Subgroups — C₂×#69051 #10947820

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#798	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#4689	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#4502	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#4461	C₂	1	1	yes	maximal, characteristic, complemented
8	#69051	C₂	1	8	yes	maximal, direct product factor
9	C₂×#2108	—	3	1	no	maximal, Sylow
10	C₂²×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
11	C₂²×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×#872	C₄	1	1	yes	characteristic
13	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×#815	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×#803	C₄	1	1	yes	characteristic
21	C₂×#798	C₄	1	8	yes	complemented
22	C₂×#798	C₂²	1	4	yes	—
23	C₂×D₃×C₁₆	C₄	1	8	yes	complemented
24	C₂×D₃×C₁₆	C₂²	1	4	yes	—
25	#4689	C₄	1	4	yes	—
26	#4689	C₂²	1	4	yes	—
27	C₃×C₄×C₁₆	C₂²	1	4	yes	complemented
28	#4502	C₂²	1	4	yes	complemented
29	Dic₃×C₁₆	C₂²	1	4	yes	complemented
30	#4461	C₂²	1	4	yes	complemented
31	C₂²×M₃₂	—	3	1	no	—
32	C₂³×C₁₆	—	3	1	no	—
33	C₂×#405	—	3	1	no	—
34	#2108	—	3	8	no	—
35	C₂×#346	—	3	1	no	—
36	C₂×#346	—	3	1	no	—
37	C₂×C₄×C₁₆	—	3	1	no	—
38	C₂×C₄×C₁₆	D₃	1	1	yes	characteristic, complemented
39	C₂²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
43	C₂×(C₂×D₃)⋊C₄	C₈	1	2	yes	—
44	C₂×Dic₃⋊C₄	C₈	1	2	yes	—
45	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×ΓL(1,𝔽₂₅)	C₈	1	8	yes	—
49	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	4	yes	—
50	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
51	C₂×D₃×C₈	C₂×C₄	1	4	yes	—
52	C₂×D₃×C₈	C₂×C₄	1	4	yes	—
53	C₂×D₃×C₈	C₂³	1	2	yes	—
54	#872	C₈	1	4	yes	—
55	#872	C₂×C₄	1	4	yes	—
56	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
57	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
58	C₂×C₃×C₁₆	C₂³	1	2	yes	—
59	C₂×C₃×C₁₆	C₂³	1	2	yes	—
60	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
61	C₃×C₄×C₈	C₂³	1	2	yes	—
62	#815	C₂×C₄	1	4	yes	—
63	C₈×Dic₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₃	C₂³	1	2	yes	—
65	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
66	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
67	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
68	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
69	#803	C₂×C₄	1	4	yes	—
70	#798	C₂×C₄	1	16	yes	complemented
71	D₃×C₁₆	C₂×C₄	1	16	yes	complemented
72	C₂²×M₁₆	—	3	1	no	—
73	C₂³×C₈	—	3	1	no	—
74	C₂×#117	—	3	1	no	—
75	C₂×M₃₂	—	3	8	no	—
76	C₂×M₃₂	—	3	4	no	—
77	C₂²×C₁₆	—	3	1	no	—
78	C₂²×C₁₆	—	3	1	no	—
79	C₂²×C₁₆	—	3	8	no	—
80	C₂²×C₁₆	—	3	4	no	—
81	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
83	#405	—	3	4	no	—
84	#405	—	3	4	no	—
85	C₂×#97	—	3	1	no	—
86	C₂×#97	—	3	1	no	—
87	C₂×C₄×C₈	—	3	1	no	—
88	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
89	#346	—	3	4	no	—
90	#346	—	3	4	no	—
91	C₄×C₁₆	—	3	4	no	—
92	C₄×C₁₆	C₂×D₃	1	4	yes	complemented
93	C₂³×D₃	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
