Subgroups — C₂×#20243082 #26213633

Order 12032.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625329	C₂	1	2	yes	maximal
2	#20243082	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688878	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	8	no	—
6	C₂²×#10625329	—	2	8	no	—
7	C₂²×#10625329	C₄	1	16	yes	complemented
8	C₂²×#10625329	C₂²	1	8	yes	—
9	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
10	#15688878	C₂²	1	16	yes	complemented
11	C₂⁴×C₄×C₄₇	C₄	1	2	yes	—
12	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₄₇	—	2	8	no	—
14	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
15	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
18	C₂×#10625329	—	2	64	no	—
19	C₂×#10625329	—	2	16	no	—
20	C₂×#10625329	—	2	16	no	—
21	C₂×#10625329	C₂×C₄	1	32	yes	complemented
22	C₂²×C₄₇⋊C₈	—	2	8	no	—
23	C₂²×C₄₇⋊C₈	—	2	8	no	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
25	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
26	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	—	2	8	no	—
28	C₂³×C₄×C₄₇	—	2	16	no	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₄₇	D₄	1	4	yes	—
35	C₂³×C₄×C₄₇	D₄	1	8	yes	—
36	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	—	2	16	no	—
40	C₂×C₄²×C₄₇	—	2	8	no	—
41	C₂×C₄²×C₄₇	—	2	8	no	—
42	C₂×C₄²×C₄₇	—	2	16	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
46	C₂⁴×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	C₄²	1	4	yes	—
48	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
49	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	64	no	—
53	C₂²×C₄×C₄₇	—	2	16	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	32	no	—
57	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
58	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	32	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	32	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	16	yes	complemented
66	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₄²×C₄₇	—	2	8	no	—
79	C₄²×C₄₇	—	2	16	no	—
80	C₄²×C₄₇	—	2	8	no	—
81	#10625329	—	2	64	no	—
82	C₂×C₄₇⋊C₈	—	2	32	no	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	complemented
86	C₂³×C₄₇	—	2	8	no	—
87	C₂³×C₄₇	—	2	16	no	—
88	C₂³×C₄₇	—	2	16	no	—
89	C₂³×C₄₇	—	2	16	no	—
90	C₂³×C₄₇	#95	1	16	yes	—
91	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
92	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
98	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	64	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	32	no	—
103	C₂×C₄×C₄₇	#95	1	32	yes	—
104	C₂×C₄×C₄₇	#95	1	16	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×M₁₆	1	16	yes	complemented
116	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₄₇⋊C₈	—	2	16	no	—
121	C₂×#1728	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	8	no	—
123	C₂²×C₄₇	—	2	8	no	—
124	C₂²×C₄₇	—	2	8	no	—
125	C₂²×C₄₇	—	2	32	no	—
126	C₂²×C₄₇	C₂×#95	1	8	yes	—
127	C₂²×C₄₇	C₂×#95	1	4	yes	—
128	C₂²×C₄₇	C₄×M₁₆	1	8	yes	—
129	C₂²×C₄₇	#407	1	8	yes	—
130	C₂²×C₄₇	#423	1	8	yes	—
131	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
135	C₄×C₄₇	—	2	16	no	—
136	C₄×C₄₇	C₂×#95	1	4	yes	—
137	C₄×C₄₇	C₂×#95	1	8	yes	—
138	C₄×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
139	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
140	C₂³×M₁₆	—	47	2	no	—
141	C₂³×C₄²	D₄₇	1	1	yes	characteristic
142	#1728	—	47	8	no	—
143	C₂×#336	—	47	4	no	—
144	C₂×C₄₇	—	2	8	no	—
145	C₂×C₄₇	#1728	1	8	yes	complemented
146	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
147	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
148	C₂×C₄₇	C₂×#407	1	2	yes	—
149	C₂×C₄₇	C₂×#423	1	2	yes	—
150	C₂⁴×C₄	Dic₄₇	1	2	yes	—
151	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
152	C₂²×M₁₆	—	47	16	no	—
153	C₂²×M₁₆	—	47	8	no	—
154	C₂²×M₁₆	—	94	8	no	—
155	C₂²×M₁₆	—	94	8	no	—
156	C₂³×C₈	—	47	4	no	—
157	C₂²×C₄²	—	2	8	no	—
158	C₂²×C₄²	Dic₄₇	1	4	yes	—
159	C₂²×C₄²	Dic₄₇	1	2	yes	—
160	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
161	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
162	#336	—	47	16	no	—
163	C₄₇	C₂×#1728	1	1	yes	characteristic, complemented, Sylow
164	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
165	C₂³×C₄	—	2	8	no	—
166	C₂³×C₄	—	2	16	no	—
167	C₂³×C₄	Dic₉₄	1	4	yes	—
168	C₂³×C₄	C₄×D₄₇	1	8	yes	—
169	C₂³×C₄	D₁₈₈	1	4	yes	—
170	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
171	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
172	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
173	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
174	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
175	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂×M₁₆	—	47	32	no	—
178	C₂×M₁₆	—	94	64	no	—
