Subgroups — C₂×#20256391 #27097204

Order 12032.

1894 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×#15689852	C₂	1	1	yes	maximal, characteristic
2	#20256391	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689502	C₂	1	3	yes	maximal
4	C₂×#15689333	C₂	1	3	yes	maximal
5	C₂×#15689330	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#10625337	C₂	1	3	yes	maximal
8	C₂×C₄₇×#557	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625337	C₂²	1	3	yes	—
10	C₂²×#10625337	C₂²	1	6	yes	—
11	#15689852	—	2	4	no	—
12	#15689852	C₂²	1	8	yes	—
13	C₂×#13094024	C₂²	1	3	yes	—
14	C₂×#13094015	—	2	3	no	—
15	C₂×Q₈×Dic₄₇	—	2	1	no	—
16	C₂×C₄×Dic₉₄	C₂²	1	3	yes	—
17	#15689502	C₂²	1	24	yes	—
18	#15689333	C₂²	1	24	yes	—
19	#15689330	C₂²	1	24	yes	—
20	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	—
21	C₄×#10625337	C₂²	1	24	yes	—
22	C₂×#13093957	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
24	C₂×#13093948	C₂²	1	3	yes	—
25	C₂×#13093947	C₂²	1	3	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
27	C₄₇×#557	—	2	4	no	—
28	C₄₇×#557	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
30	C₂×#125×C₄₇	—	2	3	no	—
31	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
33	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂²×Dic₉₄	C₂³	1	3	yes	—
35	C₂×#10625337	—	2	12	no	—
36	C₂×#10625337	—	4	12	no	—
37	C₂×#10625337	C₂³	1	12	yes	—
38	C₂×#10625337	C₂³	1	24	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	#13094024	—	2	6	no	—
42	#13094024	C₂³	1	12	yes	—
43	#13094015	—	2	24	no	—
44	Q₈×Dic₄₇	—	2	8	no	—
45	C₄×Dic₉₄	—	2	6	no	—
46	C₄×Dic₉₄	C₂³	1	12	yes	—
47	C₂×C₄⋊Dic₄₇	—	2	3	no	—
48	C₂×C₄⋊Dic₄₇	—	2	3	no	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×Dic₄₇⋊C₄	—	2	3	no	—
51	C₂×C₄×Dic₄₇	—	2	3	no	—
52	#13093957	C₂³	1	4	yes	—
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	12	yes	—
54	#13093948	C₂³	1	12	yes	—
55	#13093947	C₂³	1	12	yes	—
56	C₄×C₄₇⋊C₈	C₂³	1	12	yes	—
57	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₄₇	C₂³	1	3	yes	—
59	C₄⋊Q₈×C₄₇	—	2	6	no	—
60	C₄⋊Q₈×C₄₇	C₂³	1	12	yes	—
61	#125×C₄₇	—	2	24	no	—
62	C₄×Q₈×C₄₇	—	2	8	no	—
63	C₄×Q₈×C₄₇	—	2	6	no	—
64	C₄×Q₈×C₄₇	D₄	1	4	yes	—
65	C₄×Q₈×C₄₇	C₂³	1	12	yes	—
66	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
68	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
69	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
70	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	—	2	3	no	—
73	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
74	C₂×Q₈×C₄₇	—	2	3	no	—
75	C₂×Q₈×C₄₇	—	2	6	no	—
76	C₂×Q₈×C₄₇	—	2	6	no	—
77	C₂×Q₈×C₄₇	Q₁₆	1	8	yes	complemented
78	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
79	C₂×Q₈×C₄₇	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₄₇	—	2	1	no	—
81	C₂²×C₄×C₄₇	—	2	3	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	1	no	—
86	C₂²×Dic₄₇	—	2	3	no	—
87	C₂×Dic₉₄	—	2	6	no	—
88	C₂×Dic₉₄	—	2	6	no	—
89	C₂×Dic₉₄	—	2	3	no	—
90	C₂×Dic₉₄	C₂⁴	1	6	yes	—
91	C₄⋊C₄×C₄₇	—	2	12	no	—
92	C₄⋊C₄×C₄₇	—	2	12	no	—
93	C₄⋊C₄×C₄₇	—	2	12	no	—
94	C₄⋊C₄×C₄₇	—	2	1	no	—
95	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
96	C₄⋊C₄×C₄₇	C₂⁴	1	2	yes	—
97	C₄²×C₄₇	—	2	12	no	—
98	C₄²×C₄₇	C₂×D₄	1	6	yes	—
99	C₄²×C₄₇	C₂⁴	1	6	yes	—
100	#10625337	—	2	24	no	—
101	#10625337	—	4	24	no	—
102	C₄⋊Dic₄₇	—	2	12	no	—
