Subgroups — C₂×#20257198 #27124462

Order 12032.

6826 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20257198	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈×Dic₉₄	C₂	1	4	yes	maximal, complemented
3	C₂×#15689852	C₂	1	4	yes	maximal, complemented
4	C₂×#15689849	C₂	1	8	yes	maximal, complemented
5	C₂×#15689754	C₂	1	6	yes	maximal
6	C₂×#15689680	C₂	1	6	yes	maximal
7	C₂×#15689601	C₂	1	2	yes	maximal
8	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	Q₈×Dic₉₄	C₂²	1	64	yes	complemented
10	#15689852	C₂²	1	64	yes	complemented
11	#15689849	C₂²	1	128	yes	complemented
12	#15689754	C₂²	1	96	yes	—
13	#15689680	C₂²	1	96	yes	—
14	C₂×#13094024	C₂²	1	6	yes	—
15	C₂×#13094024	C₂²	1	6	yes	—
16	C₂×#13094023	C₂²	1	24	yes	—
17	C₂×#13094016	C₂²	1	12	yes	—
18	C₂×#13094015	C₂²	1	24	yes	—
19	C₂×#13094003	C₂²	1	12	yes	—
20	C₂×#13094003	C₂²	1	4	yes	—
21	#15689601	C₂²	1	32	yes	—
22	C₂×Q₈×Dic₄₇	C₂²	1	8	yes	—
23	C₂×Q₈×Dic₄₇	C₂²	1	8	yes	—
24	C₂×C₄×Dic₉₄	C₂²	1	12	yes	—
25	C₂×C₄×Dic₉₄	C₂²	1	12	yes	—
26	C₂×#13093983	C₂²	1	12	yes	—
27	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₄₇	C₂²	1	16	yes	—
29	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
31	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
32	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
33	C₂²×Dic₉₄	C₂³	1	4	yes	—
34	C₂²×Dic₉₄	C₂³	1	12	yes	—
35	#13094024	C₂³	1	48	yes	—
36	#13094024	C₂³	1	48	yes	—
37	#13094023	C₂³	1	192	yes	—
38	#13094016	C₂³	1	96	yes	—
39	#13094015	C₂³	1	192	yes	—
40	#13094003	C₂³	1	96	yes	—
41	#13094003	C₂³	1	32	yes	—
42	Q₈×Dic₄₇	C₂³	1	64	yes	—
43	Q₈×Dic₄₇	C₂³	1	64	yes	—
44	C₄×Dic₉₄	C₂³	1	96	yes	—
45	C₄×Dic₉₄	C₂³	1	96	yes	—
46	#13093983	C₂³	1	96	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
50	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
51	C₂×Dic₄₇⋊C₄	C₂³	1	8	yes	—
52	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
53	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
54	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
55	C₂²×Q₈×C₄₇	Q₈	1	16	yes	complemented
56	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
57	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
58	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
61	C₄×Q₈×C₄₇	C₂³	1	32	yes	—
62	C₄×Q₈×C₄₇	C₂³	1	32	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	24	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	24	yes	—
68	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	C₂×Q₈	1	32	yes	—
70	C₂×Q₈×C₄₇	C₂×Q₈	1	64	yes	complemented
71	C₂×Q₈×C₄₇	C₂⁴	1	16	yes	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	4	6	no	—
74	C₂²×C₄×C₄₇	C₂×Q₈	1	24	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
83	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
84	C₂×Dic₉₄	—	2	16	no	—
85	C₂×Dic₉₄	—	4	24	no	—
86	C₂×Dic₉₄	C₂⁴	1	16	yes	—
87	C₂×Dic₉₄	C₂⁴	1	48	yes	—
88	C₄⋊C₄×C₄₇	C₂⁴	1	48	yes	—
89	C₄²×C₄₇	C₂⁴	1	24	yes	—
90	C₄²×C₄₇	C₂⁴	1	24	yes	—
91	C₂²⋊Dic₄₇	C₂⁴	1	32	yes	—
92	C₄⋊Dic₄₇	C₂⁴	1	48	yes	—
93	C₄⋊Dic₄₇	C₂⁴	1	48	yes	—
94	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
95	C₄⋊Dic₄₇	C₂⁴	1	48	yes	—
96	Dic₄₇⋊C₄	C₂⁴	1	32	yes	—
