Subgroups — C₂×#20256393 #27097206

Order 12032.

1894 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689851	C₂	1	2	yes	maximal
2	#20256393	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689501	C₂	1	2	yes	maximal
5	C₂×#15689333	C₂	1	2	yes	maximal
6	C₂×#15689332	C₂	1	4	yes	maximal
7	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#15689217	C₂	1	2	yes	maximal
9	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
11	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
12	C₂²×#10625337	C₂²	1	4	yes	—
13	#15689851	—	2	8	no	—
14	#15689851	C₂²	1	16	yes	—
15	C₂×#13094024	C₂²	1	2	yes	—
16	C₂×#13094022	C₂²	1	4	yes	—
17	C₂×#13094016	—	2	2	no	—
18	C₂×#13094014	—	2	4	no	—
19	C₂×C₄×Dic₉₄	—	2	2	no	—
20	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
21	C₂×#13093989	C₂²	1	4	yes	—
22	#15689503	C₂²	1	8	yes	—
23	#15689501	C₂²	1	16	yes	—
24	#15689333	C₂²	1	16	yes	—
25	#15689332	C₂²	1	32	yes	—
26	C₈×Dic₉₄	C₂²	1	8	yes	—
27	#15689217	C₂²	1	16	yes	—
28	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
30	C₂×#13093947	C₂²	1	4	yes	—
31	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
34	C₄×Q₁₆×C₄₇	C₂²	1	8	yes	—
35	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#103×C₄₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
39	C₂²×Dic₉₄	C₂³	1	4	yes	—
40	C₂×Dic₁₈₈	—	2	8	no	—
41	C₂×Dic₁₈₈	—	4	4	no	—
42	C₂×Dic₁₈₈	C₂³	1	8	yes	—
43	C₂×Dic₁₈₈	C₂³	1	8	yes	—
44	C₂×#10625337	—	4	8	no	—
45	C₂×#10625337	C₂³	1	16	yes	—
46	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
47	#13094024	—	2	4	no	—
48	#13094024	C₂³	1	8	yes	—
49	#13094022	—	2	8	no	—
50	#13094022	C₂³	1	16	yes	—
51	#13094016	—	2	16	no	—
52	#13094014	—	2	32	no	—
53	C₄×Dic₉₄	—	2	16	no	—
54	C₄×Dic₉₄	D₄	1	4	yes	—
55	C₄×Dic₉₄	C₂³	1	4	yes	—
56	#13093989	—	2	8	no	—
57	#13093989	C₂³	1	16	yes	—
58	C₂×C₄⋊Dic₄₇	—	2	2	no	—
59	C₂×C₄⋊Dic₄₇	—	2	2	no	—
60	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
61	C₂×Dic₄₇⋊C₄	—	2	4	no	—
62	C₂×Dic₄₇⋊C₄	—	2	4	no	—
63	C₂×Dic₄₇⋊C₄	—	2	4	no	—
64	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
65	C₂×C₄×Dic₄₇	—	2	4	no	—
66	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
67	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
68	Dic₄₇⋊C₈	C₂³	1	8	yes	—
69	#13093947	C₂³	1	16	yes	—
70	C₈×Dic₄₇	C₂³	1	8	yes	—
71	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₄₇	—	2	4	no	—
73	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₄₇	—	2	4	no	—
77	C₄×Q₈×C₄₇	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₄₇	—	2	2	no	—
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	#107×C₄₇	C₂³	1	4	yes	—
83	#103×C₄₇	C₂³	1	8	yes	—
84	C₄×C₈×C₄₇	C₂³	1	4	yes	—
85	C₂×Q₈×C₄₇	—	2	2	no	—
86	C₂×Q₈×C₄₇	—	2	4	no	—
