	no	—
17	#15688988	C₂²	1	8	yes	—
18	C₂×C₄₇⋊#95	—	2	4	no	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×#107×C₄₇	—	2	4	no	—
22	C₂×#107×C₄₇	C₂²	1	4	yes	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂³×Dic₄₇	D₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
37	#13093958	—	2	8	no	—
38	#13093955	—	2	16	no	—
39	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
40	C₄₇⋊#95	—	2	16	no	—
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₄₇	D₄	1	4	yes	—
45	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
51	#107×C₄₇	—	2	8	no	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	8	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	4	4	no	—
68	C₂²×Dic₄₇	—	4	8	no	—
69	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
70	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	4	4	no	—
81	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
82	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
84	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#127	1	2	yes	—
90	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
91	C₂³×C₄₇	#135	1	4	yes	—
92	C₂³×C₄₇	#135	1	4	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	8	no	—
96	C₂×C₄×C₄₇	—	8	8	no	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₄₇	#127	1	2	yes	—
101	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	2	16	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	8	no	—
115	C₂×Dic₄₇	—	8	8	no	—
116	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
117	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
118	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
119	C₂×#1846	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	#384	1	4	yes	—
121	C₂²×C₄₇	#427	1	8	yes	—
122	C₂²×C₄₇	#495	1	4	yes	—
123	C₂²×C₄₇	#495	1	4	yes	—
124	C₂²×C₄₇	#499	1	8	yes	—
125	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#135	1	2	yes	—
129	C₂²×C₄₇	C₂×#135	1	2	yes	—
130	C₄×C₄₇	—	8	4	no	—
131	C₄×C₄₇	#384	1	4	yes	—
132	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	Dic₄₇	—	2	8	no	—
137	Dic₄₇	—	8	4	no	—
138	C₂²×#107	—	47	1	no	—
139	C₂²×#107	D₄₇	1	1	yes	characteristic
140	C₂²×#106	—	47	2	no	—
141	C₂²×C₄×C₈	—	47	1	no	—
142	C₂×#382	—	47	2	no	—
143	#1846	—	47	8	no	—
144	C₂×C₄₇	#1846	1	8	yes	complemented
145	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#427	1	2	yes	—
147	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
149	C₂×C₄₇	C₂×#499	1	2	yes	—
150	C₂³×C₈	—	47	1	no	—
151	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	—	47	2	no	—
153	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
154	C₂²×C₄²	—	47	1	no	—
155	C₂×#107	—	2	4	no	—
156	C₂×#107	—	47	4	no	—
157	C₂×#107	—	94	4	no	—
158	C₂×#107	C₂×D₄₇	1	4	yes	—
159	C₂×#106	—	47	8	no	—
160	C₂×#106	—	94	8	no	—
161	C₂×C₄×C₈	—	47	8	no	—
162	C₂×C₄×C₈	—	47	4	no	—
163	#382	—	47	8	no	—
164	#382	—	94	4	no	—
165	C₂×#95	—	94	4	no	—
166	C₄₇	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	—	47	2	no	—
169	C₂³×C₄	—	94	2	no	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	—	47	4	no	—
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	47	2	no	—
175	C₂²×C₈	Dic₉₄	1	4	yes	—
176	C₂²×C₈	C₄×D₄₇	1	4	yes	—
177	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
178	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	47	4	no	—
183	C₂×C₄⋊C₄	—	94	4	no	—
184	C₂×C₄⋊C₄	—	94	4	no	—
185	C₂×C₄⋊C₄	—	94	2	no	—
186	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
187	C₂×C₄²	—	47	4	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	47	4	no	—
190	C₂×C₄²	—	47	2	no	—
191	#107	—	2	8	no	—
192	#107	—	94	8	no	—
193	#106	—	94	16	no	—
194	C₄×C₈	—	47	16	no	—
195	#95	—	94	16	no	—
196	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	47	8	no	—
201	C₂²×C₄	—	47	4	no	—
202	C₂²×C₄	—	94	4	no	—
203	C₂²×C₄	—	94	4	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	—	188	4	no	—
206	C₂²×C₄	—	188	8	no	—
207	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
209	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₄₇	1	2	yes	—
211	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
212	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	C₂×C₈	—	47	16	no	—
216	C₂×C₈	—	47	4	no	—
217	C₂×C₈	—	47	4	no	—
218	C₂×C₈	—	47	4	no	—
219	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	—
220	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
221	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
222	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	94	4	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₄⋊C₄	—	188	4	no	—
229	C₄²	—	47	8	no	—
230	C₄²	—	47	4	no	—
231	C₄²	—	47	4	no	—
232	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
233	C₂³	#13094020	1	2	yes	—
234	C₂³	#13094023	1	2	yes	—
235	C₂³	#13094043	1	4	yes	—
236	C₂³	#13094047	1	4	yes	—
237	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	8	8	no	—
241	C₂×C₄	—	47	8	no	—
242	C₂×C₄	—	94	4	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	—	94	16	no	—
245	C₂×C₄	—	376	8	no	—
246	C₂×C₄	—	188	8	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
249	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
250	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
251	C₂×C₄	#13094020	1	2	yes	—
252	C₂×C₄	#13094022	1	4	yes	—
253	C₂×C₄	#13094023	1	2	yes	—
254	C₂×C₄	#13094023	1	4	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
261	C₈	—	47	8	no	—
262	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	—
263	C₂²	#15688992	1	4	yes	—
264	C₂²	#15689160	1	8	yes	—
265	C₂²	#15689506	1	4	yes	—
266	C₂²	#15689508	1	4	yes	—
267	C₂²	#15689524	1	8	yes	—
268	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
269	C₂²	C₂×#13094020	1	1	yes	characteristic
270	C₂²	C₂×#13094023	1	1	yes	characteristic
271	C₂²	C₂×#13094043	1	2	yes	—
272	C₂²	C₂×#13094047	1	2	yes	—
273	C₄	—	8	4	no	—
274	C₄	—	94	8	no	—
275	C₄	—	376	4	no	—
276	C₄	#15688992	1	4	yes	—
277	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
278	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
279	C₄	C₂×#13094022	1	1	yes	characteristic
280	C₄	C₂×#13094023	1	1	yes	characteristic
281	C₂	#20243551	1	8	yes	direct product factor
282	C₂	C₂×#15688992	1	1	yes	characteristic
283	C₂	C₂×#15689160	1	2	yes	—
284	C₂	C₂×#15689506	1	1	yes	characteristic
285	C₂	C₂×#15689508	1	1	yes	characteristic
286	C₂	C₂×#15689524	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	C₂×#1846
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
5. #20243551	C₂	direct (N × H)
14. C₂×C₈×Dic₄₇	C₄	semidirect
39. C₈×Dic₄₇	C₂×C₄	semidirect
144. C₂×C₄₇	#1846	semidirect
166. C₄₇	C₂×#1846	semidirect
281. C₂	#20243551	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×#13093958	2
C₂²×#13093955	2
C₂²×C₈×Dic₄₇	2
C₂×#15688988	2
#20243551	2
C₂²×#107×C₄₇	2
C₂×#1846	47

Subgroups — C2×#20243551 #26216684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243551 #26216684