Subgroups — C₂×#20242758 #26198612

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689408	C₂	1	2	yes	maximal
3	#20242758	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688870	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₈₈	—	2	1	no	—
8	C₂²×Dic₁₈₈	—	2	1	no	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	#15689503	—	2	2	no	—
11	#15689503	C₂²	1	4	yes	—
12	#15689408	—	2	4	no	—
13	#15689408	C₂²	1	8	yes	—
14	C₂×#13093955	—	2	2	no	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093948	—	2	2	no	—
17	#15688876	C₂²	1	4	yes	—
18	#15688870	C₂²	1	8	yes	—
19	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₄₇	—	2	2	no	—
21	C₈×Q₈×C₄₇	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	—	2	1	no	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂×Dic₁₈₈	—	2	4	no	—
28	C₂×Dic₁₈₈	—	2	4	no	—
29	C₂×Dic₁₈₈	—	4	4	no	—
30	C₂×Dic₁₈₈	—	4	4	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	#13094024	—	2	2	no	—
33	#13094024	—	2	2	no	—
34	#13094024	—	2	2	no	—
35	#13094024	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	#13093955	—	2	8	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
41	#13093948	—	2	8	no	—
42	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	—	2	1	no	—
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₄₇	—	2	1	no	—
47	C₄×Q₈×C₄₇	—	2	1	no	—
48	C₄×Q₈×C₄₇	D₄	1	2	yes	—
49	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	—	2	4	no	—
55	C₄⋊C₈×C₄₇	—	2	1	no	—
56	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
57	C₄×C₈×C₄₇	—	2	4	no	—
58	C₄×C₈×C₄₇	D₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂×Q₈×C₄₇	—	2	1	no	—
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	Q₁₆	1	4	yes	complemented
63	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂²×C₄×C₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂×Dic₉₄	—	2	4	no	—
71	C₂×Dic₉₄	—	4	4	no	—
72	C₂×Dic₉₄	—	4	4	no	—
73	C₂×Dic₉₄	—	4	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
78	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₄₇	—	2	2	no	—
81	C₄⋊C₄×C₄₇	—	2	1	no	—
82	C₄⋊C₄×C₄₇	—	2	2	no	—
83	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₇	#41	1	1	yes	characteristic
89	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	4	4	no	—
96	Dic₁₈₈	—	4	8	no	—
97	Dic₁₈₈	—	4	8	no	—
98	C₂³×C₄₇	#121	1	1	yes	characteristic
99	Q₈×C₄₇	—	2	2	no	—
100	Q₈×C₄₇	—	2	4	no	—
101	Q₈×C₄₇	C₂×Q₁₆	1	4	yes	complemented
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₄₇	#135	1	2	yes	—
113	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	8	4	no	—
119	Dic₉₄	—	4	4	no	—
120	Dic₉₄	—	4	4	no	—
121	Dic₉₄	—	8	4	no	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	—	4	2	no	—
124	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
125	C₄₇⋊C₈	—	4	4	no	—
126	C₂×#1670	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	#460	1	2	yes	—
128	C₂²×C₄₇	#467	1	2	yes	—
129	C₂²×C₄₇	#473	1	2	yes	—
130	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	4	2	no	—
135	C₄×C₄₇	#460	1	2	yes	—
136	C₄×C₄₇	#467	1	2	yes	—
137	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
140	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	Dic₄₇	—	8	4	no	—
142	C₂×#494	—	47	1	no	—
143	C₂×#475	—	47	2	no	—
144	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
145	#1670	—	47	8	no	—
146	C₂×#335	—	47	1	no	—
147	C₂×#332	—	47	2	no	—
148	C₂×C₄₇	#1670	1	4	yes	complemented
149	C₂×C₄₇	C₂×#460	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
152	C₂²×Q₁₆	—	94	1	no	—
153	C₂²×Q₁₆	—	94	1	no	—
154	C₂×C₄⋊Q₈	—	47	2	no	—
155	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
156	#494	—	47	4	no	—
157	#494	—	94	2	no	—
158	#475	—	47	8	no	—
159	#475	—	94	4	no	—
160	C₈×Q₈	—	2	2	no	—
161	C₈×Q₈	C₂×D₄₇	1	4	yes	—
162	C₂×#106	—	94	2	no	—
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	—	47	2	no	—
165	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
