Subgroups — C₂×#20242126 #26184837

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	#20242126	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688869	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	2	no	—
6	#15689147	—	2	4	no	—
7	#15689147	C₂²	1	8	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15688869	C₂²	1	16	yes	complemented
11	C₄₇×#533	—	2	4	no	—
12	C₄₇×#533	C₄	1	4	yes	—
13	C₄₇×#533	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
15	C₂×#125×C₄₇	C₄	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
17	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
18	C₂×#117×C₄₇	C₄	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	8	no	—
21	C₂×#10625329	—	4	8	no	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
26	C₂²×Q₈×C₄₇	—	2	2	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₇	—	2	2	no	—
30	C₄⋊Q₈×C₄₇	—	2	2	no	—
31	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
33	#125×C₄₇	—	2	4	no	—
34	#125×C₄₇	C₂×C₄	1	8	yes	—
35	C₂²⋊Q₈×C₄₇	—	2	8	no	—
36	C₂²⋊Q₈×C₄₇	—	2	8	no	—
37	C₄×Q₈×C₄₇	—	2	8	no	—
38	C₄×Q₈×C₄₇	—	2	8	no	—
39	#117×C₄₇	—	2	4	no	—
40	#117×C₄₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	D₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	—	2	4	no	—
55	C₂×Q₈×C₄₇	—	4	4	no	—
56	C₂×Q₈×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₄⋊C₄×C₄₇	—	2	16	no	—
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂²⋊C₄×C₄₇	—	2	8	no	—
75	C₄²×C₄₇	—	2	8	no	—
76	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	4	yes	—
81	C₄²×C₄₇	C₂×D₄	1	2	yes	—
82	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
83	#10625329	—	4	16	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	—	4	8	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₄₇	—	4	4	no	—
91	Q₈×C₄₇	—	4	4	no	—
92	Q₈×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	4	8	no	—
102	C₂×C₄×C₄₇	#101	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
108	C₂×C₄×C₄₇	#137	1	4	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
111	C₄₇⋊C₈	—	4	8	no	—
112	C₂×#1561	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	4	4	no	—
116	C₂²×C₄₇	C₂×#101	1	2	yes	—
117	C₂²×C₄₇	#417	1	4	yes	—
118	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	—	4	8	no	—
123	C₄×C₄₇	C₂×#101	1	4	yes	—
124	C₄×C₄₇	#417	1	4	yes	—
125	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₄₇	C₂×#137	1	2	yes	—
127	C₂×#533	D₄₇	1	1	yes	characteristic
128	C₂×#423	—	47	2	no	—
129	#1561	—	47	8	no	—
130	C₂×#332	—	47	4	no	—
131	C₂×C₄₇	—	4	2	no	—
132	C₂×C₄₇	#1561	1	4	yes	complemented
133	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#417	1	2	yes	—
135	C₂²×M₁₆	—	94	2	no	—
136	#533	—	2	4	no	—
137	#533	Dic₄₇	1	4	yes	—
138	#533	C₂×D₄₇	1	4	yes	—
139	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
140	C₂×#125	Dic₄₇	1	2	yes	—
141	C₂×C₂²⋊Q₈	—	2	2	no	—
142	C₂×C₄×Q₈	—	2	2	no	—
143	C₂×#117	Dic₄₇	1	2	yes	—
144	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
145	#423	—	47	8	no	—
146	#423	—	94	4	no	—
147	C₂×C₄⋊C₈	—	47	4	no	—
148	C₂×C₄⋊C₈	—	94	2	no	—
149	#332	—	47	16	no	—
150	C₄₇	C₂×#1561	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	2	2	no	—
152	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
153	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
154	C₂×M₁₆	—	94	8	no	—
155	C₂×M₁₆	—	188	8	no	—
156	C₂²×C₈	—	94	4	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
160	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
161	#125	—	2	4	no	—
162	#125	C₂×Dic₄₇	1	8	yes	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₂²⋊Q₈	—	2	8	no	—
165	C₄×Q₈	—	2	8	no	—
166	C₄×Q₈	—	2	8	no	—
167	#117	—	2	4	no	—
168	#117	C₂×Dic₄₇	1	8	yes	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
176	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
177	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
180	C₄⋊C₈	—	47	8	no	—
181	C₄⋊C₈	—	94	4	no	—
182	C₄⋊C₈	—	94	8	no	—
183	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
193	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
194	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
195	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
200	M₁₆	—	188	16	no	—
201	C₂×C₈	—	94	8	no	—
202	C₂×C₈	—	188	8	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
208	C₂²⋊C₄	—	2	8	no	—
209	C₄²	—	2	8	no	—
210	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
211	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
212	C₄²	C₂²×Dic₄₇	1	2	yes	—
213	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
215	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
216	C₄²	C₂³×D₄₇	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
220	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	#13093943	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
238	C₂×C₄	#13094049	1	4	yes	—
239	C₂×C₄	#13094054	1	4	yes	—
240	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
242	C₈	—	188	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×#13093943	1	2	yes	—
247	C₂²	#15689124	1	4	yes	—
248	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×#13093943	1	4	yes	—
254	C₄	#15689124	1	4	yes	—
255	C₄	C₂×#13094049	1	2	yes	—
256	C₄	C₂×#13094054	1	2	yes	—
257	C₂	—	4	2	no	—
258	C₂	#20242126	1	4	yes	direct product factor
259	C₂	C₂²×#13093943	1	1	yes	characteristic
260	C₂	C₂×#15689124	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#1561
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20242126	C₂	direct (N × H)
3. C₂×#15688869	C₂	semidirect
10. #15688869	C₂²	semidirect
132. C₂×C₄₇	#1561	semidirect
150. C₄₇	C₂×#1561	semidirect
258. C₂	#20242126	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×#15689147	2
#20242126	2
C₂×#15688869	2
C₂×C₄₇×#533	2
C₂×#1561	47

Subgroups — C2×#20242126 #26184837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242126 #26184837