Subgroups — C₂×#20241214 #26104221

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688878	C₂	1	2	yes	maximal
2	#20241214	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄₇×#391	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	2	no	—
5	#15688878	—	2	2	no	—
6	#15688878	C₂²	1	4	yes	—
7	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#391	—	2	2	no	—
11	C₄₇×#391	C₄	1	2	yes	—
12	C₄₇×#391	C₂²	1	2	yes	—
13	C₂×#95×C₄₇	—	2	2	no	—
14	C₂²×C₄₇⋊C₈	—	2	8	no	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₂²×C₄₇⋊C₈	—	4	4	no	—
17	C₂²×C₄₇⋊C₈	—	4	4	no	—
18	C₂²×Q₈×C₄₇	—	2	1	no	—
19	C₂²×Q₈×C₄₇	—	2	2	no	—
20	C₂²×Q₈×C₄₇	—	4	2	no	—
21	C₂²×Q₈×C₄₇	—	4	2	no	—
22	C₂²×Q₈×C₄₇	C₈	1	4	yes	complemented
23	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₇	—	2	1	no	—
25	C₂³×C₄×C₄₇	—	2	1	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	—	4	2	no	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	#95×C₄₇	—	2	4	no	—
36	#95×C₄₇	—	2	4	no	—
37	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Q₈×C₄₇	—	2	4	no	—
39	C₂×Q₈×C₄₇	—	2	2	no	—
40	C₂×Q₈×C₄₇	—	4	8	no	—
41	C₂×Q₈×C₄₇	—	4	2	no	—
42	C₂×Q₈×C₄₇	—	4	2	no	—
43	C₂×Q₈×C₄₇	—	4	8	no	—
44	C₂×Q₈×C₄₇	—	4	2	no	—
45	C₂×Q₈×C₄₇	—	4	2	no	—
46	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	complemented
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
62	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
63	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	—	4	1	no	—
69	C₄²×C₄₇	—	4	2	no	—
70	C₄²×C₄₇	—	4	1	no	—
71	C₂×C₄₇⋊C₈	—	2	8	no	—
72	C₂×C₄₇⋊C₈	—	4	16	no	—
73	C₂×C₄₇⋊C₈	—	4	4	no	—
74	C₂×C₄₇⋊C₈	—	4	4	no	—
75	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
76	C₂³×C₄₇	#99	1	2	yes	—
77	C₂³×C₄₇	#100	1	2	yes	—
78	C₂³×C₄₇	#103	1	4	yes	—
79	C₂³×C₄₇	#104	1	4	yes	—
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₄₇	—	4	4	no	—
82	Q₈×C₄₇	—	4	8	no	—
83	Q₈×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	8	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	8	4	no	—
103	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₄₇	#103	1	4	yes	—
105	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
109	C₄₇⋊C₈	—	4	8	no	—
110	C₂×#1272	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	#323	1	4	yes	—
112	C₂²×C₄₇	Q₈⋊C₈	1	8	yes	—
113	C₂²×C₄₇	#328	1	8	yes	—
114	C₂²×C₄₇	#329	1	4	yes	—
115	C₂²×C₄₇	#333	1	4	yes	—
116	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#103	1	2	yes	—
120	C₂²×C₄₇	C₂×#104	1	2	yes	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	4	4	no	—
128	C₄×C₄₇	—	8	2	no	—
129	C₂×#391	D₄₇	1	1	yes	characteristic
130	C₂×#336	—	47	2	no	—
131	#1272	—	47	4	no	—
132	C₂×C₄₇	#1272	1	8	yes	complemented
133	C₂×C₄₇	C₂×#323	1	1	yes	characteristic
134	C₂×C₄₇	C₂×Q₈⋊C₈	1	2	yes	—
135	C₂×C₄₇	C₂×#328	1	2	yes	—
136	C₂×C₄₇	C₂×#329	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
138	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
139	C₂³×C₈	—	94	2	no	—
140	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
141	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
142	#391	—	2	2	no	—
143	#391	Dic₄₇	1	2	yes	—
144	#391	C₂×D₄₇	1	2	yes	—
145	C₂×#95	—	2	2	no	—
146	#336	—	47	4	no	—
147	#336	—	94	2	no	—
148	C₄₇	C₂×#1272	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×Q₈	—	2	1	no	—
151	C₂²×Q₈	—	4	2	no	—
152	C₂²×Q₈	—	4	2	no	—
153	C₂²×Q₈	C₄₇⋊C₈	1	4	yes	complemented
154	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
155	C₂³×C₄	—	2	1	no	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
158	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
159	C₂²×C₈	—	94	8	no	—
160	C₂²×C₈	—	94	4	no	—
161	C₂²×C₈	—	188	4	no	—
162	C₂²×C₈	—	188	4	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	4	2	no	—
165	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
166	C₂×C₄²	—	4	2	no	—
167	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	8	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	complemented
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
194	C₂²×C₄	#10625329	1	2	yes	—
195	C₂²×C₄	#10625336	1	4	yes	—
196	C₂²×C₄	#10625337	1	4	yes	—
197	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
201	C₂×C₈	—	94	8	no	—
202	C₂×C₈	—	188	4	no	—
203	C₂×C₈	—	188	16	no	—
204	C₂×C₈	—	188	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	1	no	—
210	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
211	C₂³	#13093939	1	2	yes	—
212	C₂³	#13093941	1	2	yes	—
213	C₂³	#13093948	1	4	yes	—
214	C₂³	#13093951	1	4	yes	—
215	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
216	Q₈	—	4	4	no	—
217	Q₈	—	4	8	no	—
218	Q₈	—	4	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
239	C₂×C₄	#13093948	1	4	yes	—
240	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
241	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
242	C₂×C₄	C₂×#10625336	1	2	yes	—
243	C₂×C₄	C₂×#10625337	1	2	yes	—
244	C₈	—	188	8	no	—
245	C₂²	#15688847	1	4	yes	—
246	C₂²	Q₈⋊(C₄₇⋊C₈)	1	8	yes	—
247	C₂²	#15688860	1	8	yes	—
248	C₂²	#15688863	1	4	yes	—
249	C₂²	#15688872	1	4	yes	—
250	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
251	C₂²	C₂×#13093939	1	1	yes	characteristic
252	C₂²	C₂×#13093941	1	1	yes	characteristic
253	C₂²	C₂×#13093948	1	2	yes	—
254	C₂²	C₂×#13093951	1	2	yes	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₂	#20241214	1	8	yes	direct product factor
264	C₂	C₂×#15688847	1	1	yes	characteristic
265	C₂	C₂×Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
266	C₂	C₂×#15688860	1	2	yes	—
267	C₂	C₂×#15688863	1	1	yes	characteristic
268	C₂	C₂×#15688872	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#391	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#391	D₄₇
hypocenter	C₄₇	C₂×#1272
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#391

Complements

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

N	H ≅ G/N	type
2. #20241214	C₂	direct (N × H)
22. C₂²×Q₈×C₄₇	C₈	semidirect
46. C₂×Q₈×C₄₇	C₂×C₈	semidirect
132. C₂×C₄₇	#1272	semidirect
148. C₄₇	C₂×#1272	semidirect
153. C₂²×Q₈	C₄₇⋊C₈	semidirect
181. C₂×Q₈	C₂×C₄₇⋊C₈	semidirect
263. C₂	#20241214	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×#15688878	2
#20241214	2
C₂×C₄₇×#391	2
C₂×#1272	47

Subgroups — C2×#20241214 #26104221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241214 #26104221