95	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×Dic₃	C₂×C₈	1	2	yes	—
97	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₃	C₄²	1	4	yes	—
99	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
100	C₂×C₄×D₃	C₂×C₈	1	4	yes	—
101	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
102	C₂×C₃×C₈	C₂×C₈	1	4	yes	—
103	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
108	C₃×C₄²	C₂×C₈	1	2	yes	—
109	C₃×C₄²	C₂²×C₄	1	2	yes	—
110	(C₂×D₃)⋊C₄	C₂×C₈	1	8	yes	—
111	Dic₃⋊C₄	C₂×C₈	1	8	yes	—
112	C₄×Dic₃	C₂×C₈	1	2	yes	—
113	C₄×Dic₃	C₂²×C₄	1	2	yes	—
114	C₂×C₃⋊C₈	C₄²	1	4	yes	—
115	C₂×C₃⋊C₈	C₂×C₈	1	4	yes	—
116	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
118	ΓL(1,𝔽₂₅)	C₂×C₈	1	16	yes	—
119	D₃×C₈	C₄²	1	8	yes	—
120	D₃×C₈	C₂²×C₄	1	4	yes	—
121	D₃×C₈	C₂²×C₄	1	4	yes	—
122	C₃×C₁₆	C₂²×C₄	1	4	yes	—
123	C₃×C₁₆	C₂²×C₄	1	4	yes	—
124	C₃⋊C₁₆	C₂²×C₄	1	4	yes	—
125	C₃⋊C₁₆	C₂²×C₄	1	4	yes	—
126	C₂³×C₄	—	3	1	no	—
127	C₂×M₁₆	—	3	8	no	—
128	C₂×M₁₆	—	3	4	no	—
129	C₂²×C₈	—	3	1	no	—
130	C₂²×C₈	—	3	1	no	—
131	C₂²×C₈	—	3	2	no	—
132	C₂²×C₈	—	3	4	no	—
133	C₂²×C₈	—	3	4	no	—
134	C₂²×C₈	—	3	2	no	—
135	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
137	#117	—	3	4	no	—
138	#117	—	3	4	no	—
139	C₂×C₄⋊C₄	—	3	2	no	—
140	C₂×C₂²⋊C₄	—	3	2	no	—
141	C₂×C₄²	—	3	1	no	—
142	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
143	M₃₂	—	3	16	no	—
144	C₂×C₁₆	—	3	4	no	—
145	C₂×C₁₆	—	3	4	no	—
146	C₂×C₁₆	—	3	2	no	—
147	C₂×C₁₆	—	3	2	no	—
148	C₂×C₁₆	—	3	16	no	—
149	C₂×C₁₆	C₄×D₃	1	4	yes	—
150	C₂×C₁₆	C₄×D₃	1	4	yes	—
151	C₂×C₁₆	C₂²×D₃	1	2	yes	—
152	C₂×C₁₆	C₂²×D₃	1	2	yes	—
153	#97	—	3	4	no	—
154	#97	—	3	4	no	—
155	C₄×C₈	—	3	2	no	—
156	C₄×C₈	—	3	2	no	—
157	C₄×C₈	C₄×D₃	1	2	yes	—
158	C₄×C₈	C₂²×D₃	1	2	yes	—
159	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
160	C₂²×D₃	—	2	4	no	—
161	C₂²×D₃	C₄×C₈	1	4	yes	—
162	C₂²×D₃	C₂²×C₈	1	2	yes	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	C₂×C₄²	1	2	yes	—
165	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
166	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
167	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
169	C₂×Dic₃	—	2	4	no	—
170	C₂×Dic₃	C₄×C₈	1	4	yes	—
171	C₂×Dic₃	C₂²×C₈	1	4	yes	—
172	C₂×Dic₃	C₂²×C₈	1	2	yes	—
173	C₄×D₃	C₄×C₈	1	8	yes	—
174	C₄×D₃	C₂×C₄²	1	4	yes	—
175	C₄×D₃	C₂²×C₈	1	4	yes	—
176	C₃×C₈	C₂×C₄²	1	2	yes	—
177	C₃×C₈	C₂²×C₈	1	4	yes	—
178	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
179	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
180	C₃⋊C₈	C₂×C₄²	1	4	yes	—
181	C₃⋊C₈	C₂²×C₈	1	4	yes	—
182	C₂⁴	—	3	1	no	—
183	C₂²×C₄	—	3	1	no	—
184	C₂²×C₄	—	3	4	no	—
185	C₂²×C₄	—	3	2	no	—
186	C₂²×C₄	—	3	4	no	—
187	C₂²×C₄	—	3	2	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₂²×C₄	D₃×C₈	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
191	M₁₆	—	3	16	no	—
192	C₂×C₈	—	3	4	no	—
193	C₂×C₈	—	3	4	no	—
194	C₂×C₈	—	3	2	no	—
195	C₂×C₈	—	3	2	no	—
196	C₂×C₈	—	3	8	no	—
197	C₂×C₈	—	3	4	no	—
198	C₂×C₈	—	3	4	no	—
199	C₂×C₈	D₃×C₈	1	4	yes	—
