Subgroups — C₂×#5768 #87183

Order 384.

1484 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×#41	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#950	C₂	1	2	yes	maximal, complemented
4	#5768	C₂	1	16	yes	maximal, direct product factor
5	C₂×#918	C₂	1	2	yes	maximal, complemented
6	C₂×#915	C₂	1	2	yes	maximal, complemented
7	C₂×#914	C₂	1	2	yes	maximal, complemented
8	C₂×#911	C₂	1	2	yes	maximal, complemented
9	C₂×#904	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#575	—	3	1	no	maximal, Sylow
12	C₂×C₃×#41	C₂²	1	2	yes	complemented
13	D₃×#41	—	2	8	no	—
14	D₃×#41	C₂²	1	16	yes	complemented
15	C₂×#239	C₂²	1	2	yes	—
16	C₂×D₃×Q₈	C₂²	1	2	yes	complemented
17	C₂×#237	—	2	2	no	—
18	C₂×#237	C₂²	1	2	yes	—
19	C₂×D₃×D₄	—	2	2	no	—
20	C₂×D₃×D₄	C₂²	1	2	yes	complemented
21	C₂×#235	—	2	2	no	—
22	C₂×#235	C₂²	1	2	yes	complemented
23	C₂²×C₄×D₃	—	2	2	no	—
24	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
25	C₃×#135	C₂²	1	8	yes	complemented
26	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
28	C₂×C₃×D₈	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
30	#950	C₂²	1	16	yes	complemented
31	C₂×#216	C₂²	1	2	yes	complemented
32	C₂×#215	C₂²	1	2	yes	complemented
33	C₂×#214	C₂²	1	2	yes	complemented
34	C₂×#213	C₂²	1	2	yes	complemented
35	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	#918	C₂²	1	16	yes	complemented
37	#915	C₂²	1	16	yes	complemented
38	#914	C₂²	1	16	yes	complemented
39	#911	C₂²	1	16	yes	complemented
40	C₂×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
41	#904	C₂²	1	8	yes	complemented
42	C₂×D₂₄	C₂²	1	1	yes	characteristic, complemented
43	C₂×#204	C₂²	1	2	yes	complemented
44	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
45	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
46	C₂×ΓL(1,𝔽₂₅)	C₂²	1	8	yes	complemented
47	C₂²×#41	—	3	2	no	—
48	#575	—	3	16	no	—
49	C₂×#137	—	3	2	no	—
50	C₂×#137	—	3	2	no	—
51	C₂×AGL(1,ℤ/8)	—	3	2	no	—
52	C₂×AGL(1,ℤ/8)	—	3	2	no	—
53	C₂×#135	—	3	2	no	—
54	C₂×#135	—	3	1	no	—
55	C₂×#135	D₃	1	1	yes	characteristic, complemented
56	C₂²×M₁₆	—	3	1	no	—
57	C₂³×D₃	—	2	2	no	—
58	C₂³×D₃	D₄	1	1	yes	characteristic
59	C₃×#41	—	2	4	no	—
60	C₃×#41	C₂³	1	8	yes	complemented
61	C₂×C₃×Q₈	C₂³	1	2	yes	—
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂×#67	—	2	2	no	—
67	C₂×#67	—	2	2	no	—
68	C₂×#67	—	2	2	no	—
69	C₂²×Dic₃	—	2	2	no	—
70	C₂²×Dic₃	D₄	1	1	yes	characteristic
71	#239	—	2	4	no	—
72	#239	C₂³	1	8	yes	—
73	D₃×Q₈	—	2	4	no	—
74	D₃×Q₈	C₂³	1	8	yes	complemented
75	#237	—	2	16	no	—
76	#237	—	2	4	no	—
77	#237	C₂³	1	8	yes	—
78	D₃×D₄	—	2	16	no	—
79	D₃×D₄	—	2	4	no	—
80	D₃×D₄	C₂³	1	8	yes	complemented
81	#235	—	2	4	no	—
82	#235	—	2	16	no	—
83	#235	C₂³	1	8	yes	complemented
84	C₂×D₁₂	—	2	2	no	—
85	C₂×D₁₂	C₂³	1	2	yes	—
86	C₂×C₄×D₃	—	2	2	no	—
87	C₂×C₄×D₃	—	2	2	no	—
88	C₂×C₄×D₃	—	2	2	no	—
89	C₂×C₄×D₃	—	2	16	no	—