97	Dic₄₇⋊C₄	C₂⁴	1	96	yes	—
98	C₄×Dic₄₇	C₂⁴	1	96	yes	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
102	C₂³×C₄₇	C₂⁵	1	2	yes	—
103	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
104	Q₈×C₄₇	C₂²×Q₈	1	64	yes	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	12	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	—	4	12	no	—
109	C₂×C₄×C₄₇	C₂²×Q₈	1	24	yes	—
110	C₂×C₄×C₄₇	C₂²×Q₈	1	48	yes	—
111	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
112	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
113	C₂×C₄×C₄₇	C₂⁵	1	12	yes	—
114	C₂×C₄×C₄₇	C₂⁵	1	12	yes	—
115	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
116	C₂×Dic₄₇	—	2	16	no	—
117	C₂×Dic₄₇	—	2	16	no	—
118	C₂×Dic₄₇	C₂⁵	1	16	yes	—
119	C₂×Dic₄₇	C₂⁵	1	16	yes	—
120	Dic₉₄	—	2	32	no	—
121	Dic₉₄	—	4	48	no	—
122	C₂×#3502	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	8	no	—
126	C₂²×C₄₇	C₂³×Q₈	1	4	yes	—
127	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
128	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
129	C₂²×C₄₇	#585	1	2	yes	—
130	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	12	no	—
133	C₄×C₄₇	C₂³×Q₈	1	24	yes	—
134	C₄×C₄₇	C₂×2₋¹⁺⁴	1	4	yes	—
135	Dic₄₇	—	2	16	no	—
136	Dic₄₇	—	2	16	no	—
137	#3502	—	47	32	no	—
138	C₂×Q₈²	—	47	4	no	—
139	C₂×#557	—	47	8	no	—
140	C₂×#557	—	47	4	no	—
141	C₂×#544	—	47	6	no	—
142	C₂×#533	—	47	6	no	—
143	C₂×#519	—	47	2	no	—
144	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
145	C₂×C₄₇	—	2	4	no	—
146	C₂×C₄₇	#3502	1	4	yes	complemented
147	C₂×C₄₇	C₂⁴×Q₈	1	1	yes	characteristic
148	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
149	C₂×C₄₇	C₂×#585	1	1	yes	characteristic
150	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
151	Q₈²	—	47	64	no	—
152	#557	—	47	128	no	—
153	#557	—	47	64	no	—
154	#544	—	47	96	no	—
155	#533	—	47	96	no	—
156	C₂×C₄⋊Q₈	—	47	24	no	—
157	C₂×C₄⋊Q₈	—	47	6	no	—
158	C₂×C₄⋊Q₈	—	47	6	no	—
159	C₂×#125	—	47	24	no	—
160	C₂×#125	—	47	12	no	—
161	C₂×C₂²⋊Q₈	—	47	4	no	—
162	C₂×C₂²⋊Q₈	—	47	12	no	—
163	#519	—	47	32	no	—
164	C₂×C₄×Q₈	—	47	8	no	—
165	C₂×C₄×Q₈	—	47	8	no	—
166	C₂×C₄×Q₈	—	47	12	no	—
167	C₂×C₄×Q₈	—	47	12	no	—
168	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
169	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
170	C₂×C₄×Q₈	C₂×D₄₇	1	16	yes	—
171	C₂×#117	—	47	12	no	—
172	C₂²×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
173	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
174	C₄₇	C₂×#3502	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	47	4	no	—
176	C₂²×Q₈	—	47	12	no	—
177	C₂²×Q₈	Dic₉₄	1	16	yes	complemented
178	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
179	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
180	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
183	C₄⋊Q₈	—	47	192	no	—
184	C₄⋊Q₈	—	47	48	no	—
185	C₄⋊Q₈	—	47	48	no	—
186	#125	—	47	192	no	—
187	#125	—	47	96	no	—
188	C₂²⋊Q₈	—	47	32	no	—
189	C₂²⋊Q₈	—	47	96	no	—
190	C₄×Q₈	—	47	64	no	—
191	C₄×Q₈	—	47	64	no	—
192	C₄×Q₈	—	47	96	no	—
193	C₄×Q₈	—	47	96	no	—
194	C₄×Q₈	C₂²×D₄₇	1	32	yes	—
195	C₄×Q₈	C₂²×D₄₇	1	32	yes	—
196	#117	—	47	96	no	—
197	C₂×C₄⋊C₄	—	47	12	no	—