87	C₂×Q₈×C₄₇	—	2	4	no	—
88	C₂×Q₈×C₄₇	C₂⁴	1	4	yes	—
89	C₂²×C₄×C₄₇	—	2	2	no	—
90	C₂²×C₄×C₄₇	—	2	2	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₇	—	2	4	no	—
95	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
96	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
97	C₂×Dic₉₄	—	2	8	no	—
98	C₂×Dic₉₄	—	2	8	no	—
99	C₂×Dic₉₄	—	2	4	no	—
100	C₂×Dic₉₄	Q₁₆	1	8	yes	complemented
101	C₂×Dic₉₄	C₂×D₄	1	4	yes	—
102	C₂×Dic₉₄	C₂⁴	1	8	yes	—
103	Q₁₆×C₄₇	—	2	8	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	#41	1	4	yes	—
106	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
107	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
108	C₄⋊C₄×C₄₇	—	2	8	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	C₂⁴	1	4	yes	—
111	C₄²×C₄₇	—	2	8	no	—
112	C₄²×C₄₇	C₂×D₄	1	2	yes	—
113	C₄²×C₄₇	C₂⁴	1	2	yes	—
114	#10625337	—	4	16	no	—
115	C₄⋊Dic₄₇	—	2	8	no	—
116	C₄⋊Dic₄₇	—	2	8	no	—
117	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
118	Dic₄₇⋊C₄	—	2	16	no	—
119	Dic₄₇⋊C₄	—	2	16	no	—
120	Dic₄₇⋊C₄	—	2	16	no	—
121	Dic₄₇⋊C₄	C₂×D₄	1	8	yes	—
122	C₄×Dic₄₇	—	2	16	no	—
123	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
124	C₄×Dic₄₇	C₂⁴	1	4	yes	—
125	C₂×C₄₇⋊C₈	—	2	4	no	—
126	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
127	Dic₁₈₈	—	2	16	no	—
128	Dic₁₈₈	—	4	8	no	—
129	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	Q₈×C₄₇	—	2	4	no	—
131	Q₈×C₄₇	—	2	4	no	—
132	Q₈×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	4	no	—
135	C₂×C₄×C₄₇	—	2	4	no	—
136	C₂×C₄×C₄₇	—	4	4	no	—
137	C₂×C₄×C₄₇	—	4	4	no	—
138	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
139	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
144	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
145	C₂×Dic₄₇	—	2	4	no	—
146	C₂×Dic₄₇	—	2	8	no	—
147	C₂×Dic₄₇	—	4	8	no	—
148	C₂×Dic₄₇	C₂×Q₁₆	1	8	yes	—
149	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
150	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
151	Dic₉₄	—	2	8	no	—
152	Dic₉₄	—	2	8	no	—
153	Dic₉₄	—	4	8	no	—
154	Dic₉₄	C₂×Q₁₆	1	16	yes	complemented
155	C₈×C₄₇	—	2	2	no	—
156	C₈×C₄₇	C₂×#41	1	4	yes	—
157	C₄₇⋊C₈	—	2	4	no	—
158	C₂×#3395	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	#550	1	2	yes	—
160	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
161	C₂²×C₄₇	#578	1	2	yes	—
162	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	4	4	no	—
165	C₄×C₄₇	—	4	4	no	—
166	C₄×C₄₇	#550	1	2	yes	—
167	C₄×C₄₇	C₂²×Q₁₆	1	4	yes	—
168	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
169	C₄×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
170	Dic₄₇	—	2	4	no	—
171	Dic₄₇	—	4	8	no	—
172	Dic₄₇	C₂²×Q₁₆	1	8	yes	—
173	C₂×#557	—	47	2	no	—
174	#3395	—	47	16	no	—
