Subgroups — C₂×#20241363 #26126068

Order 12032.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689241	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241363	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13094046	C₄	1	1	yes	characteristic
6	C₂×#13094030	—	2	1	no	—
7	#15689241	—	2	1	no	—
8	#15689241	—	2	1	no	—
9	#15689241	C₄	1	2	yes	—
10	#15689241	C₂²	1	2	yes	—
11	C₂×#13093960	C₄	1	1	yes	characteristic
12	C₂×#13093952	—	2	1	no	—
13	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
14	#15688844	—	2	1	no	—
15	#15688844	C₂²	1	2	yes	complemented
16	C₂×M₃₂×C₄₇	—	2	1	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#364	—	2	1	no	—
19	C₄₇×#364	C₂²	1	2	yes	complemented
20	C₂×#10625350	—	2	1	no	—
21	#13094046	—	2	1	no	—
22	#13094046	—	2	1	no	—
23	#13094046	C₈	1	2	yes	complemented
24	#13094046	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₈₈	C₈	1	1	yes	characteristic, complemented
26	C₂×#10625325	—	2	1	no	—
27	C₂×D₃₇₆	C₈	1	1	yes	characteristic, complemented
28	#13094030	—	2	2	no	—
29	#13094030	—	2	2	no	—
30	#13094030	—	4	1	no	—
31	#13094030	—	4	1	no	—
32	C₂×#10625329	—	2	1	no	—
33	C₂×#10625324	—	4	1	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₂×C₈×D₄₇	—	4	1	no	—
36	#13093960	—	2	1	no	—
37	#13093960	C₂×C₄	1	2	yes	—
38	#13093952	—	2	2	no	—
39	#13093952	—	2	2	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₄×C₄₇⋊C₈	—	2	2	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	M₃₂×C₄₇	—	2	2	no	—
46	M₃₂×C₄₇	—	4	1	no	—
47	C₂×C₁₆×C₄₇	—	4	1	no	—
48	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₇	—	4	1	no	—
55	#10625350	—	2	2	no	—
56	#10625350	—	2	2	no	—
57	#10625350	—	4	1	no	—
58	#10625350	—	4	1	no	—
59	C₂×D₁₈₈	—	2	1	no	—
60	C₂×C₄×D₄₇	—	4	1	no	—
61	C₂×Dic₉₄	—	2	1	no	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
67	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
68	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
69	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
70	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
71	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	#10625329	—	2	2	no	—
78	#10625329	—	4	1	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	C₂×C₄₇⋊C₈	—	4	1	no	—
82	Dic₁₈₈	—	2	1	no	—
83	Dic₁₈₈	C₂×C₈	1	2	yes	complemented
84	D₃₇₆	—	2	1	no	—
85	D₃₇₆	C₂×C₈	1	2	yes	complemented
86	#10625325	—	2	2	no	—
87	#10625325	—	2	2	no	—
88	#10625324	—	4	2	no	—
89	#10625324	—	4	2	no	—
90	C₈×D₄₇	—	4	2	no	—
91	C₈×D₄₇	—	4	2	no	—
92	C₁₆×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	#104	1	1	yes	characteristic
94	C₂²×D₄₇	—	4	1	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²⋊D₄₇	—	4	2	no	—
105	C₂²⋊D₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	4	1	no	—
107	D₁₈₈	—	2	2	no	—
108	D₁₈₈	—	4	1	no	—
109	C₄×D₄₇	—	4	2	no	—
110	C₄×D₄₇	—	4	2	no	—
111	Dic₉₄	—	2	2	no	—
112	Dic₉₄	—	4	1	no	—
113	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
114	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₄₇	#102	1	1	yes	characteristic
116	C₈×C₄₇	#102	1	1	yes	characteristic
117	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
118	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
119	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
120	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₄₇⋊C₈	—	4	4	no	—
122	C₄₇⋊C₈	—	4	2	no	—
123	C₂×#1336	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
128	C₂×D₄₇	—	4	2	no	—
129	C₂×D₄₇	—	8	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₄₇	D₄⋊C₈	1	1	yes	characteristic
134	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
136	Dic₄₇	—	4	2	no	—
137	C₂×#443	—	47	1	no	—
138	C₂×#364	D₄₇	1	1	yes	characteristic, complemented
139	C₂×#322	—	47	1	no	—
140	#1336	—	47	4	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	—	2	1	no	—
143	C₂×C₄₇	#1336	1	2	yes	complemented
144	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
145	D₄₇	—	8	2	no	—
146	C₂×#135	—	47	1	no	—
147	C₂×#131	—	94	1	no	—
148	C₂×M₃₂	—	2	1	no	—
149	#443	—	47	2	no	—
150	#443	—	47	2	no	—
151	#443	—	94	1	no	—
152	#443	—	94	1	no	—
153	C₂×#108	—	47	1	no	—
154	C₂×#104	—	94	1	no	—
155	C₂×C₄×C₈	—	94	1	no	—
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	#364	—	2	1	no	—
158	#364	C₂×D₄₇	1	2	yes	complemented
159	#322	—	47	2	no	—
160	#322	—	94	1	no	—
