Subgroups — C₂×#20245795 #26374575

Order 12032.

3854 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093983	C₂	1	2	yes	maximal
2	#20245795	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15688993	C₂	1	2	yes	maximal, complemented
4	C₂×#15688989	C₂	1	2	yes	maximal, complemented
5	C₂×#15688983	C₂	1	4	yes	maximal
6	C₂×C₄×C₄⋊Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	—	2	16	no	—
9	C₂×#13093983	C₂²	1	16	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	8	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
14	#15688993	C₂²	1	16	yes	complemented
15	#15688989	C₂²	1	16	yes	complemented
16	#15688983	C₂²	1	32	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	32	yes	complemented
18	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
19	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
20	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₄	1	16	yes	complemented
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	8	yes	—
26	C₂³×Dic₄₇	C₂³	1	8	yes	—
27	#13093983	—	2	64	no	—
28	C₂×C₄⋊Dic₄₇	—	2	16	no	—
29	C₂×C₄⋊Dic₄₇	—	2	32	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	32	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
33	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
34	C₂×C₄×Dic₄₇	—	2	16	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	32	yes	—
37	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	—	2	16	no	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₄₇	D₄	1	8	yes	—
42	C₂³×C₄×C₄₇	Q₈	1	8	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	64	yes	complemented
53	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
56	C₂⁴×C₄₇	—	2	8	no	—
57	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
59	C₂⁴×C₄₇	C₂×Q₈	1	4	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
61	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	64	no	—
64	C₂²×C₄×C₄₇	—	2	32	no	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	64	yes	complemented
66	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	32	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	8	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	16	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
78	C₂²×C₄×C₄₇	#41	1	16	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
83	C₂²×Dic₄₇	—	2	16	no	—
84	C₂²×Dic₄₇	—	2	32	no	—
85	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	32	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
89	C₂²⋊Dic₄₇	—	2	32	no	—
90	C₄⋊Dic₄₇	—	2	32	no	—
91	C₄⋊Dic₄₇	—	2	64	no	—
92	C₄×Dic₄₇	—	2	32	no	—
93	C₂³×C₄₇	—	2	16	no	—
94	C₂³×C₄₇	—	2	16	no	—
95	C₂³×C₄₇	—	2	16	no	—
96	C₂³×C₄₇	—	2	8	no	—
97	C₂³×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
98	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
99	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
100	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
102	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
103	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
104	C₂³×C₄₇	C₂×#41	1	8	yes	—
105	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	16	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	—	2	64	no	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	64	yes	complemented
112	C₂×C₄×C₄₇	#117	1	32	yes	complemented
113	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
120	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	16	yes	—
122	C₂×Dic₄₇	—	2	16	no	—
123	C₂×Dic₄₇	—	2	32	no	—
124	C₂×Dic₄₇	—	4	32	no	—
125	C₂×#2285	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	8	no	—
127	C₂²×C₄₇	—	2	8	no	—
128	C₂²×C₄₇	—	2	8	no	—
129	C₂²×C₄₇	—	2	32	no	—
130	C₂²×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
131	C₂²×C₄₇	C₂²×C₄⋊C₄	1	4	yes	—
132	C₂²×C₄₇	C₂×#117	1	8	yes	—
133	C₂²×C₄₇	#532	1	8	yes	—
134	C₂²×C₄₇	#533	1	8	yes	—
135	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
136	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	C₂²×#41	1	4	yes	—
139	C₄×C₄₇	—	2	8	no	—
140	C₄×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
141	C₄×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
142	C₄×C₄₇	C₂×#117	1	16	yes	complemented
143	Dic₄₇	—	4	16	no	—
144	C₂²×#117	—	47	2	no	—
145	C₂³×C₄²	D₄₇	1	1	yes	characteristic
146	#2285	—	47	16	no	—
147	C₂×#384	—	47	2	no	—
148	C₂×#382	—	47	2	no	—
149	C₂×#380	—	47	4	no	—
150	C₂×C₄×C₄⋊C₄	—	47	4	no	—
151	C₂×C₄₇	—	2	8	no	—
152	C₂×C₄₇	#2285	1	8	yes	complemented
153	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×#117	1	2	yes	—
155	C₂×C₄₇	C₂×#532	1	2	yes	—
156	C₂×C₄₇	C₂×#533	1	2	yes	—
157	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
158	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
159	C₂×#117	—	47	16	no	—
160	C₂×#117	—	94	16	no	—
161	C₂²×C₄⋊C₄	—	47	4	no	—
162	C₂²×C₄⋊C₄	—	47	8	no	—
163	C₂²×C₂²⋊C₄	—	47	4	no	—
164	C₂²×C₄²	—	47	4	no	—
165	C₂²×C₄²	Dic₄₇	1	16	yes	complemented
166	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
167	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
168	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
169	C₂²×C₄²	C₂×D₄₇	1	8	yes	—
170	#384	—	47	16	no	—
171	#382	—	47	16	no	—
172	#380	—	47	32	no	—
173	C₄×C₄⋊C₄	—	47	32	no	—
174	C₂×#95	—	47	8	no	—
175	C₄₇	C₂×#2285	1	1	yes	characteristic, complemented, Sylow
