Subgroups — C₂×#20246108 #26375075

Order 12032.

2958 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₄₇	C₂	1	2	yes	maximal
2	#20246108	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689030	C₂	1	2	yes	maximal
4	C₂×#15689009	C₂	1	8	yes	maximal
5	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₄₇	—	2	16	no	—
9	C₂×Q₈×Dic₄₇	C₂²	1	16	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
12	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
15	#15689030	C₂²	1	16	yes	—
16	#15689009	C₂²	1	64	yes	—
17	#15688989	C₂²	1	8	yes	—
18	#15688969	C₂²	1	8	yes	—
19	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
20	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
21	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
22	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₄₇	C₄	1	16	yes	complemented
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	8	yes	—
25	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂³×Dic₄₇	C₂³	1	2	yes	—
28	C₂³×Dic₄₇	C₂³	1	4	yes	—
29	C₂³×Dic₄₇	C₂³	1	2	yes	—
30	Q₈×Dic₄₇	—	2	64	no	—
31	C₂×C₄⋊Dic₄₇	—	2	8	no	—
32	C₂×C₄⋊Dic₄₇	—	2	16	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
35	C₂×C₄×Dic₄₇	—	2	4	no	—
36	C₂×C₄×Dic₄₇	—	2	16	no	—
37	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
41	C₄₇⋊#95	C₂³	1	16	yes	—
42	C₄₇⋊#95	C₂³	1	16	yes	—
43	C₄₇⋊#95	C₂³	1	32	yes	—
44	C₂²×Q₈×C₄₇	—	2	8	no	—
45	C₂²×Q₈×C₄₇	—	2	8	no	—
46	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
47	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂×Q₈×C₄₇	—	2	64	no	—
58	C₂×Q₈×C₄₇	—	2	16	no	—
59	C₂×Q₈×C₄₇	—	2	16	no	—
60	C₂×Q₈×C₄₇	C₂²×C₄	1	32	yes	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	8	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₇	#41	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
72	C₂²×Dic₄₇	—	2	8	no	—
73	C₂²×Dic₄₇	—	2	8	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	16	no	—
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₂²×Dic₄₇	C₂×D₄	1	8	yes	—
78	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
79	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
80	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
81	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
83	C₄⋊Dic₄₇	—	2	16	no	—
84	C₄⋊Dic₄₇	—	2	32	no	—
85	C₄×Dic₄₇	—	2	8	no	—
86	C₄×Dic₄₇	—	2	32	no	—
87	C₄×Dic₄₇	C₂×D₄	1	16	yes	—
88	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
90	C₂³×C₄₇	C₂×#41	1	2	yes	—
91	C₂³×C₄₇	2₋¹⁺⁴	1	8	yes	—
92	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
93	Q₈×C₄₇	—	2	64	no	—
94	C₂×C₄×C₄₇	—	2	32	no	—
95	C₂×C₄×C₄₇	—	2	16	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	C₄×D₄	1	16	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
105	C₂×Dic₄₇	—	2	16	no	—
106	C₂×Dic₄₇	—	2	8	no	—
107	C₂×Dic₄₇	—	2	8	no	—
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	16	no	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	—	4	16	no	—
112	C₂×Dic₄₇	—	4	8	no	—
113	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
114	C₂×#2327	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	C₂×C₄×D₄	1	4	yes	—
116	C₂²×C₄₇	#519	1	8	yes	—
117	C₂²×C₄₇	#536	1	8	yes	—
118	C₂²×C₄₇	#541	1	8	yes	—
119	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	4	yes	—
123	C₄×C₄₇	—	2	16	no	—
124	C₄×C₄₇	—	2	8	no	—
125	C₄×C₄₇	C₂×C₄×D₄	1	8	yes	—
126	Dic₄₇	—	2	8	no	—
127	Dic₄₇	—	4	8	no	—
128	Dic₄₇	—	4	4	no	—
129	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
130	C₂²×C₄×Q₈	—	47	2	no	—
131	#2327	—	47	16	no	—
132	C₂×#391	—	47	2	no	—
133	C₂×#387	—	47	8	no	—
134	C₂×#382	—	47	1	no	—
135	C₂×#376	—	47	1	no	—
136	C₂×C₄₇	#2327	1	8	yes	complemented
137	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#519	1	2	yes	—
139	C₂×C₄₇	C₂×#536	1	2	yes	—
140	C₂×C₄₇	C₂×#541	1	2	yes	—
141	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
142	C₂×C₄⋊Q₈	Dic₄₇	1	16	yes	complemented
143	C₂×C₄⋊Q₈	C₂×D₄₇	1	8	yes	—
144	C₂×C₄×Q₈	—	47	16	no	—
145	C₂×C₄×Q₈	—	94	16	no	—
146	C₂²×C₄⋊C₄	—	47	2	no	—
147	C₂²×C₄⋊C₄	—	47	4	no	—
148	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
149	C₂²×C₄²	—	47	1	no	—
150	C₂²×C₄²	—	47	4	no	—