161	C₄₇	C₂×#1336	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	94	1	no	—
163	#135	—	47	2	no	—
164	#135	—	47	2	no	—
165	#135	—	94	1	no	—
166	#135	—	94	1	no	—
167	C₂×Q₁₆	—	47	1	no	—
168	C₂×SD₁₆	—	94	1	no	—
169	C₂×D₈	—	47	1	no	—
170	#131	—	94	2	no	—
171	#131	—	94	2	no	—
172	#131	—	188	1	no	—
173	#131	—	188	1	no	—
174	C₂×M₁₆	—	94	1	no	—
175	C₂×M₁₆	—	188	1	no	—
176	C₂²×C₈	—	94	2	no	—
177	C₂²×C₈	—	188	1	no	—
178	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	4	1	no	—
183	C₂×C₁₆	—	4	1	no	—
184	#108	—	47	2	no	—
185	#108	—	94	1	no	—
186	#104	—	94	2	no	—
187	#104	—	94	2	no	—
188	C₄×C₈	—	94	2	no	—
189	C₄×C₈	—	94	2	no	—
190	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
191	C₄×C₈	D₁₈₈	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	#41	—	94	2	no	—
195	#41	—	94	2	no	—
196	#41	—	188	1	no	—
197	#41	—	188	1	no	—
198	C₂×Q₈	—	94	1	no	—
199	C₂×D₄	—	94	1	no	—
200	C₂×D₄	—	188	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	188	1	no	—
203	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
204	Q₁₆	—	47	2	no	—
205	Q₁₆	—	94	1	no	—
206	SD₁₆	—	94	2	no	—
207	SD₁₆	—	94	2	no	—
208	D₈	—	47	2	no	—
209	D₈	—	94	1	no	—
210	M₁₆	—	94	2	no	—
211	M₁₆	—	188	1	no	—
212	M₁₆	—	188	2	no	—
213	M₁₆	—	188	2	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	188	4	no	—
216	C₂×C₈	—	188	1	no	—
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
220	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
221	C₂×C₈	#10625324	1	1	yes	characteristic
222	C₂×C₈	#10625324	1	1	yes	characteristic
223	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
224	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
225	C₂×C₈	#10625334	1	1	yes	characteristic
226	C₂×C₈	#10625334	1	1	yes	characteristic
227	C₂×C₈	#10625336	1	1	yes	characteristic
228	C₂×C₈	#10625336	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
234	C₁₆	—	4	2	no	—
235	C₂³	—	188	1	no	—
236	C₂³	#13093950	1	1	yes	characteristic
237	Q₈	—	94	2	no	—
238	Q₈	—	188	1	no	—
239	D₄	—	94	2	no	—
240	D₄	—	188	2	no	—
241	D₄	—	188	2	no	—
242	D₄	—	188	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	188	1	no	—
247	C₂×C₄	—	188	2	no	—
248	C₂×C₄	—	188	2	no	—
249	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
250	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
251	C₂×C₄	#13093944	1	1	yes	characteristic
252	C₂×C₄	#13093944	1	1	yes	characteristic
253	C₂×C₄	#13093950	1	1	yes	characteristic
254	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
255	C₈	—	188	4	no	—
256	C₈	—	188	2	no	—
257	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
258	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
259	C₈	#13093944	1	1	yes	characteristic
260	C₈	#13093944	1	1	yes	characteristic
261	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
262	C₈	C₂×#10625324	1	1	yes	characteristic
263	C₈	C₂×#10625334	1	1	yes	characteristic
264	C₈	C₂×#10625336	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	376	2	no	—
267	C₂²	—	188	2	no	—
268	C₂²	#15688850	1	1	yes	characteristic
269	C₂²	#15688850	1	1	yes	characteristic
270	C₂²	C₂×#13093950	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	188	2	no	—
274	C₄	#15688850	1	1	yes	characteristic
275	C₄	#15688850	1	1	yes	characteristic
276	C₄	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
277	C₄	C₂×#13093944	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	376	2	no	—
281	C₂	#20241363	1	2	yes	direct product factor
282	C₂	C₂×#15688850	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093944
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	D₄⋊C₈
prime radical	C₂×#13094046	C₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	C₂×#1336
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#15688844	C₂	semidirect
3. #20241363	C₂	direct (N × H)
4. C₂×C₄₇×#364	C₂	semidirect
15. #15688844	C₂²	semidirect
19. C₄₇×#364	C₂²	semidirect
23. #13094046	C₈	semidirect
25. C₂×Dic₁₈₈	C₈	semidirect
27. C₂×D₃₇₆	C₈	semidirect
83. Dic₁₈₈	C₂×C₈	semidirect
85. D₃₇₆	C₂×C₈	semidirect
138. C₂×#364	D₄₇	semidirect
143. C₂×C₄₇	#1336	semidirect
158. #364	C₂×D₄₇	semidirect
161. C₄₇	C₂×#1336	semidirect
281. C₂	#20241363	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×#15689241	2
C₂×#15688844	2
#20241363	2
C₂×C₄₇×#364	2
C₂×#1336	47

Subgroups — C2×#20241363 #26126068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241363 #26126068