90	C₂×C₄×D₃	D₄	1	2	yes	—
91	C₂×C₄×D₃	D₄	1	4	yes	—
92	C₂×C₄×D₃	C₂³	1	2	yes	—
93	C₂×C₄×D₃	C₂³	1	4	yes	—
94	C₂×Dic₆	—	2	2	no	—
95	C₂×Dic₆	C₂³	1	2	yes	—
96	C₃×Q₁₆	C₂³	1	4	yes	complemented
97	C₃×SD₁₆	C₂³	1	8	yes	complemented
98	C₃×D₈	C₂³	1	4	yes	complemented
99	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂³	1	4	yes	complemented
102	#216	C₂³	1	8	yes	complemented
103	#215	C₂³	1	8	yes	complemented
104	#214	C₂³	1	8	yes	complemented
105	#213	C₂³	1	8	yes	complemented
106	C₂×C₃⋊C₈	C₂³	1	2	yes	—
107	C₂×C₃⋊C₈	C₂³	1	4	yes	complemented
108	Dic₁₂	C₂³	1	4	yes	complemented
109	D₂₄	C₂³	1	4	yes	complemented
110	#204	C₂³	1	8	yes	complemented
111	ΓL(1,𝔽₂₅)	C₂³	1	8	yes	complemented
112	ΓL(1,𝔽₂₅)	C₂³	1	8	yes	complemented
113	C₂×#41	—	3	16	no	—
114	C₂×#41	—	3	2	no	—
115	C₂×#41	—	3	2	no	—
116	C₂×#41	—	3	2	no	—
117	C₂×#41	—	6	2	no	—
118	C₂×#41	—	6	2	no	—
119	C₂×#41	—	6	8	no	—
120	C₂×#41	C₂×D₃	1	2	yes	complemented
121	C₂²×Q₈	—	3	2	no	—
122	C₂²×D₄	—	3	2	no	—
123	C₂²×D₄	—	6	2	no	—
124	C₂³×C₄	—	3	1	no	—
125	C₂³×C₄	—	6	2	no	—
126	#137	—	3	16	no	—
127	#137	—	3	16	no	—
128	AGL(1,ℤ/8)	—	3	16	no	—
129	AGL(1,ℤ/8)	—	3	16	no	—
130	#135	—	3	16	no	—
131	#135	—	3	8	no	—
132	#135	C₂×D₃	1	8	yes	complemented
133	C₂×Q₁₆	—	3	2	no	—
134	C₂×Q₁₆	—	3	1	no	—
135	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
136	C₂×SD₁₆	—	3	2	no	—
137	C₂×SD₁₆	—	3	2	no	—
138	C₂×SD₁₆	—	3	2	no	—
139	C₂×SD₁₆	C₂×D₃	1	2	yes	complemented
140	C₂×D₈	—	3	2	no	—
141	C₂×D₈	—	3	1	no	—
142	C₂×D₈	C₂×D₃	1	1	yes	characteristic, complemented
143	C₂×M₁₆	—	3	2	no	—
144	C₂×M₁₆	—	3	2	no	—
145	C₂×M₁₆	—	3	8	no	—
146	C₂²×C₈	—	3	1	no	—
147	C₂²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×D₃	—	2	2	no	—
151	C₂²×D₃	—	2	8	no	—
152	C₂²×D₃	—	4	2	no	—
153	C₂²×D₃	—	4	8	no	—
154	C₂²×D₃	C₂×D₄	1	2	yes	—
155	C₂²×D₃	C₂×D₄	1	4	yes	—
156	C₃×Q₈	—	2	2	no	—
157	C₃×Q₈	C₂⁴	1	4	yes	—
158	C₃×D₄	—	2	8	no	—
159	C₃×D₄	—	2	2	no	—
160	C₃×D₄	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
164	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂⁴	1	2	yes	—
167	#67	—	2	8	no	—
168	#67	—	2	8	no	—
169	#67	—	2	8	no	—
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	—	2	8	no	—
172	C₂×Dic₃	C₂×D₄	1	2	yes	—
173	C₂×Dic₃	C₂×D₄	1	4	yes	—
174	D₁₂	—	2	8	no	—
175	D₁₂	—	2	2	no	—
176	D₁₂	C₂⁴	1	4	yes	—
177	C₄×D₃	—	2	8	no	—
178	C₄×D₃	—	2	8	no	—
179	C₄×D₃	—	2	8	no	—
180	C₄×D₃	C₂×D₄	1	8	yes	—
181	C₄×D₃	C₂×D₄	1	4	yes	—
182	C₄×D₃	C₂⁴	1	4	yes	—
183	Dic₆	—	2	2	no	—
184	Dic₆	—	2	8	no	—
185	Dic₆	C₂⁴	1	4	yes	—
186	C₃×C₈	C₂⁴	1	2	yes	—
187	C₃×C₈	C₂⁴	1	2	yes	—
188	C₃⋊C₈	C₂⁴	1	4	yes	—
189	C₂⁴	—	3	1	no	—
190	C₂⁴	—	6	2	no	—
191	#41	—	2	4	no	—
192	#41	—	3	8	no	—
193	#41	—	3	8	no	—
194	#41	—	3	8	no	—
195	#41	—	6	16	no	—
196	#41	—	6	16	no	—
197	#41	—	6	4	no	—
198	#41	—	6	4	no	—
199	#41	—	6	4	no	—
200	#41	C₂²×D₃	1	8	yes	complemented
201	C₂×Q₈	—	3	8	no	—
202	C₂×Q₈	—	3	2	no	—