151	C₂²×C₄²	—	47	1	no	—
152	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
153	#391	—	47	16	no	—
154	#387	—	47	64	no	—
155	#382	—	47	8	no	—
156	#376	—	47	8	no	—
157	C₂×#95	—	47	4	no	—
158	C₂×#95	—	47	4	no	—
159	C₂×#95	—	47	8	no	—
160	C₄₇	C₂×#2327	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	—	2	8	no	—
163	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
164	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
165	C₂³×C₄	—	47	2	no	—
166	C₂³×C₄	—	47	4	no	—
167	C₂³×C₄	—	47	2	no	—
168	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₄⋊Q₈	C₂×Dic₄₇	1	64	yes	complemented
172	C₄×Q₈	—	94	64	no	—
173	C₂×C₄⋊C₄	—	47	8	no	—
174	C₂×C₄⋊C₄	—	47	16	no	—
175	C₂×C₄⋊C₄	—	94	16	no	—
176	C₂×C₄⋊C₄	—	94	8	no	—
177	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
178	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
179	C₂×C₄²	—	47	8	no	—
180	C₂×C₄²	—	47	4	no	—
181	C₂×C₄²	—	47	16	no	—
182	C₂×C₄²	—	47	4	no	—
183	C₂×C₄²	—	94	16	no	—
184	C₂×C₄²	—	94	4	no	—
185	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
186	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
187	#95	—	47	16	no	—
188	#95	—	47	16	no	—
189	#95	—	47	32	no	—
190	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
191	C₂×Q₈	—	2	64	no	—
192	C₂×Q₈	—	2	16	no	—
193	C₂×Q₈	—	2	16	no	—
194	C₂×Q₈	C₂²×Dic₄₇	1	32	yes	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	47	8	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	4	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	—	94	8	no	—
203	C₂²×C₄	—	94	8	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	—	94	8	no	—
206	C₂²×C₄	—	94	16	no	—
207	C₂²×C₄	D₄×D₄₇	1	4	yes	—
208	C₂²×C₄	#10625352	1	4	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
210	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
214	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
215	C₄⋊C₄	—	94	32	no	—
216	C₄⋊C₄	—	94	16	no	—
217	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
218	C₄²	—	47	16	no	—
219	C₄²	—	94	32	no	—
220	C₄²	—	94	8	no	—
221	C₄²	C₂²×Dic₄₇	1	16	yes	—
222	C₂³	C₂³×Dic₄₇	1	2	yes	—
223	C₂³	C₂×D₄×D₄₇	1	2	yes	—
224	C₂³	C₂×#10625352	1	2	yes	—
225	C₂³	#13094072	1	8	yes	—
226	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
227	Q₈	—	2	64	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	47	8	no	—
234	C₂×C₄	—	94	16	no	—
235	C₂×C₄	—	94	8	no	—
236	C₂×C₄	—	94	8	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	94	16	no	—
239	C₂×C₄	—	94	8	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	—	188	16	no	—
242	C₂×C₄	D₄×Dic₄₇	1	16	yes	—
243	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
244	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
245	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
246	C₂×C₄	C₂×D₄×D₄₇	1	4	yes	—
247	C₂×C₄	C₂×#10625352	1	4	yes	—
248	C₂²	C₂×D₄×Dic₄₇	1	4	yes	—
249	C₂²	#15689601	1	8	yes	—
250	C₂²	#15689698	1	8	yes	—
251	C₂²	#15689734	1	8	yes	—
252	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
253	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
254	C₂²	C₂²×#10625352	1	1	yes	characteristic
255	C₂²	C₂×#13094072	1	4	yes	—
256	C₄	—	2	16	no	—
257	C₄	—	2	8	no	—
258	C₄	—	94	8	no	—
259	C₄	—	188	8	no	—
260	C₄	—	188	4	no	—
261	C₄	C₂×D₄×Dic₄₇	1	8	yes	—
262	C₂	#20246108	1	8	yes	direct product factor
263	C₂	C₂²×D₄×Dic₄₇	1	1	yes	characteristic
264	C₂	C₂×#15689601	1	2	yes	—
265	C₂	C₂×#15689698	1	2	yes	—
266	C₂	C₂×#15689734	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#2327
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20246108	C₂	direct (N × H)
23. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
51. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
136. C₂×C₄₇	#2327	semidirect
142. C₂×C₄⋊Q₈	Dic₄₇	semidirect
160. C₄₇	C₂×#2327	semidirect
171. C₄⋊Q₈	C₂×Dic₄₇	semidirect
262. C₂	#20246108	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×Q₈×Dic₄₇	2
#20246108	2
C₂×#15689030	2
C₂×#15689009	2
C₂×#15688989	2
C₂×#15688969	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#2327	47

Subgroups — C2×#20246108 #26375075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246108 #26375075