200	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
204	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
205	C₄⋊C₄	—	3	8	no	—
206	C₂²⋊C₄	—	3	8	no	—
207	C₄²	—	3	2	no	—
208	C₄²	—	3	2	no	—
209	C₄²	D₃×C₈	1	2	yes	—
210	C₄²	C₂×C₄×D₃	1	2	yes	—
211	C₁₆	—	3	4	no	—
212	C₁₆	—	3	4	no	—
213	C₁₆	C₂×C₄×D₃	1	4	yes	—
214	C₁₆	C₂×C₄×D₃	1	4	yes	—
215	C₂²×C₃	C₂×C₄×C₈	1	2	yes	—
216	C₂²×C₃	#504	1	4	yes	—
217	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
218	C₂×D₃	—	2	8	no	—
219	C₂×D₃	—	2	4	no	—
220	C₂×D₃	C₂×C₄×C₈	1	4	yes	—
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	C₂×C₄×C₈	1	2	yes	—
223	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
224	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
225	Dic₃	—	2	4	no	—
226	Dic₃	C₂×C₄×C₈	1	4	yes	—
227	C₂³	—	3	4	no	—
228	C₂³	—	3	2	no	—
229	C₂³	—	6	4	no	—
230	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	3	4	no	—
233	C₂×C₄	—	3	2	no	—
234	C₂×C₄	—	3	8	no	—
235	C₂×C₄	—	3	4	no	—
236	C₂×C₄	—	3	4	no	—
237	C₂×C₄	—	3	4	no	—
238	C₂×C₄	—	6	4	no	—
239	C₂×C₄	C₄²×D₃	1	2	yes	—
240	C₂×C₄	C₂×D₃×C₈	1	4	yes	—
241	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
243	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
244	C₈	—	3	4	no	—
245	C₈	—	3	4	no	—
246	C₈	C₄²×D₃	1	2	yes	—
247	C₈	C₂×D₃×C₈	1	4	yes	—
248	C₈	C₂²×C₄×D₃	1	1	yes	characteristic
249	C₈	C₂²×C₄×D₃	1	1	yes	characteristic
250	C₂×C₃	#2108	1	4	yes	complemented
251	C₂×C₃	C₂²×C₄×C₈	1	1	yes	characteristic
252	C₂×C₃	C₂×#504	1	2	yes	—
253	D₃	—	2	4	no	—
254	C₂²	—	3	4	no	—
255	C₂²	—	6	4	no	—
256	C₂²	—	6	8	no	—
257	C₂²	C₄×D₃×C₈	1	2	yes	—
258	C₂²	#4901	1	4	yes	—
259	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	3	4	no	—
262	C₄	—	6	4	no	—
263	C₄	C₄×D₃×C₈	1	2	yes	—
264	C₄	C₂×C₄²×D₃	1	1	yes	characteristic
265	C₄	C₂²×D₃×C₈	1	1	yes	characteristic
266	C₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	6	4	no	—
268	C₂	#69051	1	4	yes	direct product factor
269	C₂	C₂×C₄×D₃×C₈	1	1	yes	characteristic
270	C₂	C₂×#4901	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×C₄×C₁₆	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂³×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₃
hypocenter	C₃	C₂×#2108
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×C₄×C₁₆	C₂	semidirect
5. C₂×#4502	C₂	semidirect
6. C₂×Dic₃×C₁₆	C₂	semidirect
7. C₂×#4461	C₂	semidirect
8. #69051	C₂	direct (N × H)
21. C₂×#798	C₄	semidirect
23. C₂×D₃×C₁₆	C₄	semidirect
27. C₃×C₄×C₁₆	C₂²	semidirect
28. #4502	C₂²	semidirect
29. Dic₃×C₁₆	C₂²	semidirect
30. #4461	C₂²	semidirect
38. C₂×C₄×C₁₆	D₃	semidirect
70. #798	C₂×C₄	semidirect
71. D₃×C₁₆	C₂×C₄	semidirect
92. C₄×C₁₆	C₂×D₃	semidirect
250. C₂×C₃	#2108	semidirect
266. C₃	C₂×#2108	semidirect
268. C₂	#69051	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#798	2
C₂²×D₃×C₁₆	2
C₂×#4689	2
C₂×C₃×C₄×C₁₆	2
C₂×#4502	2
C₂×Dic₃×C₁₆	2
C₂×#4461	2
#69051	2
C₂×#2108	3

Subgroups — C2×#69051 #10947820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69051 #10947820