179	C₂×M₁₆	—	94	16	no	—
180	C₂×M₁₆	—	94	16	no	—
181	C₂²×C₈	—	47	16	no	—
182	C₂²×C₈	—	47	8	no	—
183	C₂²×C₈	—	94	8	no	—
184	C₂²×C₈	—	94	8	no	—
185	C₂×C₄²	—	2	16	no	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	2	8	no	—
188	C₂×C₄²	—	2	16	no	—
189	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
190	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
192	C₂⁴	—	2	8	no	—
193	C₂⁴	C₄×Dic₄₇	1	4	yes	—
194	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
195	C₂⁴	C₂²⋊Dic₄₇	1	4	yes	—
196	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
197	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	64	no	—
199	C₂²×C₄	—	2	16	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	#10625329	1	16	yes	complemented
204	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
205	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
206	C₂²×C₄	Dic₄₇⋊C₄	1	32	yes	—
207	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
208	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	32	yes	—
210	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
211	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
212	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
213	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
216	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
217	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
219	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	M₁₆	—	94	64	no	—
225	C₂×C₈	—	47	16	no	—
226	C₂×C₈	—	94	32	no	—
227	C₂×C₈	—	94	8	no	—
228	C₂×C₈	—	94	8	no	—
229	C₄²	—	2	16	no	—
230	C₄²	—	2	8	no	—
231	C₄²	—	2	8	no	—
232	C₂³	—	2	16	no	—
233	C₂³	—	2	16	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	16	no	—
236	C₂³	C₄₇⋊#95	1	16	yes	—
237	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
238	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
239	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
240	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
241	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
242	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
243	C₂³	C₂×#10625329	1	8	yes	—
244	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
245	C₂×C₄	—	2	64	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	32	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	C₄₇⋊#95	1	32	yes	—
250	C₂×C₄	C₄₇⋊#95	1	16	yes	—
251	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
253	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	8	yes	—
254	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	16	yes	—
255	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
256	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
257	C₂×C₄	C₂×Dic₄₇⋊C₄	1	16	yes	—
258	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
261	C₂×C₄	C₂×#10625329	1	16	yes	complemented
262	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
263	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
264	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
265	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
266	C₈	—	94	16	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	32	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	8	no	—
271	C₂²	C₂×C₄₇⋊#95	1	8	yes	—
272	C₂²	C₂×C₄₇⋊#95	1	4	yes	—
273	C₂²	C₄×#10625329	1	8	yes	—
274	C₂²	#15689087	1	8	yes	—
275	C₂²	#15689147	1	8	yes	—
276	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
277	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
278	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
279	C₂²	C₂²×#10625329	1	4	yes	—
280	C₄	—	2	16	no	—
281	C₄	C₂×C₄₇⋊#95	1	4	yes	—
282	C₄	C₂×C₄₇⋊#95	1	8	yes	—
283	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	2	yes	—
284	C₄	C₂²×Dic₄₇⋊C₄	1	2	yes	—
285	C₂	—	2	8	no	—
286	C₂	#20243082	1	8	yes	direct product factor
287	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic
288	C₂	C₂×C₄×#10625329	1	2	yes	—
289	C₂	C₂×#15689087	1	2	yes	—
290	C₂	C₂×#15689147	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₂×#1728
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. #20243082	C₂	direct (N × H)
3. C₂×#15688878	C₂	semidirect
7. C₂²×#10625329	C₄	semidirect
10. #15688878	C₂²	semidirect
21. C₂×#10625329	C₂×C₄	semidirect
24. C₂²×C₄₇⋊C₈	C₂×C₄	semidirect
65. C₂²×C₄×C₄₇	M₁₆	semidirect
85. C₂×C₄₇⋊C₈	C₂²×C₄	semidirect
115. C₂×C₄×C₄₇	C₂×M₁₆	semidirect
145. C₂×C₄₇	#1728	semidirect
163. C₄₇	C₂×#1728	semidirect
203. C₂²×C₄	#10625329	semidirect
261. C₂×C₄	C₂×#10625329	semidirect
286. C₂	#20243082	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#10625329	2
#20243082	2
C₂×#15688878	2
C₂³×C₄²×C₄₇	2
C₂×#1728	47

Subgroups — C2×#20243082 #26213633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243082 #26213633