Subgroups — C₂×#20244894 #26245387

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689163	C₂	1	2	yes	maximal
2	C₂×#15689149	C₂	1	1	yes	maximal, characteristic
3	#20244894	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688892	C₂	1	2	yes	maximal
5	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	1	yes	characteristic
10	#15689163	—	2	4	no	—
11	#15689163	C₂²	1	8	yes	—
12	C₂×#13093957	—	2	2	no	—
13	C₂×#13093955	—	2	2	no	—
14	#15689149	—	2	2	no	—
15	#15689149	C₂²	1	4	yes	—
16	C₂×Dic₄₇⋊C₈	—	2	2	no	—
17	#15688892	C₂²	1	8	yes	—
18	#15688888	C₂²	1	4	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×#107×C₄₇	—	2	2	no	—
22	C₂×#107×C₄₇	—	2	2	no	—
23	C₂×#107×C₄₇	C₂²	1	4	yes	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	1	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	Q₈	1	8	yes	—
29	C₂×#10625329	C₂³	1	2	yes	—
30	C₂×#10625329	C₂³	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	—	2	1	no	—
33	#13093983	—	2	1	no	—
34	#13093983	—	2	1	no	—
35	#13093983	D₄	1	2	yes	—
36	#13093983	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	#13093957	—	2	8	no	—
42	#13093955	—	2	8	no	—
43	Dic₄₇⋊C₈	—	2	8	no	—
44	C₂³×C₄×C₄₇	—	2	2	no	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	—	2	4	no	—
47	C₂²×C₈×C₄₇	D₄	1	2	yes	—
48	C₂²×C₈×C₄₇	D₄	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
56	#107×C₄₇	—	2	8	no	—
57	C₂⁴×C₄₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	8	4	no	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	1	no	—
71	C₂²×Dic₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	4	2	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₂²⋊Dic₄₇	—	2	2	no	—
81	C₂²⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	1	no	—
83	C₄⋊Dic₄₇	—	2	1	no	—
84	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
85	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
86	C₄×Dic₄₇	—	2	2	no	—
87	C₄×Dic₄₇	—	2	2	no	—
88	#10625329	—	2	8	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	2	4	no	—
91	#10625329	—	4	2	no	—
92	#10625329	C₂×Q₈	1	8	yes	—
93	C₂×C₄₇⋊C₈	—	2	8	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	#121	1	1	yes	characteristic
100	C₂³×C₄₇	#121	1	1	yes	characteristic
101	C₂³×C₄₇	#125	1	4	yes	—
102	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	8	4	no	—
105	C₂×C₄×C₄₇	—	8	4	no	—
106	C₂×C₄×C₄₇	—	8	4	no	—
107	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	#123	1	2	yes	—
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	#125	1	4	yes	—
114	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₈×C₄₇	—	2	4	no	—
125	C₄₇⋊C₈	—	2	8	no	—
126	C₄₇⋊C₈	—	4	2	no	—
127	C₂×#2089	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	8	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	#398	1	2	yes	—
134	C₂²×C₄₇	#398	1	2	yes	—
135	C₂²×C₄₇	#472	1	2	yes	—
136	C₂²×C₄₇	#473	1	2	yes	—
137	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#125	1	2	yes	—
139	C₄×C₄₇	—	8	4	no	—
140	C₄×C₄₇	#398	1	2	yes	—
141	C₄×C₄₇	#398	1	2	yes	—
142	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
143	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
144	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	Dic₄₇	—	4	2	no	—
146	Dic₄₇	—	4	2	no	—
147	C₂×#428	—	47	2	no	—
148	C₂²×#107	D₄₇	1	1	yes	characteristic
149	C₂×#424	—	47	1	no	—
150	#2089	—	47	8	no	—
151	C₂×#343	—	47	2	no	—
152	C₂×#341	—	47	1	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#2089	1	4	yes	complemented
156	C₂×C₄₇	C₂×#398	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
159	C₂²×M₁₆	—	47	2	no	—
160	C₂²×M₁₆	—	47	1	no	—
161	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#117	—	47	1	no	—
163	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
