Subgroups — C₂×#20253762 #27085545

Order 12032.

2806 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689680	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253762	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689487	C₂	1	4	yes	maximal, complemented
4	C₂×#15689469	C₂	1	4	yes	maximal, complemented
5	C₂×#15689128	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	#15689680	—	2	4	no	—
10	#15689680	C₂²	1	8	yes	complemented
11	C₂×#13094024	C₂²	1	2	yes	complemented
12	C₂×#13094016	C₂²	1	2	yes	complemented
13	C₂×#13094003	—	2	2	no	—
14	C₂×C₄×Dic₉₄	—	2	2	no	—
15	C₂×#13093983	C₂²	1	2	yes	—
16	#15689487	C₂²	1	32	yes	complemented
17	#15689469	C₂²	1	32	yes	complemented
18	#15689128	C₂²	1	32	yes	complemented
19	C₂×#13093945	C₂²	1	16	yes	complemented
20	C₄×#10625329	C₂²	1	8	yes	complemented
21	C₂×#13093935	C₂²	1	2	yes	complemented
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	complemented
23	C₂³×D₄×C₄₇	—	2	2	no	—
24	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
25	C₂×#127×C₄₇	—	2	2	no	—
26	C₂×#127×C₄₇	—	2	4	no	—
27	C₂×#127×C₄₇	C₂²	1	2	yes	—
28	C₂×#127×C₄₇	C₂²	1	8	yes	—
29	C₂×#127×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×Dic₉₄	—	2	2	no	—
32	C₂×#10625329	—	2	8	no	—
33	C₂×#10625329	C₂³	1	8	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
35	#13094024	—	2	4	no	—
36	#13094024	C₂³	1	8	yes	complemented
37	#13094016	—	2	4	no	—
38	#13094016	C₂³	1	8	yes	complemented
39	#13094003	—	2	16	no	—
40	C₄×Dic₉₄	—	2	16	no	—
41	#13093983	—	2	4	no	—
42	#13093983	C₂³	1	8	yes	—
43	C₂×C₄⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
45	C₂×Dic₄₇⋊C₄	—	2	4	no	—
46	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	#13093945	C₂³	1	64	yes	complemented
49	#13093935	C₂³	1	8	yes	complemented
50	C₄×C₄₇⋊C₈	C₂³	1	8	yes	complemented
51	C₂⁵×C₄₇	—	2	2	no	—
52	C₂²×D₄×C₄₇	—	2	2	no	—
53	C₂²×D₄×C₄₇	—	2	4	no	—
54	C₂²×D₄×C₄₇	—	2	4	no	—
55	C₂²×D₄×C₄₇	—	2	16	no	—
56	C₂²×D₄×C₄₇	—	2	4	no	—
57	C₂²×D₄×C₄₇	—	2	16	no	—
58	C₂²×D₄×C₄₇	—	4	16	no	—
59	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
60	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
61	C₂³×C₄×C₄₇	D₄	1	2	yes	—
62	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
63	#127×C₄₇	—	2	16	no	—
64	#127×C₄₇	—	2	4	no	—
65	#127×C₄₇	—	2	4	no	—
66	#127×C₄₇	C₂³	1	8	yes	—
67	#127×C₄₇	C₂³	1	8	yes	—
68	C₂×C₄²×C₄₇	D₄	1	2	yes	—
69	C₂×C₄²×C₄₇	D₄	1	4	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
73	C₂⁴×C₄₇	—	2	8	no	—
74	C₂⁴×C₄₇	—	2	4	no	—
75	C₂⁴×C₄₇	—	4	16	no	—
76	C₂⁴×C₄₇	—	4	8	no	—
77	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×D₄×C₄₇	—	2	8	no	—
79	C₂×D₄×C₄₇	—	2	4	no	—
80	C₂×D₄×C₄₇	—	2	16	no	—
81	C₂×D₄×C₄₇	—	2	16	no	—