198	C₂×C₄⋊C₄	—	47	12	no	—
199	C₂×C₄⋊C₄	—	47	8	no	—
200	C₂×C₄⋊C₄	—	47	4	no	—
201	C₂×C₄⋊C₄	—	47	12	no	—
202	C₂×C₄⋊C₄	—	47	24	no	—
203	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
204	C₂×C₄⋊C₄	C₂²×D₄₇	1	24	yes	—
205	C₂×C₂²⋊C₄	—	47	8	no	—
206	C₂×C₄²	—	47	24	no	—
207	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
209	C₂×C₄²	C₂²×D₄₇	1	24	yes	—
210	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
211	C₂×Q₈	—	47	16	no	—
212	C₂×Q₈	—	47	48	no	—
213	C₂×Q₈	—	94	16	no	—
214	C₂×Q₈	—	188	24	no	—
215	C₂×Q₈	C₂×Dic₉₄	1	32	yes	—
216	C₂×Q₈	C₂×Dic₉₄	1	64	yes	complemented
217	C₂×Q₈	C₂³×D₄₇	1	16	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	6	no	—
220	C₂²×C₄	—	47	8	no	—
221	C₂²×C₄	—	47	8	no	—
222	C₂²×C₄	C₂×Dic₉₄	1	24	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
226	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
227	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
229	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
230	C₄⋊C₄	—	47	32	no	—
231	C₄⋊C₄	—	47	96	no	—
232	C₄⋊C₄	—	47	48	no	—
233	C₄⋊C₄	—	47	48	no	—
234	C₄⋊C₄	—	47	16	no	—
235	C₄⋊C₄	—	47	48	no	—
236	C₄⋊C₄	C₂³×D₄₇	1	48	yes	—
237	C₂²⋊C₄	—	47	32	no	—
238	C₄²	—	47	96	no	—
239	C₄²	C₂³×D₄₇	1	24	yes	—
240	C₄²	C₂³×D₄₇	1	24	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂²×Dic₉₄	1	4	yes	—
244	C₂³	C₂⁴×D₄₇	1	2	yes	—
245	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
246	Q₈	—	94	32	no	—
247	Q₈	—	188	48	no	—
248	Q₈	C₂²×Dic₉₄	1	64	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	12	no	—
253	C₂×C₄	—	47	16	no	—
254	C₂×C₄	—	47	16	no	—
255	C₂×C₄	—	94	16	no	—
256	C₂×C₄	—	94	16	no	—
257	C₂×C₄	C₂²×Dic₉₄	1	24	yes	—
258	C₂×C₄	C₂²×Dic₉₄	1	48	yes	—
259	C₂×C₄	#13094072	1	4	yes	—
260	C₂×C₄	C₂⁴×D₄₇	1	2	yes	—
261	C₂×C₄	C₂⁴×D₄₇	1	12	yes	—
262	C₂×C₄	C₂⁴×D₄₇	1	12	yes	—
263	C₂×C₄	C₂⁴×D₄₇	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	C₂³×Dic₉₄	1	4	yes	—
268	C₂²	C₂³×Dic₉₄	1	2	yes	—
269	C₂²	C₂×#13094072	1	2	yes	—
270	C₂²	#15689959	1	2	yes	—
271	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	12	no	—
274	C₄	—	94	16	no	—
275	C₄	—	94	16	no	—
276	C₄	C₂³×Dic₉₄	1	24	yes	—
277	C₄	C₂×#13094072	1	4	yes	—
278	C₂	—	2	4	no	—
279	C₂	#20257198	1	4	yes	direct product factor
280	C₂	C₂⁴×Dic₉₄	1	1	yes	characteristic
281	C₂	C₂²×#13094072	1	1	yes	characteristic
282	C₂	C₂×#15689959	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#3502
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20257198	C₂	direct (N × H)
2. C₂×Q₈×Dic₉₄	C₂	semidirect
3. C₂×#15689852	C₂	semidirect
4. C₂×#15689849	C₂	semidirect
9. Q₈×Dic₉₄	C₂²	semidirect
10. #15689852	C₂²	semidirect
11. #15689849	C₂²	semidirect
55. C₂²×Q₈×C₄₇	Q₈	semidirect
70. C₂×Q₈×C₄₇	C₂×Q₈	semidirect
146. C₂×C₄₇	#3502	semidirect
174. C₄₇	C₂×#3502	semidirect
177. C₂²×Q₈	Dic₉₄	semidirect
216. C₂×Q₈	C₂×Dic₉₄	semidirect
279. C₂	#20257198	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20257198	2
C₂×Q₈×Dic₉₄	2
C₂×#15689852	2
C₂×#15689849	2
C₂×#15689754	2
C₂×#15689680	2
C₂×#15689601	2
C₂²×C₄×Q₈×C₄₇	2
C₂×#3502	47

Subgroups — C2×#20257198 #27124462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20257198 #27124462