164	#428	—	47	8	no	—
165	#428	—	94	4	no	—
166	C₂×#107	—	2	2	no	—
167	C₂×#107	—	2	2	no	—
168	C₂×#107	—	94	2	no	—
169	C₂×#107	C₂×D₄₇	1	4	yes	—
170	C₂×#106	—	94	2	no	—
171	#424	—	47	4	no	—
172	#424	—	94	2	no	—
173	C₂×C₄⋊C₈	—	94	2	no	—
174	#343	—	47	8	no	—
175	#341	—	47	4	no	—
176	C₄₇	C₂×#2089	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
179	C₂×M₁₆	—	47	8	no	—
180	C₂×M₁₆	—	47	4	no	—
181	C₂×M₁₆	—	47	2	no	—
182	C₂×M₁₆	—	94	4	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	94	1	no	—
185	C₂×M₁₆	—	94	4	no	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	94	2	no	—
188	C₂²×C₈	—	94	1	no	—
189	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
191	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
192	#117	—	47	2	no	—
193	#117	—	47	2	no	—
194	#117	—	94	1	no	—
195	#117	—	94	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	47	1	no	—
202	C₂×C₄⋊C₄	—	47	1	no	—
203	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
204	C₂×C₂²⋊C₄	—	94	1	no	—
205	C₂×C₄²	—	94	1	no	—
206	#107	—	2	8	no	—
207	#107	—	94	8	no	—
208	#106	—	94	8	no	—
209	C₄⋊C₈	—	94	8	no	—
210	C₂⁴	#10625352	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	—	94	1	no	—
216	C₂²×C₄	—	94	1	no	—
217	C₂²×C₄	#10625350	1	4	yes	—
218	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	#10625352	1	1	yes	characteristic
221	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	M₁₆	—	47	8	no	—
226	M₁₆	—	94	8	no	—
227	M₁₆	—	94	4	no	—
228	M₁₆	—	94	4	no	—
229	M₁₆	—	188	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	94	8	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	47	2	no	—
243	C₄⋊C₄	—	47	2	no	—
244	C₄⋊C₄	—	94	1	no	—
245	C₄⋊C₄	—	94	1	no	—
246	C₂²⋊C₄	—	94	2	no	—
247	C₂²⋊C₄	—	94	2	no	—
248	C₄²	—	94	2	no	—
249	C₄²	—	94	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	#13093995	1	1	yes	characteristic
253	C₂³	#13093995	1	1	yes	characteristic
254	C₂³	#13094014	1	4	yes	—
255	C₂³	C₂×#10625352	1	1	yes	characteristic
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	—	94	2	no	—
261	C₂×C₄	—	94	2	no	—
262	C₂×C₄	—	188	2	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	#13093995	1	1	yes	characteristic
265	C₂×C₄	#13093995	1	1	yes	characteristic
266	C₂×C₄	#13094002	1	4	yes	—
267	C₂×C₄	#13094002	1	4	yes	—
268	C₂×C₄	#13094007	1	2	yes	—
269	C₂×C₄	#13094007	1	2	yes	—
270	C₂×C₄	#13094014	1	4	yes	—
271	C₂×C₄	#13094022	1	2	yes	—
272	C₂×C₄	#13094022	1	2	yes	—
273	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
276	C₂×C₄	C₂×#10625350	1	2	yes	—
277	C₈	—	2	4	no	—
278	C₈	—	94	8	no	—
279	C₈	—	188	2	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#15689054	1	2	yes	—
286	C₂²	#15689054	1	2	yes	—
287	C₂²	#15689390	1	2	yes	—
288	C₂²	#15689395	1	2	yes	—
289	C₂²	C₂×#13093995	1	1	yes	characteristic
290	C₂²	C₂×#13094014	1	2	yes	—
291	C₄	—	8	4	no	—
292	C₄	—	188	2	no	—
293	C₄	—	188	2	no	—
294	C₄	#15689054	1	2	yes	—
295	C₄	#15689054	1	2	yes	—
296	C₄	C₂×#13094002	1	2	yes	—
297	C₄	C₂×#13094007	1	1	yes	characteristic
298	C₄	C₂×#13094022	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#20244894	1	4	yes	direct product factor
302	C₂	C₂×#15689054	1	1	yes	characteristic
303	C₂	C₂×#15689390	1	1	yes	characteristic
304	C₂	C₂×#15689395	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	C₂×#2089
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #20244894	C₂	direct (N × H)
155. C₂×C₄₇	#2089	semidirect
176. C₄₇	C₂×#2089	semidirect
301. C₂	#20244894	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×#15689163	2
C₂×#15689149	2
#20244894	2
C₂×#15688892	2
C₂×#15688888	2
C₂²×#107×C₄₇	2
C₂×#2089	47

Subgroups — C2×#20244894 #26245387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244894 #26245387