175	C₂×#494	—	47	1	no	—
176	C₂×#494	—	47	2	no	—
177	C₂×#462	—	47	4	no	—
178	C₂×#462	—	47	2	no	—
179	C₂×C₈×Q₈	—	47	1	no	—
180	C₂×C₄×Q₁₆	—	47	2	no	—
181	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
182	C₂×C₄₇	#3395	1	4	yes	complemented
183	C₂×C₄₇	C₂×#550	1	1	yes	characteristic
184	C₂×C₄₇	C₂³×Q₁₆	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
186	C₂²×Q₁₆	—	47	4	no	—
187	C₂²×Q₁₆	—	47	2	no	—
188	C₂²×Q₁₆	—	47	2	no	—
189	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
190	#557	—	47	16	no	—
191	#557	—	94	8	no	—
192	C₂×C₄⋊Q₈	—	47	4	no	—
193	C₂×C₄⋊Q₈	—	47	2	no	—
194	C₂×#125	—	94	2	no	—
195	C₂×#125	—	94	4	no	—
196	C₂×C₄×Q₈	—	47	1	no	—
197	C₂×C₄×Q₈	—	47	4	no	—
198	C₂×C₄×Q₈	—	94	2	no	—
199	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
200	#494	—	47	16	no	—
201	#494	—	47	8	no	—
202	#462	—	47	16	no	—
203	#462	—	47	32	no	—
204	C₈×Q₈	—	47	8	no	—
205	C₄×Q₁₆	—	47	16	no	—
206	C₄×Q₁₆	C₂×D₄₇	1	8	yes	—
207	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
208	C₂×C₄⋊C₈	—	47	2	no	—
209	C₂×C₄⋊C₈	—	47	1	no	—
210	C₂×#103	—	47	4	no	—
211	C₂×#103	C₂×D₄₇	1	2	yes	—
212	C₂×C₄×C₈	—	47	2	no	—
213	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
214	C₄₇	C₂×#3395	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	47	1	no	—
216	C₂²×Q₈	—	47	4	no	—
217	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
218	C₂×Q₁₆	—	2	4	no	—
219	C₂×Q₁₆	—	47	16	no	—
220	C₂×Q₁₆	—	47	8	no	—
221	C₂×Q₁₆	—	47	8	no	—
222	C₂×Q₁₆	—	94	8	no	—
223	C₂×Q₁₆	—	188	4	no	—
224	C₂×Q₁₆	—	188	8	no	—
225	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
226	C₂²×C₈	—	47	2	no	—
227	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₄⋊Q₈	—	47	16	no	—
230	C₄⋊Q₈	—	47	8	no	—
231	C₄⋊Q₈	—	94	8	no	—
232	C₄⋊Q₈	—	94	4	no	—
233	#125	—	94	16	no	—
234	#125	—	94	32	no	—
235	C₄×Q₈	—	2	4	no	—
236	C₄×Q₈	—	47	4	no	—
237	C₄×Q₈	—	47	4	no	—
238	C₄×Q₈	—	47	16	no	—
239	C₄×Q₈	—	94	16	no	—
240	C₄×Q₈	—	94	8	no	—
241	C₄×Q₈	C₂²×D₄₇	1	8	yes	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	47	2	no	—
244	C₂×C₄⋊C₄	—	47	1	no	—
245	C₂×C₄⋊C₄	—	94	4	no	—
246	C₂×C₄⋊C₄	—	94	4	no	—
247	C₂×C₄⋊C₄	—	94	4	no	—
248	C₂×C₄⋊C₄	—	94	2	no	—
249	C₂×C₄⋊C₄	—	94	2	no	—
250	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	47	2	no	—
253	C₂×C₄²	—	94	4	no	—
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#107	C₂²×D₄₇	1	4	yes	—
256	C₄⋊C₈	—	47	8	no	—
257	C₄⋊C₈	—	47	4	no	—
258	#103	—	47	16	no	—
259	#103	C₂²×D₄₇	1	8	yes	—
260	C₄×C₈	—	47	8	no	—
261	C₄×C₈	C₂²×D₄₇	1	4	yes	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	4	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	47	8	no	—
266	C₂×Q₈	—	47	4	no	—
267	C₂×Q₈	—	47	8	no	—
268	C₂×Q₈	—	94	4	no	—
269	C₂×Q₈	—	94	8	no	—