176	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
177	C₂³×C₄	—	2	16	no	—
178	C₂³×C₄	—	47	8	no	—
179	C₂³×C₄	Dic₉₄	1	8	yes	—
180	C₂³×C₄	D₁₈₈	1	8	yes	—
181	C₂³×C₄	C₂×Dic₄₇	1	16	yes	—
182	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
184	C₂³×C₄	C₂²×D₄₇	1	8	yes	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
187	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	#117	—	94	64	no	—
190	C₂×C₄⋊C₄	—	47	16	no	—
191	C₂×C₄⋊C₄	—	47	32	no	—
192	C₂×C₄⋊C₄	—	94	16	no	—
193	C₂×C₄⋊C₄	—	94	32	no	—
194	C₂×C₂²⋊C₄	—	47	16	no	—
195	C₂×C₂²⋊C₄	—	94	16	no	—
196	C₂×C₄²	—	47	16	no	—
197	C₂×C₄²	—	94	16	no	—
198	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
199	C₂×C₄²	C₂×Dic₄₇	1	16	yes	—
200	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
201	C₂×C₄²	C₂×Dic₄₇	1	64	yes	complemented
202	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
205	#95	—	47	32	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	C₂×Dic₉₄	1	4	yes	—
208	C₂⁴	C₂×D₁₈₈	1	4	yes	—
209	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
210	C₂⁴	C₂³×D₄₇	1	2	yes	—
211	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	64	no	—
214	C₂²×C₄	—	2	32	no	—
215	C₂²×C₄	—	47	16	no	—
216	C₂²×C₄	—	94	16	no	—
217	C₂²×C₄	—	94	32	no	—
218	C₂²×C₄	C₄⋊Dic₄₇	1	64	yes	complemented
219	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
220	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
221	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
222	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
223	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
224	C₂²×C₄	C₂×D₁₈₈	1	16	yes	—
225	C₂²×C₄	#10625350	1	16	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	32	yes	—
228	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
229	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
232	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
235	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
236	C₄⋊C₄	—	94	32	no	—
237	C₄⋊C₄	—	94	64	no	—
238	C₂²⋊C₄	—	94	32	no	—
239	C₄²	—	94	32	no	—
240	C₄²	C₂²×Dic₄₇	1	16	yes	—
241	C₄²	C₂²×Dic₄₇	1	32	yes	—
242	C₄²	C₂²×Dic₄₇	1	16	yes	—
243	C₂³	—	2	16	no	—
244	C₂³	—	2	16	no	—
245	C₂³	—	2	16	no	—
246	C₂³	—	2	8	no	—
247	C₂³	C₂×C₄⋊Dic₄₇	1	16	yes	—
248	C₂³	C₂³×Dic₄₇	1	4	yes	—
249	C₂³	C₂³×Dic₄₇	1	2	yes	—
250	C₂³	C₂²×D₁₈₈	1	4	yes	—
251	C₂³	C₂²×D₁₈₈	1	2	yes	—
252	C₂³	C₂²×Dic₉₄	1	4	yes	—
253	C₂³	C₂²×Dic₉₄	1	2	yes	—
254	C₂³	C₂×#10625350	1	8	yes	—
255	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	64	no	—
259	C₂×C₄	—	94	16	no	—
260	C₂×C₄	—	94	32	no	—
261	C₂×C₄	—	188	32	no	—
262	C₂×C₄	C₂×C₄⋊Dic₄₇	1	16	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₄₇	1	16	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₄₇	1	64	yes	complemented
265	C₂×C₄	#13093983	1	32	yes	complemented
266	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
267	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
268	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
269	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
270	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
271	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
272	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
273	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
274	C₂×C₄	C₂×#10625350	1	16	yes	—
275	C₂²	—	2	32	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	C₂²×C₄⋊Dic₄₇	1	8	yes	—
280	C₂²	C₂²×C₄⋊Dic₄₇	1	4	yes	—
281	C₂²	C₂×#13093983	1	8	yes	—
282	C₂²	#15689673	1	8	yes	—
283	C₂²	#15689680	1	8	yes	—
284	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
285	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
286	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
287	C₂²	C₂²×#10625350	1	4	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	188	16	no	—
290	C₄	C₂²×C₄⋊Dic₄₇	1	8	yes	—
291	C₄	C₂²×C₄⋊Dic₄₇	1	8	yes	—
292	C₄	C₂×#13093983	1	16	yes	complemented
293	C₂	—	2	8	no	—
294	C₂	#20245795	1	8	yes	direct product factor
295	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
296	C₂	C₂²×#13093983	1	2	yes	—
297	C₂	C₂×#15689673	1	2	yes	—
298	C₂	C₂×#15689680	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₂×#2285
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. #20245795	C₂	direct (N × H)
3. C₂×#15688993	C₂	semidirect
4. C₂×#15688989	C₂	semidirect
6. C₂×C₄×C₄⋊Dic₄₇	C₂	semidirect
14. #15688993	C₂²	semidirect
15. #15688989	C₂²	semidirect
17. C₄×C₄⋊Dic₄₇	C₂²	semidirect
21. C₂²×C₄²×C₄₇	C₄	semidirect
52. C₂×C₄²×C₄₇	C₂×C₄	semidirect
65. C₂²×C₄×C₄₇	C₄⋊C₄	semidirect
111. C₂×C₄×C₄₇	C₂×C₄⋊C₄	semidirect
112. C₂×C₄×C₄₇	#117	semidirect
142. C₄×C₄₇	C₂×#117	semidirect
152. C₂×C₄₇	#2285	semidirect
165. C₂²×C₄²	Dic₄₇	semidirect
175. C₄₇	C₂×#2285	semidirect
201. C₂×C₄²	C₂×Dic₄₇	semidirect
218. C₂²×C₄	C₄⋊Dic₄₇	semidirect
264. C₂×C₄	C₂×C₄⋊Dic₄₇	semidirect
265. C₂×C₄	#13093983	semidirect
292. C₄	C₂×#13093983	semidirect
294. C₂	#20245795	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×#13093983	2
#20245795	2
C₂×#15688993	2
C₂×#15688989	2
C₂×#15688983	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂³×C₄²×C₄₇	2
C₂×#2285	47

Subgroups — C2×#20245795 #26374575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245795 #26374575