103	C₄⋊Dic₄₇	—	2	12	no	—
104	C₄⋊Dic₄₇	—	2	1	no	—
105	C₄⋊Dic₄₇	C₂⁴	1	2	yes	—
106	Dic₄₇⋊C₄	—	2	12	no	—
107	C₄×Dic₄₇	—	2	12	no	—
108	C₂×C₄₇⋊C₈	—	2	6	no	—
109	C₂×C₄₇⋊C₈	#41	1	4	yes	—
110	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
111	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₄₇	—	2	6	no	—
114	Q₈×C₄₇	—	2	6	no	—
115	Q₈×C₄₇	—	4	6	no	—
116	Q₈×C₄₇	C₂×Q₁₆	1	16	yes	complemented
117	C₂×C₄×C₄₇	—	2	6	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	6	no	—
120	C₂×C₄×C₄₇	—	4	6	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	C₂×Q₁₆	1	12	yes	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	2	6	no	—
131	C₂×Dic₄₇	—	4	6	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	Dic₉₄	—	2	6	no	—
134	Dic₉₄	—	2	6	no	—
135	Dic₉₄	—	4	6	no	—
136	C₄₇⋊C₈	—	2	6	no	—
137	C₄₇⋊C₈	C₂×#41	1	4	yes	—
138	C₂×#3395	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	#550	1	2	yes	—
140	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
141	C₂²×C₄₇	#578	1	2	yes	—
142	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₄₇	—	2	6	no	—
144	C₄×C₄₇	—	4	6	no	—
145	C₄×C₄₇	—	4	2	no	—
146	C₄×C₄₇	#550	1	2	yes	—
147	C₄×C₄₇	C₂²×Q₁₆	1	12	yes	—
148	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
149	C₄×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	Dic₄₇	—	4	6	no	—
151	Dic₄₇	—	4	2	no	—
152	C₂×#557	—	47	1	no	—
153	C₂×#557	D₄₇	1	1	yes	characteristic
154	#3395	—	47	16	no	—
155	C₂×#494	—	47	3	no	—
156	C₂×#462	—	47	3	no	—
157	C₂×#462	—	47	3	no	—
158	C₂×C₈×Q₈	—	47	1	no	—
159	C₂×C₄×Q₁₆	—	47	3	no	—
160	C₂×C₄₇	#3395	1	4	yes	complemented
161	C₂×C₄₇	C₂×#550	1	1	yes	characteristic
162	C₂×C₄₇	C₂³×Q₁₆	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
164	C₂²×Q₁₆	—	47	3	no	—
165	C₂²×Q₁₆	—	47	6	no	—
166	#557	—	2	4	no	—
167	#557	—	47	8	no	—
168	#557	—	94	4	no	—
169	#557	C₂×D₄₇	1	8	yes	—
170	C₂×C₄⋊Q₈	—	47	3	no	—
171	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
172	C₂×#125	—	2	3	no	—
173	C₂×#125	—	94	3	no	—
174	C₂×C₄×Q₈	—	2	1	no	—
175	C₂×C₄×Q₈	—	47	3	no	—
176	C₂×C₄×Q₈	—	94	1	no	—
177	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
178	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
179	#494	—	47	24	no	—
180	#462	—	47	24	no	—
181	#462	—	47	24	no	—
182	C₈×Q₈	—	47	8	no	—
183	C₄×Q₁₆	—	47	24	no	—
184	C₂×#107	—	47	1	no	—
185	C₂×C₄⋊C₈	—	47	3	no	—
186	C₂×#103	—	47	3	no	—
187	C₂×#103	—	47	3	no	—
188	C₂×C₄×C₈	—	47	3	no	—
189	C₄₇	C₂×#3395	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	47	3	no	—
191	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂²×Q₈	C₂²×D₄₇	1	3	yes	—
193	C₂×Q₁₆	—	47	12	no	—
194	C₂×Q₁₆	—	47	24	no	—
195	C₂×Q₁₆	—	94	12	no	—
196	C₂×Q₁₆	—	188	12	no	—
197	C₂²×C₈	—	47	1	no	—
198	C₂²×C₈	—	47	3	no	—
199	C₄⋊Q₈	—	2	6	no	—
200	C₄⋊Q₈	—	47	12	no	—
201	C₄⋊Q₈	—	94	6	no	—
202	C₄⋊Q₈	C₂²×D₄₇	1	12	yes	—
203	#125	—	2	24	no	—
204	#125	—	94	24	no	—
205	C₄×Q₈	—	2	8	no	—
206	C₄×Q₈	—	2	6	no	—
207	C₄×Q₈	—	47	12	no	—
208	C₄×Q₈	—	94	8	no	—
209	C₄×Q₈	—	94	6	no	—
210	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
211	C₄×Q₈	C₂²×D₄₇	1	12	yes	—
212	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
213	C₂×C₄⋊C₄	—	2	3	no	—
214	C₂×C₄⋊C₄	—	2	3	no	—
215	C₂×C₄⋊C₄	—	2	3	no	—
216	C₂×C₄⋊C₄	—	47	1	no	—
217	C₂×C₄⋊C₄	—	94	3	no	—
218	C₂×C₄⋊C₄	—	94	3	no	—
219	C₂×C₄⋊C₄	—	94	3	no	—