82	C₂×D₄×C₄₇	—	2	16	no	—
83	C₂×D₄×C₄₇	—	2	32	no	—
84	C₂×D₄×C₄₇	—	4	16	no	—
85	C₂×D₄×C₄₇	—	4	64	no	—
86	C₂×D₄×C₄₇	—	4	16	no	—
87	C₂×D₄×C₄₇	—	4	8	no	—
88	C₂×D₄×C₄₇	—	4	16	no	—
89	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
90	C₂²×C₄×C₄₇	—	2	8	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₇	#41	1	8	yes	—
97	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
99	C₂²×Dic₄₇	—	2	4	no	—
100	C₂×Dic₉₄	—	2	8	no	—
101	C₂×Dic₉₄	—	4	8	no	—
102	C₄²×C₄₇	C₂×D₄	1	8	yes	—
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	C₄²×C₄₇	C₂⁴	1	2	yes	—
106	C₄²×C₄₇	C₂⁴	1	2	yes	—
107	C₂²⋊Dic₄₇	—	2	8	no	—
108	C₄⋊Dic₄₇	—	2	8	no	—
109	C₄⋊Dic₄₇	—	2	4	no	—
110	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
111	Dic₄₇⋊C₄	—	2	16	no	—
112	C₄×Dic₄₇	—	2	8	no	—
113	#10625329	—	2	16	no	—
114	C₂×C₄₇⋊C₈	—	2	8	no	—
115	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
116	C₂³×C₄₇	—	2	4	no	—
117	C₂³×C₄₇	—	2	8	no	—
118	C₂³×C₄₇	—	4	32	no	—
119	C₂³×C₄₇	—	4	16	no	—
120	C₂³×C₄₇	—	4	8	no	—
121	C₂³×C₄₇	—	4	16	no	—
122	C₂³×C₄₇	—	8	32	no	—
123	C₂³×C₄₇	#124	1	4	yes	—
124	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
125	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	D₄×C₄₇	—	2	16	no	—
127	D₄×C₄₇	—	4	32	no	—
128	D₄×C₄₇	—	4	16	no	—
129	D₄×C₄₇	—	4	8	no	—
130	D₄×C₄₇	—	4	32	no	—
131	C₂×C₄×C₄₇	—	2	4	no	—
132	C₂×C₄×C₄₇	—	2	16	no	—
133	C₂×C₄×C₄₇	#124	1	8	yes	—
134	C₂×C₄×C₄₇	#124	1	4	yes	—
135	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
141	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
142	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
143	C₂×Dic₄₇	—	2	8	no	—
144	C₂×Dic₄₇	—	4	8	no	—
145	Dic₉₄	—	4	16	no	—
146	C₄₇⋊C₈	—	2	8	no	—
147	C₂×#3138	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	8	no	—
149	C₂²×C₄₇	—	2	4	no	—
150	C₂²×C₄₇	—	4	8	no	—
151	C₂²×C₄₇	—	4	16	no	—
152	C₂²×C₄₇	—	8	32	no	—
153	C₂²×C₄₇	—	8	16	no	—
154	C₂²×C₄₇	C₂×#124	1	4	yes	—
155	C₂²×C₄₇	C₂×#124	1	2	yes	—
156	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
157	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
158	C₄×C₄₇	—	2	4	no	—
159	C₄×C₄₇	C₂×#124	1	4	yes	—
160	C₄×C₄₇	C₂×#124	1	2	yes	—
161	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	8	yes	—
162	C₄×C₄₇	C₂²×#41	1	2	yes	—
163	Dic₄₇	—	4	8	no	—
164	C₂×#533	—	47	1	no	—
165	C₂²×#127	D₄₇	1	1	yes	characteristic
166	#3138	—	47	16	no	—
167	C₂×#490	—	47	4	no	—
168	C₂×#486	—	47	4	no	—
169	C₂×#418	—	47	4	no	—
170	C₂×C₄×M₁₆	—	47	1	no	—
171	C₂×C₄₇	—	2	4	no	—
172	C₂×C₄₇	—	8	8	no	—
173	C₂×C₄₇	#3138	1	4	yes	complemented
174	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
175	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
176	C₂³×D₄	—	2	2	no	—
177	C₂³×D₄	C₂×D₄₇	1	2	yes	—