270	C₂×Q₈	—	94	8	no	—
271	C₂×Q₈	C₂³×D₄₇	1	4	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	47	2	no	—
275	C₂²×C₄	—	47	2	no	—
276	C₂²×C₄	—	94	4	no	—
277	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
278	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
279	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	Q₁₆	—	2	8	no	—
281	Q₁₆	—	94	16	no	—
282	Q₁₆	—	188	8	no	—
283	Q₁₆	—	188	16	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	47	4	no	—
286	C₂×C₈	—	94	4	no	—
287	C₂×C₈	#10625350	1	4	yes	—
288	C₂×C₈	C₂³×D₄₇	1	2	yes	—
289	C₂×C₈	C₂³×D₄₇	1	2	yes	—
290	C₄⋊C₄	—	2	8	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	47	8	no	—
293	C₄⋊C₄	—	47	4	no	—
294	C₄⋊C₄	—	94	16	no	—
295	C₄⋊C₄	—	94	16	no	—
296	C₄⋊C₄	—	94	16	no	—
297	C₄⋊C₄	—	94	8	no	—
298	C₄⋊C₄	—	94	8	no	—
299	C₄⋊C₄	C₂³×D₄₇	1	4	yes	—
300	C₄²	—	2	8	no	—
301	C₄²	—	47	4	no	—
302	C₄²	—	47	4	no	—
303	C₄²	—	94	16	no	—
304	C₄²	D₄×D₄₇	1	2	yes	—
305	C₄²	C₂³×D₄₇	1	2	yes	—
306	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	47	16	no	—
311	Q₈	—	94	8	no	—
312	Q₈	—	94	8	no	—
313	Q₈	—	188	8	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	47	8	no	—
320	C₂×C₄	—	47	4	no	—
321	C₂×C₄	—	47	4	no	—
322	C₂×C₄	—	94	4	no	—
323	C₂×C₄	—	94	8	no	—
324	C₂×C₄	—	188	8	no	—
325	C₂×C₄	Q₁₆×D₄₇	1	4	yes	—
326	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
327	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
329	C₂×C₄	C₂×#10625350	1	2	yes	—
330	C₂×C₄	#13094070	1	2	yes	—
331	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	94	4	no	—
334	C₈	C₂×#10625350	1	4	yes	—
335	C₂²	#15689794	1	2	yes	—
336	C₂²	C₂×Q₁₆×D₄₇	1	2	yes	—
337	C₂²	#15689937	1	2	yes	—
338	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	4	4	no	—
341	C₄	—	4	4	no	—
342	C₄	—	47	8	no	—
343	C₄	—	94	4	no	—
344	C₄	—	188	8	no	—
345	C₄	#15689794	1	2	yes	—
346	C₄	C₂×Q₁₆×D₄₇	1	4	yes	—
347	C₄	C₂²×#10625350	1	1	yes	characteristic
348	C₄	C₂×#13094070	1	1	yes	characteristic
349	C₂	#20256393	1	4	yes	direct product factor
350	C₂	C₂×#15689794	1	1	yes	characteristic
351	C₂	C₂²×Q₁₆×D₄₇	1	1	yes	characteristic
352	C₂	C₂×#15689937	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#3395
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20256393	C₂	direct (N × H)
100. C₂×Dic₉₄	Q₁₆	semidirect
154. Dic₉₄	C₂×Q₁₆	semidirect
182. C₂×C₄₇	#3395	semidirect
214. C₄₇	C₂×#3395	semidirect
349. C₂	#20256393	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×#15689851	2
#20256393	2
C₂×#15689503	2
C₂×#15689501	2
C₂×#15689333	2
C₂×#15689332	2
C₂×C₈×Dic₉₄	2
C₂×#15689217	2
C₂×C₄×Q₁₆×C₄₇	2
C₂×#3395	47

Subgroups — C2×#20256393 #27097206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256393 #27097206