220	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
221	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	—	2	3	no	—
223	C₂×C₄²	—	94	3	no	—
224	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
225	#107	—	47	4	no	—
226	C₄⋊C₈	—	47	12	no	—
227	#103	—	47	12	no	—
228	#103	—	47	12	no	—
229	C₄×C₈	—	47	12	no	—
230	C₂×Q₈	—	2	3	no	—
231	C₂×Q₈	—	2	6	no	—
232	C₂×Q₈	—	2	6	no	—
233	C₂×Q₈	—	47	6	no	—
234	C₂×Q₈	—	94	3	no	—
235	C₂×Q₈	—	94	6	no	—
236	C₂×Q₈	—	94	6	no	—
237	C₂×Q₈	#10625337	1	8	yes	complemented
238	C₂×Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
239	C₂×Q₈	C₂³×D₄₇	1	6	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	3	no	—
242	C₂²×C₄	—	94	1	no	—
243	C₂²×C₄	—	94	3	no	—
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	Q₁₆	—	94	24	no	—
248	Q₁₆	—	188	24	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	47	2	no	—
251	C₂×C₈	—	47	6	no	—
252	C₂×C₈	—	94	6	no	—
253	C₄⋊C₄	—	2	12	no	—
254	C₄⋊C₄	—	2	12	no	—
255	C₄⋊C₄	—	2	12	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	47	2	no	—
258	C₄⋊C₄	—	94	12	no	—
259	C₄⋊C₄	—	94	1	no	—
260	C₄⋊C₄	—	94	12	no	—
261	C₄⋊C₄	—	94	12	no	—
262	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
263	C₄⋊C₄	C₂³×D₄₇	1	2	yes	—
264	C₄²	—	2	12	no	—
265	C₄²	—	94	12	no	—
266	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
267	C₄²	C₂³×D₄₇	1	6	yes	—
268	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
269	Q₈	—	2	6	no	—
270	Q₈	—	2	6	no	—
271	Q₈	—	4	6	no	—
272	Q₈	—	94	6	no	—
273	Q₈	—	94	6	no	—
274	Q₈	—	188	6	no	—
275	Q₈	C₂×#10625337	1	16	yes	complemented
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	6	no	—
279	C₂×C₄	—	4	6	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	94	6	no	—
283	C₂×C₄	—	188	6	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	C₂×#10625337	1	12	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
287	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
289	C₂×C₄	#41×D₄₇	1	2	yes	—
290	C₂×C₄	#13094070	1	2	yes	—
291	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
292	C₈	—	47	4	no	—
293	C₈	—	94	6	no	—
294	C₂²	#15689792	1	2	yes	—
295	C₂²	C₂²×#10625337	1	2	yes	—
296	C₂²	#15689938	1	2	yes	—
297	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	4	6	no	—
300	C₄	—	4	2	no	—
301	C₄	—	188	6	no	—
302	C₄	—	188	2	no	—
303	C₄	#15689792	1	2	yes	—
304	C₄	C₂²×#10625337	1	12	yes	—
305	C₄	C₂×#41×D₄₇	1	1	yes	characteristic
306	C₄	C₂×#13094070	1	1	yes	characteristic
307	C₂	#20256391	1	4	yes	direct product factor
308	C₂	C₂×#15689792	1	1	yes	characteristic
309	C₂	C₂³×#10625337	1	1	yes	characteristic
310	C₂	C₂×#15689938	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#557	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#557	D₄₇
hypocenter	C₄₇	C₂×#3395
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
2. #20256391	C₂	direct (N × H)
77. C₂×Q₈×C₄₇	Q₁₆	semidirect
116. Q₈×C₄₇	C₂×Q₁₆	semidirect
160. C₂×C₄₇	#3395	semidirect
189. C₄₇	C₂×#3395	semidirect
237. C₂×Q₈	#10625337	semidirect
275. Q₈	C₂×#10625337	semidirect
307. C₂	#20256391	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×#15689852	2
#20256391	2
C₂×#15689502	2
C₂×#15689333	2
C₂×#15689330	2
C₂×Q₈×C₄₇⋊C₈	2
C₂×C₄×#10625337	2
C₂×C₄₇×#557	2
C₂×#3395	47

Subgroups — C2×#20256391 #27097204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256391 #27097204