203	C₂×Q₈	—	6	2	no	—
204	C₂×Q₈	—	6	4	no	—
205	C₂×Q₈	C₂²×D₃	1	2	yes	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	3	8	no	—
208	C₂×D₄	—	3	2	no	—
209	C₂×D₄	—	6	2	no	—
210	C₂×D₄	—	6	4	no	—
211	C₂×D₄	—	6	2	no	—
212	C₂×D₄	—	6	16	no	—
213	C₂×D₄	—	6	2	no	—
214	C₂×D₄	—	6	2	no	—
215	C₂×D₄	C₂²×D₃	1	2	yes	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	3	1	no	—
218	C₂²×C₄	—	3	4	no	—
219	C₂²×C₄	—	3	2	no	—
220	C₂²×C₄	—	3	2	no	—
221	C₂²×C₄	—	3	4	no	—
222	C₂²×C₄	—	6	2	no	—
223	C₂²×C₄	—	6	16	no	—
224	C₂²×C₄	—	6	2	no	—
225	C₂²×C₄	—	6	2	no	—
226	C₂²×C₄	—	6	2	no	—
227	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
228	Q₁₆	—	3	8	no	—
229	Q₁₆	—	3	4	no	—
230	Q₁₆	C₂²×D₃	1	4	yes	complemented
231	SD₁₆	—	3	8	no	—
232	SD₁₆	—	3	8	no	—
233	SD₁₆	—	3	8	no	—
234	SD₁₆	C₂²×D₃	1	8	yes	complemented
235	D₈	—	3	8	no	—
236	D₈	—	3	4	no	—
237	D₈	C₂²×D₃	1	4	yes	complemented
238	M₁₆	—	3	8	no	—
239	M₁₆	—	3	8	no	—
240	C₂×C₈	—	3	2	no	—
241	C₂×C₈	—	3	4	no	—
242	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×D₃	1	4	yes	complemented
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	4	no	—
247	C₂²×C₃	—	4	4	no	—
248	C₂²×C₃	C₂²×D₄	1	2	yes	—
249	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×D₃	—	2	4	no	—
251	C₂×D₃	—	2	4	no	—
252	C₂×D₃	—	4	4	no	—
253	C₂×D₃	—	4	16	no	—
254	C₂×D₃	—	4	4	no	—
255	C₂×D₃	C₂²×D₄	1	4	yes	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
258	C₃×C₄	C₂²×D₄	1	2	yes	—
259	C₃×C₄	C₂⁵	1	1	yes	characteristic
260	Dic₃	—	2	4	no	—
261	Dic₃	C₂²×D₄	1	4	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	—	3	2	no	—
264	C₂³	—	3	4	no	—
265	C₂³	—	6	8	no	—
266	C₂³	—	6	2	no	—
267	C₂³	—	12	2	no	—
268	C₂³	—	12	8	no	—
269	C₂³	D₃×D₄	1	1	yes	characteristic
270	Q₈	—	2	2	no	—
271	Q₈	—	3	4	no	—
272	Q₈	—	6	8	no	—
273	Q₈	—	6	2	no	—
274	Q₈	C₂³×D₃	1	4	yes	—
275	D₄	—	2	8	no	—
276	D₄	—	2	2	no	—
277	D₄	—	3	4	no	—
278	D₄	—	6	8	no	—
279	D₄	—	6	8	no	—
280	D₄	—	6	8	no	—
281	D₄	—	6	8	no	—
282	D₄	—	6	2	no	—
283	D₄	C₂³×D₃	1	4	yes	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	3	2	no	—
287	C₂×C₄	—	3	4	no	—
288	C₂×C₄	—	3	8	no	—
289	C₂×C₄	—	3	4	no	—
290	C₂×C₄	—	3	4	no	—
291	C₂×C₄	—	6	8	no	—
292	C₂×C₄	—	6	8	no	—
293	C₂×C₄	—	6	8	no	—
294	C₂×C₄	—	6	2	no	—
295	C₂×C₄	—	6	8	no	—
296	C₂×C₄	D₃×D₄	1	2	yes	—
297	C₂×C₄	D₃×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
299	C₂×C₄	C₂³×D₃	1	2	yes	—
300	C₈	—	3	4	no	—
301	C₈	C₂³×D₃	1	2	yes	—
302	C₈	C₂³×D₃	1	2	yes	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	4	4	no	—
305	C₂×C₃	#575	1	2	yes	complemented
306	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
307	D₃	—	2	4	no	—
308	D₃	—	4	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	3	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	6	4	no	—
314	C₂²	—	6	4	no	—
315	C₂²	—	12	4	no	—
316	C₂²	—	12	16	no	—
317	C₂²	—	12	4	no	—
318	C₂²	C₂×D₃×D₄	1	2	yes	—