178	C₂²×M₁₆	—	47	2	no	—
179	#533	—	47	8	no	—
180	#533	—	94	4	no	—
181	C₂×C₄⋊Q₈	—	47	2	no	—
182	C₂×#127	—	2	2	no	—
183	C₂×#127	—	2	4	no	—
184	C₂×#127	C₂×D₄₇	1	2	yes	—
185	C₂×#127	C₂×D₄₇	1	2	yes	—
186	C₂×#127	C₂×D₄₇	1	8	yes	—
187	C₂×#125	—	47	2	no	—
188	C₂×C₂²⋊Q₈	—	94	2	no	—
189	C₂×C₄×Q₈	—	94	2	no	—
190	C₂×#117	—	47	2	no	—
191	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
192	#490	—	47	32	no	—
193	#486	—	47	32	no	—
194	#418	—	47	32	no	—
195	C₂×#102	—	47	16	no	—
196	C₄×M₁₆	—	47	8	no	—
197	C₂×#97	—	47	2	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₄₇	C₂×#3138	1	1	yes	characteristic, complemented, Sylow
200	C₂⁵	—	2	2	no	—
201	C₂²×Q₈	—	94	2	no	—
202	C₂²×D₄	—	2	16	no	—
203	C₂²×D₄	—	2	4	no	—
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	—	2	16	no	—
206	C₂²×D₄	—	2	4	no	—
207	C₂²×D₄	—	2	4	no	—
208	C₂²×D₄	—	4	16	no	—
209	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
210	C₂²×D₄	C₂²×D₄₇	1	8	yes	—
211	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×M₁₆	—	47	8	no	—
214	C₂×M₁₆	—	94	8	no	—
215	C₂²×C₈	—	47	4	no	—
216	C₄⋊Q₈	—	47	8	no	—
217	C₄⋊Q₈	—	94	4	no	—
218	#127	—	2	16	no	—
219	#127	—	2	4	no	—
220	#127	—	2	4	no	—
221	#127	C₂²×D₄₇	1	8	yes	—
222	#127	C₂²×D₄₇	1	8	yes	—
223	#125	—	47	8	no	—
224	#125	—	94	4	no	—
225	C₂²⋊Q₈	—	94	16	no	—
226	C₄×Q₈	—	94	16	no	—
227	#117	—	47	8	no	—
228	#117	—	94	4	no	—
229	C₂×C₄⋊C₄	—	47	4	no	—
230	C₂×C₄⋊C₄	—	94	4	no	—
231	C₂×C₄⋊C₄	—	94	2	no	—
232	C₂×C₂²⋊C₄	—	94	2	no	—
233	C₂×C₄²	—	94	2	no	—
234	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
235	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
239	#102	—	47	64	no	—
240	#97	—	47	8	no	—
241	C₄×C₈	—	47	8	no	—
242	C₂⁴	—	2	8	no	—
243	C₂⁴	—	2	4	no	—
244	C₂⁴	—	4	16	no	—
245	C₂⁴	—	4	8	no	—
246	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂×Q₈	—	94	8	no	—
248	C₂×Q₈	—	188	8	no	—
249	C₂×D₄	—	2	8	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	16	no	—
252	C₂×D₄	—	2	16	no	—
253	C₂×D₄	—	2	16	no	—
254	C₂×D₄	—	2	32	no	—
255	C₂×D₄	—	4	16	no	—
256	C₂×D₄	—	4	64	no	—
257	C₂×D₄	—	4	16	no	—
258	C₂×D₄	—	4	8	no	—
259	C₂×D₄	—	4	16	no	—
260	C₂×D₄	C₂³×D₄₇	1	8	yes	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	94	4	no	—
263	C₂²×C₄	#10625352	1	8	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
269	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
270	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
271	M₁₆	—	94	16	no	—
272	C₂×C₈	—	47	8	no	—
273	C₂×C₈	—	94	8	no	—
274	C₄⋊C₄	—	47	8	no	—
275	C₄⋊C₄	—	94	16	no	—
276	C₄⋊C₄	—	94	4	no	—
277	C₄⋊C₄	—	94	8	no	—
278	C₂²⋊C₄	—	94	8	no	—
279	C₄²	—	94	8	no	—
280	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
