Subgroups — C₂×#20242207 #26185671

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₄₇⋊C₈	C₂	1	2	yes	maximal
2	#20242207	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
5	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	Q₈×C₄₇⋊C₈	—	2	4	no	—
8	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15688874	C₂²	1	4	yes	—
15	Q₈⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
16	#15688844	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₄₇	C₄	1	4	yes	—
20	C₄×Q₁₆×C₄₇	C₂²	1	4	yes	—
21	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
22	C₂×#103×C₄₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
29	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
30	C₄×C₄₇⋊C₈	—	2	1	no	—
31	C₄×C₄₇⋊C₈	—	2	8	no	—
32	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
33	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×Q₁₆×C₄₇	C₈	1	8	yes	complemented
37	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₄₇	—	2	2	no	—
41	C₄×Q₈×C₄₇	—	2	2	no	—
42	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#107×C₄₇	C₂×C₄	1	4	yes	—
49	#103×C₄₇	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂×Q₈×C₄₇	—	2	2	no	—
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	C₂×C₈	1	8	yes	—
55	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
61	Q₁₆×C₄₇	C₂×C₈	1	16	yes	complemented
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄⋊C₄×C₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	2	2	no	—
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₄₇	—	2	4	no	—
71	C₄²×C₄₇	—	2	4	no	—
72	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	#41	1	1	yes	characteristic
75	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
82	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₄₇	—	2	4	no	—
84	Q₈×C₄₇	—	2	8	no	—
85	Q₈×C₄₇	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₄₇	#131	1	2	yes	—
96	C₂×C₄×C₄₇	#137	1	4	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₈×C₄₇	—	2	2	no	—
101	C₈×C₄₇	C₂²×C₈	1	4	yes	—
102	C₄₇⋊C₈	—	2	4	no	—
103	C₄₇⋊C₈	—	4	2	no	—
104	C₄₇⋊C₈	—	4	4	no	—
105	C₂×#1579	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	C₈×D₄	1	2	yes	—
107	C₂²×C₄₇	#441	1	2	yes	—
108	C₂²×C₄₇	#444	1	2	yes	—
109	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₄₇	—	2	8	no	—
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	2	1	no	—
113	C₄×C₄₇	—	4	4	no	—
114	C₄×C₄₇	C₈×D₄	1	2	yes	—
115	C₄×C₄₇	#441	1	2	yes	—
116	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
117	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
118	C₄×C₄₇	C₂×#137	1	2	yes	—
119	C₂×C₈×Q₈	—	47	2	no	—
120	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
121	#1579	—	47	8	no	—
122	C₂×#334	—	47	1	no	—
123	C₂×Q₈⋊C₈	—	47	2	no	—
124	C₂×#322	—	47	1	no	—
125	C₂×C₄₇	#1579	1	4	yes	complemented
126	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
128	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
131	C₈×Q₈	—	47	8	no	—
132	C₈×Q₈	—	94	4	no	—
133	C₄×Q₁₆	Dic₄₇	1	4	yes	—
134	C₄×Q₁₆	C₂×D₄₇	1	4	yes	—
135	C₂×#107	Dic₄₇	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	47	2	no	—
137	C₂×C₄⋊C₈	—	94	2	no	—
138	C₂×#103	Dic₄₇	1	2	yes	—
139	C₂×C₄×C₈	—	47	1	no	—
140	C₂×C₄×C₈	—	47	1	no	—
141	C₂×C₄×C₈	—	94	2	no	—
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	#334	—	47	4	no	—
144	Q₈⋊C₈	—	47	8	no	—
145	#322	—	47	4	no	—
146	C₄₇	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
148	C₂×Q₁₆	C₄₇⋊C₈	1	8	yes	complemented
149	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
150	C₂²×C₈	—	47	2	no	—
151	C₂²×C₈	—	94	2	no	—
152	C₂²×C₈	—	94	1	no	—
153	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
158	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
163	#107	C₂×Dic₄₇	1	4	yes	—
164	C₄⋊C₈	—	47	4	no	—
165	C₄⋊C₈	—	94	8	no	—
166	C₄⋊C₈	—	94	2	no	—
167	#103	C₂×Dic₄₇	1	8	yes	—
168	C₄×C₈	—	47	2	no	—
169	C₄×C₈	—	47	2	no	—
170	C₄×C₈	—	47	2	no	—
171	C₄×C₈	—	94	8	no	—
172	C₄×C₈	—	94	1	no	—
173	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
174	C₄×C₈	C₂²×D₄₇	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	C₂×C₄₇⋊C₈	1	8	yes	—
178	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
182	C₂²×C₄	#10625352	1	1	yes	characteristic
183	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
184	Q₁₆	C₂×C₄₇⋊C₈	1	16	yes	complemented
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	47	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	—	94	4	no	—
189	C₂×C₈	—	94	2	no	—
190	C₂×C₈	—	188	2	no	—
191	C₂×C₈	—	188	4	no	—
192	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
193	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
194	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	D₄×D₄₇	1	1	yes	characteristic
202	C₄²	#10625352	1	1	yes	characteristic
203	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₄²	C₂³×D₄₇	1	1	yes	characteristic
205	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	2	8	no	—
208	Q₈	C₂²×C₄₇⋊C₈	1	8	yes	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
218	C₂×C₄	#13094031	1	2	yes	—
219	C₂×C₄	#13094056	1	4	yes	—
220	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	94	4	no	—
225	C₈	—	188	2	no	—
226	C₈	—	188	4	no	—
227	C₈	C₂²×C₄₇⋊C₈	1	4	yes	—
228	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
229	C₂²	#15689230	1	2	yes	—
230	C₂²	#15689245	1	2	yes	—
231	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	8	no	—
235	C₄	—	4	4	no	—
236	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
237	C₄	#15689230	1	2	yes	—
238	C₄	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
239	C₄	C₂×#13094031	1	1	yes	characteristic
240	C₄	C₂×#13094056	1	2	yes	—
241	C₂	#20242207	1	4	yes	direct product factor
242	C₂	C₂×D₄×C₄₇⋊C₈	1	1	yes	characteristic
243	C₂	C₂×#15689230	1	1	yes	characteristic
244	C₂	C₂×#15689245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#1579
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20242207	C₂	direct (N × H)
36. C₂×Q₁₆×C₄₇	C₈	semidirect
61. Q₁₆×C₄₇	C₂×C₈	semidirect
125. C₂×C₄₇	#1579	semidirect
146. C₄₇	C₂×#1579	semidirect
148. C₂×Q₁₆	C₄₇⋊C₈	semidirect
184. Q₁₆	C₂×C₄₇⋊C₈	semidirect
241. C₂	#20242207	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×Q₈×C₄₇⋊C₈	2
#20242207	2
C₂×#15688874	2
C₂×Q₈⋊(C₄₇⋊C₈)	2
C₂×#15688844	2
C₂×C₄×Q₁₆×C₄₇	2
C₂×#1579	47

Subgroups — C2×#20242207 #26185671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242207 #26185671