319	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	3	4	no	—
322	C₄	—	6	4	no	—
323	C₄	C₂×D₃×D₄	1	2	yes	—
324	C₄	C₂×D₃×D₄	1	1	yes	characteristic
325	C₄	C₂⁴×D₃	1	1	yes	characteristic
326	C₃	C₂×#575	1	1	yes	characteristic, complemented, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	4	4	no	—
329	C₂	—	6	4	no	—
330	C₂	—	12	4	no	—
331	C₂	#5768	1	2	yes	direct product factor
332	C₂	C₂²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂×C₃×#135	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₂×#135	D₃
hypocenter	C₃	C₂×#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×#41	C₂	semidirect
2. C₂×C₃×#135	C₂	semidirect
3. C₂×#950	C₂	semidirect
4. #5768	C₂	direct (N × H)
5. C₂×#918	C₂	semidirect
6. C₂×#915	C₂	semidirect
7. C₂×#914	C₂	semidirect
8. C₂×#911	C₂	semidirect
9. C₂×#904	C₂	semidirect
10. C₂²×ΓL(1,𝔽₂₅)	C₂	semidirect
12. C₂×C₃×#41	C₂²	semidirect
14. D₃×#41	C₂²	semidirect
16. C₂×D₃×Q₈	C₂²	semidirect
20. C₂×D₃×D₄	C₂²	semidirect
22. C₂×#235	C₂²	semidirect
25. C₃×#135	C₂²	semidirect
26. C₂×C₃×Q₁₆	C₂²	semidirect
27. C₂×C₃×SD₁₆	C₂²	semidirect
28. C₂×C₃×D₈	C₂²	semidirect
29. C₂²×C₃×C₈	C₂²	semidirect
30. #950	C₂²	semidirect
31. C₂×#216	C₂²	semidirect
32. C₂×#215	C₂²	semidirect
33. C₂×#214	C₂²	semidirect
34. C₂×#213	C₂²	semidirect
35. C₂²×C₃⋊C₈	C₂²	semidirect
36. #918	C₂²	semidirect
37. #915	C₂²	semidirect
38. #914	C₂²	semidirect
39. #911	C₂²	semidirect
40. C₂×Dic₁₂	C₂²	semidirect
41. #904	C₂²	semidirect
42. C₂×D₂₄	C₂²	semidirect
43. C₂×#204	C₂²	semidirect
44. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
45. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
46. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
55. C₂×#135	D₃	semidirect
60. C₃×#41	C₂³	semidirect
74. D₃×Q₈	C₂³	semidirect
80. D₃×D₄	C₂³	semidirect
83. #235	C₂³	semidirect
96. C₃×Q₁₆	C₂³	semidirect
97. C₃×SD₁₆	C₂³	semidirect
98. C₃×D₈	C₂³	semidirect
101. C₂×C₃×C₈	C₂³	semidirect
102. #216	C₂³	semidirect
103. #215	C₂³	semidirect
104. #214	C₂³	semidirect
105. #213	C₂³	semidirect
107. C₂×C₃⋊C₈	C₂³	semidirect
108. Dic₁₂	C₂³	semidirect
109. D₂₄	C₂³	semidirect
110. #204	C₂³	semidirect
111. ΓL(1,𝔽₂₅)	C₂³	semidirect
112. ΓL(1,𝔽₂₅)	C₂³	semidirect
120. C₂×#41	C₂×D₃	semidirect
132. #135	C₂×D₃	semidirect
135. C₂×Q₁₆	C₂×D₃	semidirect
139. C₂×SD₁₆	C₂×D₃	semidirect
142. C₂×D₈	C₂×D₃	semidirect
147. C₂²×C₈	C₂×D₃	semidirect
200. #41	C₂²×D₃	semidirect
230. Q₁₆	C₂²×D₃	semidirect
234. SD₁₆	C₂²×D₃	semidirect
237. D₈	C₂²×D₃	semidirect
244. C₂×C₈	C₂²×D₃	semidirect
305. C₂×C₃	#575	semidirect
326. C₃	C₂×#575	semidirect
331. C₂	#5768	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×D₃×#41	2
C₂×C₃×#135	2
C₂×#950	2
#5768	2
C₂×#918	2
C₂×#915	2
C₂×#914	2
C₂×#911	2
C₂×#904	2
C₂²×ΓL(1,𝔽₂₅)	2
C₂×#575	3

Subgroups — C2×#5768 #87183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5768 #87183