Subgroups — C₂×#20242155 #26185296

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	#20242155	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	2	yes	maximal, complemented
5	C₂×#15688874	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	—	2	1	no	—
8	#15689180	—	2	2	no	—
9	#15689180	C₂²	1	4	yes	—
10	#15689147	—	2	2	no	—
11	#15689147	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
17	C₂×#13093935	—	2	1	no	—
18	#15688876	C₂²	1	8	yes	complemented
19	#15688874	C₂²	1	8	yes	complemented
20	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₄₇	C₄	1	4	yes	—
23	C₄×M₁₆×C₄₇	C₂²	1	4	yes	—
24	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	4	4	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
37	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
38	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
40	#13093935	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	—	2	8	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	#97×C₄₇	C₂×C₄	1	4	yes	—
57	#97×C₄₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₄₇	C₂³	1	4	yes	—
60	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	M₁₆×C₄₇	—	2	16	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
80	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
92	#10625329	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	4	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₄₇	#137	1	2	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₄₇	—	2	4	no	—
128	C₈×C₄₇	C₂×M₁₆	1	8	yes	complemented
129	C₄₇⋊C₈	—	4	2	no	—
130	C₄₇⋊C₈	—	4	4	no	—
131	C₄₇⋊C₈	—	4	2	no	—
132	C₂×#1569	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	4	4	no	—
137	C₂²×C₄₇	#423	1	2	yes	—
138	C₂²×C₄₇	#428	1	2	yes	—
139	C₂²×C₄₇	#430	1	2	yes	—
140	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	#423	1	2	yes	—
147	C₄×C₄₇	#428	1	2	yes	—
148	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
149	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
151	C₂×#432	—	47	1	no	—
152	C₂×#423	—	47	1	no	—
153	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
154	#1569	—	47	8	no	—
155	C₂×#335	—	47	2	no	—
156	C₂×#334	—	47	2	no	—
157	C₂×C₄₇	—	4	2	no	—
158	C₂×C₄₇	#1569	1	4	yes	complemented
159	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#428	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#430	1	1	yes	characteristic
162	C₂²×M₁₆	—	94	1	no	—
163	C₂²×M₁₆	Dic₄₇	1	2	yes	—
164	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	#432	—	47	4	no	—
166	#432	—	94	2	no	—
167	#423	—	47	4	no	—
168	#423	—	94	2	no	—
169	C₂×C₄⋊C₈	—	47	1	no	—
170	C₂×C₄⋊C₈	—	47	2	no	—
171	C₂×C₄⋊C₈	—	47	1	no	—
172	C₂×C₄⋊C₈	—	94	1	no	—
173	C₂×C₂²⋊C₈	—	94	1	no	—
174	C₄×M₁₆	Dic₄₇	1	4	yes	—
175	C₄×M₁₆	C₂×D₄₇	1	4	yes	—
176	C₂×#97	—	94	1	no	—
177	C₂×#97	Dic₄₇	1	1	yes	characteristic
178	C₂×#97	Dic₄₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
180	#335	—	47	8	no	—
181	#334	—	47	8	no	—
182	C₄₇	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
184	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂×M₁₆	—	2	8	no	—
187	C₂×M₁₆	—	94	4	no	—
188	C₂×M₁₆	—	188	4	no	—
189	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	—	94	1	no	—
192	C₂²×C₈	—	94	1	no	—
193	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
194	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
196	C₂×C₄²	Dic₉₄	1	2	yes	—
197	C₂×C₄²	D₁₈₈	1	2	yes	—
198	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
201	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₈	—	47	2	no	—
204	C₄⋊C₈	—	47	4	no	—
205	C₄⋊C₈	—	47	2	no	—
206	C₄⋊C₈	—	94	2	no	—
207	C₄⋊C₈	—	94	1	no	—
208	C₄⋊C₈	—	94	1	no	—
209	C₄⋊C₈	—	94	4	no	—
210	C₂²⋊C₈	—	94	4	no	—
211	#97	—	94	4	no	—
212	#97	C₂×Dic₄₇	1	4	yes	—
213	#97	C₂×Dic₄₇	1	4	yes	—
214	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
215	C₄×C₈	C₂²×D₄₇	1	4	yes	—
216	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
227	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
229	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
230	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
233	M₁₆	—	2	16	no	—
234	M₁₆	—	188	8	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	—	188	4	no	—
241	C₂×C₈	—	188	2	no	—
242	C₂×C₈	#10625329	1	8	yes	complemented
243	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
244	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
245	C₄²	C₄⋊Dic₄₇	1	2	yes	—
246	C₄²	C₄⋊Dic₄₇	1	2	yes	—
247	C₄²	C₄⋊Dic₄₇	1	2	yes	—
248	C₄²	C₂×Dic₉₄	1	2	yes	—
249	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
250	C₄²	C₂×D₁₈₈	1	2	yes	—
251	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
252	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₄²	C₂²×Dic₄₇	1	2	yes	—
254	C₄²	C₂³×D₄₇	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
259	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#10625329	1	4	yes	—
278	C₂×C₄	#13094052	1	2	yes	—
279	C₂×C₄	#13094057	1	2	yes	—
280	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
282	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	188	2	no	—
285	C₈	—	188	4	no	—
286	C₈	—	188	2	no	—
287	C₈	C₂×#10625329	1	8	yes	complemented
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#15689147	1	2	yes	—
293	C₂²	#15689165	1	2	yes	—
294	C₂²	#15689172	1	2	yes	—
295	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	#15689147	1	2	yes	—
302	C₄	#15689165	1	2	yes	—
303	C₄	C₂²×#10625329	1	2	yes	—
304	C₄	C₂×#13094052	1	1	yes	characteristic
305	C₄	C₂×#13094057	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	#20242155	1	4	yes	direct product factor
308	C₂	C₂×#15689147	1	1	yes	characteristic
309	C₂	C₂×#15689165	1	1	yes	characteristic
310	C₂	C₂×#15689172	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₂×#1569
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20242155	C₂	direct (N × H)
4. C₂×#15688876	C₂	semidirect
5. C₂×#15688874	C₂	semidirect
18. #15688876	C₂²	semidirect
19. #15688874	C₂²	semidirect
79. C₂×C₈×C₄₇	M₁₆	semidirect
128. C₈×C₄₇	C₂×M₁₆	semidirect
158. C₂×C₄₇	#1569	semidirect
182. C₄₇	C₂×#1569	semidirect
242. C₂×C₈	#10625329	semidirect
287. C₈	C₂×#10625329	semidirect
307. C₂	#20242155	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×#15689180	2
C₂×#15689147	2
#20242155	2
C₂×#15688876	2
C₂×#15688874	2
C₂×C₄×M₁₆×C₄₇	2
C₂×#1569	47

Subgroups — C2×#20242155 #26185296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242155 #26185296