166	C₂×#103	—	94	2	no	—
167	C₂×C₄×C₈	—	2	1	no	—
168	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
169	#335	—	47	4	no	—
170	#332	—	47	8	no	—
171	C₄₇	C₂×#1670	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₈	—	94	2	no	—
173	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
174	C₂×Q₁₆	—	94	4	no	—
175	C₂×Q₁₆	—	94	4	no	—
176	C₂×Q₁₆	—	188	4	no	—
177	C₂×Q₁₆	—	188	4	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	94	2	no	—
180	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₄⋊Q₈	—	47	4	no	—
183	C₄⋊Q₈	—	94	2	no	—
184	C₄⋊Q₈	—	94	2	no	—
185	C₄⋊Q₈	—	94	2	no	—
186	C₄×Q₈	—	2	1	no	—
187	C₄×Q₈	—	2	1	no	—
188	C₄×Q₈	D₁₈₈	1	2	yes	—
189	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	94	2	no	—
192	C₂×C₄⋊C₄	—	94	2	no	—
193	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	#106	—	94	8	no	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	47	4	no	—
200	C₄⋊C₈	—	94	2	no	—
201	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
202	#103	—	94	8	no	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
205	C₄×C₈	C₂²×D₄₇	1	2	yes	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	94	4	no	—
209	C₂×Q₈	—	188	4	no	—
210	C₂×Q₈	—	188	4	no	—
211	C₂×Q₈	—	188	2	no	—
212	C₂×Q₈	Dic₁₈₈	1	4	yes	complemented
213	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	188	2	no	—
217	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
218	C₂²×C₄	#10625350	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	Q₁₆	—	188	8	no	—
221	Q₁₆	—	188	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	188	4	no	—
227	C₂×C₈	#10625336	1	4	yes	—
228	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
229	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₄⋊C₄	—	188	2	no	—
236	C₄⋊C₄	—	188	2	no	—
237	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
241	C₄²	#10625350	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄²	C₂³×D₄₇	1	1	yes	characteristic
244	C₂³	#13093998	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	2	no	—
247	Q₈	—	376	4	no	—
248	Q₈	—	188	4	no	—
249	Q₈	—	188	4	no	—
250	Q₈	C₂×Dic₁₈₈	1	4	yes	complemented
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	376	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	#13093998	1	1	yes	characteristic
259	C₂×C₄	#13093998	1	1	yes	characteristic
260	C₂×C₄	#13093998	1	1	yes	characteristic
261	C₂×C₄	C₂×#10625336	1	2	yes	—
262	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
263	C₂×C₄	#13094046	1	2	yes	—
264	C₂×C₄	#13094049	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
267	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	4	2	no	—
270	C₈	—	188	4	no	—
271	C₈	C₂×#10625336	1	4	yes	—
272	C₂²	#15689323	1	2	yes	—
273	C₂²	#15689364	1	2	yes	—
274	C₂²	#15689398	1	2	yes	—
275	C₂²	C₂×#13093998	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	376	4	no	—
281	C₄	#15689323	1	2	yes	—
282	C₄	#15689364	1	2	yes	—
283	C₄	C₂²×#10625336	1	1	yes	characteristic
284	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
285	C₄	C₂×#13094046	1	1	yes	characteristic
286	C₄	C₂×#13094049	1	1	yes	characteristic
287	C₂	#20242758	1	4	yes	direct product factor
288	C₂	C₂×#15689323	1	1	yes	characteristic
289	C₂	C₂×#15689364	1	1	yes	characteristic
290	C₂	C₂×#15689398	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#121
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	C₂×#1670
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20242758	C₂	direct (N × H)
62. C₂×Q₈×C₄₇	Q₁₆	semidirect
101. Q₈×C₄₇	C₂×Q₁₆	semidirect
148. C₂×C₄₇	#1670	semidirect
171. C₄₇	C₂×#1670	semidirect
212. C₂×Q₈	Dic₁₈₈	semidirect
250. Q₈	C₂×Dic₁₈₈	semidirect
287. C₂	#20242758	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×#15689503	2
C₂×#15689408	2
#20242758	2
C₂×#15688876	2
C₂×#15688870	2
C₂×C₈×Q₈×C₄₇	2
C₂×#1670	47

Subgroups — C2×#20242758 #26198612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242758 #26198612