Subgroups — C₂×#20242136 #26185252

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20242136	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688876	C₂	1	2	yes	maximal, complemented
4	C₂×#15688874	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	—	2	4	no	—
7	#15689176	C₂²	1	8	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
11	#15688876	C₂²	1	8	yes	complemented
12	#15688874	C₂²	1	8	yes	complemented
13	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
26	C₄×C₄₇⋊C₈	—	2	8	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	—	2	4	no	—
31	C₂²×C₈×C₄₇	C₈	1	8	yes	complemented
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₄₇	C₂³	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₄₇	—	2	8	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₄₇	C₂×C₈	1	16	yes	complemented
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
87	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₄₇⋊C₈	—	2	8	no	—
90	C₂×C₄₇⋊C₈	—	4	8	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₄⋊C₈	1	4	yes	—
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
103	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
104	C₂×C₄×C₄₇	#108	1	4	yes	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₄₇	#135	1	4	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	C₂²×C₈	1	4	yes	—
122	C₈×C₄₇	C₂²×C₈	1	4	yes	—
123	C₄₇⋊C₈	—	4	8	no	—
124	C₂×#1564	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	8	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
129	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
130	C₂²×C₄₇	#427	1	2	yes	—
131	C₂²×C₄₇	C₂×#108	1	2	yes	—
132	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
137	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
138	C₄×C₄₇	#427	1	2	yes	—
139	C₄×C₄₇	C₂×#108	1	4	yes	—
140	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
141	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
142	C₄×C₄₇	C₂×#135	1	2	yes	—
143	C₂×#431	—	47	2	no	—
144	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
145	#1564	—	47	8	no	—
146	C₂×#335	—	47	2	no	—
147	C₂×#334	—	47	2	no	—
148	C₂×C₄₇	—	2	4	no	—
149	C₂×C₄₇	#1564	1	4	yes	complemented
150	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
152	C₂×C₄₇	C₂²×#108	1	1	yes	characteristic
153	C₂³×C₈	Dic₄₇	1	1	yes	characteristic
154	C₂³×C₈	Dic₄₇	1	1	yes	characteristic
155	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
156	#431	—	47	8	no	—
157	#431	—	94	4	no	—
158	C₂×C₄⋊C₈	—	47	4	no	—
159	C₂×C₂²⋊C₈	—	94	2	no	—
160	C₂×C₄×C₈	—	94	2	no	—
161	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
162	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
163	C₂×C₄×C₈	Dic₄₇	1	4	yes	—
164	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₄₇	1	4	yes	—
167	#335	—	47	8	no	—
168	#334	—	47	8	no	—
169	C₄₇	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
171	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	94	4	no	—
175	C₂²×C₈	C₄₇⋊C₈	1	8	yes	complemented
176	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
179	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
182	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
183	C₂×C₄²	Dic₉₄	1	2	yes	—
184	C₂×C₄²	D₁₈₈	1	2	yes	—
185	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
190	C₄⋊C₈	—	47	8	no	—
191	C₄⋊C₈	—	94	4	no	—
192	C₂²⋊C₈	—	94	8	no	—
193	C₄×C₈	—	94	8	no	—
194	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
195	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
196	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
197	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
198	C₄×C₈	C₂²×D₄₇	1	2	yes	—
199	C₄×C₈	C₂²×D₄₇	1	2	yes	—
200	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
204	C₂²×C₄	#10625329	1	4	yes	—
205	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
207	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
208	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
209	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
213	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
214	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	94	8	no	—
221	C₂×C₈	—	188	8	no	—
222	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
223	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
224	C₂×C₈	C₂×C₄₇⋊C₈	1	16	yes	complemented
225	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
226	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
227	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
228	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
229	C₄²	C₄⋊Dic₄₇	1	2	yes	—
230	C₄²	C₄⋊Dic₄₇	1	2	yes	—
231	C₄²	C₄⋊Dic₄₇	1	2	yes	—
232	C₄²	C₂×Dic₉₄	1	2	yes	—
233	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
234	C₄²	C₂×D₁₈₈	1	2	yes	—
235	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
236	C₄²	C₂²×Dic₄₇	1	2	yes	—
237	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
238	C₄²	C₂³×D₄₇	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
242	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
243	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
251	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
252	C₂×C₄	#13093960	1	4	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
259	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
260	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
261	C₂×C₄	C₂×#10625329	1	4	yes	—
262	C₂×C₄	C₂×#10625329	1	2	yes	—
263	C₂×C₄	#13094046	1	4	yes	—
264	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	188	8	no	—
270	C₈	C₂²×C₄₇⋊C₈	1	4	yes	—
271	C₈	C₂²×C₄₇⋊C₈	1	4	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	8	no	—
275	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
276	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
277	C₂²	#15689161	1	2	yes	—
278	C₂²	C₂×#13093960	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
284	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
285	C₄	#15689161	1	2	yes	—
286	C₄	C₂×#13093960	1	4	yes	—
287	C₄	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
288	C₄	C₂²×#10625329	1	1	yes	characteristic
289	C₄	C₂×#13094046	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	#20242136	1	4	yes	direct product factor
292	C₂	C₂²×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
293	C₂	C₂×#15689161	1	1	yes	characteristic
294	C₂	C₂²×#13093960	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₁₈₈
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#1564
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20242136	C₂	direct (N × H)
3. C₂×#15688876	C₂	semidirect
4. C₂×#15688874	C₂	semidirect
11. #15688876	C₂²	semidirect
12. #15688874	C₂²	semidirect
31. C₂²×C₈×C₄₇	C₈	semidirect
74. C₂×C₈×C₄₇	C₂×C₈	semidirect
149. C₂×C₄₇	#1564	semidirect
169. C₄₇	C₂×#1564	semidirect
175. C₂²×C₈	C₄₇⋊C₈	semidirect
224. C₂×C₈	C₂×C₄₇⋊C₈	semidirect
291. C₂	#20242136	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×#15689176	2
#20242136	2
C₂×#15688876	2
C₂×#15688874	2
C₂²×C₄×C₈×C₄₇	2
C₂×#1564	47

Subgroups — C2×#20242136 #26185252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242136 #26185252