281	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
283	C₄²	C₂³×D₄₇	1	2	yes	—
284	C₄²	C₂³×D₄₇	1	2	yes	—
285	C₂³	—	2	4	no	—
286	C₂³	—	2	8	no	—
287	C₂³	—	4	32	no	—
288	C₂³	—	4	16	no	—
289	C₂³	—	4	8	no	—
290	C₂³	—	4	16	no	—
291	C₂³	—	8	32	no	—
292	C₂³	#13094011	1	4	yes	—
293	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
294	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
295	Q₈	—	188	16	no	—
296	D₄	—	2	16	no	—
297	D₄	—	4	32	no	—
298	D₄	—	4	16	no	—
299	D₄	—	4	8	no	—
300	D₄	—	4	32	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	16	no	—
303	C₂×C₄	—	94	8	no	—
304	C₂×C₄	—	188	8	no	—
305	C₂×C₄	#13094011	1	8	yes	—
306	C₂×C₄	#13094011	1	4	yes	—
307	C₂×C₄	#13094050	1	8	yes	—
308	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
310	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
312	C₂×C₄	C₂×#10625352	1	4	yes	—
313	C₂×C₄	C₂×#10625352	1	8	yes	—
314	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
315	C₈	—	94	8	no	—
316	C₂²	—	2	8	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	4	16	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	8	32	no	—
321	C₂²	—	8	16	no	—
322	C₂²	C₂×#13094011	1	4	yes	—
323	C₂²	C₂×#13094011	1	2	yes	—
324	C₂²	C₂×#13094050	1	4	yes	—
325	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	188	8	no	—
328	C₄	C₂×#13094011	1	4	yes	—
329	C₄	C₂×#13094011	1	2	yes	—
330	C₄	C₂×#13094050	1	8	yes	—
331	C₄	C₂²×#10625352	1	2	yes	—
332	C₂	—	2	4	no	—
333	C₂	—	8	8	no	—
334	C₂	#20253762	1	4	yes	direct product factor
335	C₂	C₂²×#13094011	1	1	yes	characteristic
336	C₂	C₂²×#13094050	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	C₂×#3138
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#15689680	C₂	semidirect
2. #20253762	C₂	direct (N × H)
3. C₂×#15689487	C₂	semidirect
4. C₂×#15689469	C₂	semidirect
5. C₂×#15689128	C₂	semidirect
6. C₂×C₄×#10625329	C₂	semidirect
10. #15689680	C₂²	semidirect
11. C₂×#13094024	C₂²	semidirect
12. C₂×#13094016	C₂²	semidirect
16. #15689487	C₂²	semidirect
17. #15689469	C₂²	semidirect
18. #15689128	C₂²	semidirect
19. C₂×#13093945	C₂²	semidirect
20. C₄×#10625329	C₂²	semidirect
21. C₂×#13093935	C₂²	semidirect
22. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
36. #13094024	C₂³	semidirect
38. #13094016	C₂³	semidirect
48. #13093945	C₂³	semidirect
49. #13093935	C₂³	semidirect
50. C₄×C₄₇⋊C₈	C₂³	semidirect
173. C₂×C₄₇	#3138	semidirect
199. C₄₇	C₂×#3138	semidirect
334. C₂	#20253762	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×#15689680	2
#20253762	2
C₂×#15689487	2
C₂×#15689469	2
C₂×#15689128	2
C₂×C₄×#10625329	2
C₂²×#127×C₄₇	2
C₂×#3138	47

Subgroups — C2×#20253762 #27085545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253762 #27085545