Subgroups — C₂×#20295828 #28377782

Order 13952.

1512 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812745	C₂	1	2	yes	maximal, complemented
2	#20295828	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16812726	C₂	1	4	yes	maximal, complemented
4	C₂×#16812724	C₂	1	2	yes	maximal, complemented
5	C₂×#16812722	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₂×#16812705	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#16812745	—	2	8	no	—
11	#16812745	C₂²	1	16	yes	complemented
12	C₂×#13182030	C₂²	1	4	yes	complemented
13	C₂×#13182029	—	2	2	no	—
14	C₂×#13182029	C₂²	1	1	yes	characteristic
15	C₂²×D₄₃₆	—	2	2	no	—
16	C₂²×D₄₃₆	C₂²	1	2	yes	complemented
17	C₂×D₄×D₁₀₉	—	2	4	no	—
18	C₂×D₄×D₁₀₉	C₂²	1	4	yes	complemented
19	#16812726	C₂²	1	32	yes	complemented
20	#16812724	C₂²	1	16	yes	complemented
21	#16812722	C₂²	1	16	yes	complemented
22	C₂×#13182016	C₂²	1	4	yes	complemented
23	C₂×D₈₇₂	C₂²	1	2	yes	complemented
24	C₂×D₈₇₂	C₂²	1	2	yes	complemented
25	C₂×D₈₇₂	C₂²	1	8	yes	complemented
26	C₂×#13182014	C₂²	1	4	yes	complemented
27	D₈×D₁₀₉	C₂²	1	16	yes	complemented
28	#16812705	C₂²	1	8	yes	complemented
29	C₂×#13182013	C₂²	1	1	yes	characteristic, complemented
30	C₂×#13182012	C₂²	1	2	yes	complemented
31	C₂×C₈×D₁₀₉	C₂²	1	2	yes	complemented
32	C₂×#41×C₁₀₉	C₂²	1	2	yes	complemented
33	#135×C₁₀₉	C₂²	1	8	yes	complemented
34	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	complemented
36	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₁₀₉	—	2	2	no	—
39	C₂³×D₁₀₉	—	2	4	no	—
40	#13182030	—	2	8	no	—
41	#13182030	C₂³	1	16	yes	complemented
42	#13182029	—	2	16	no	—
43	#13182029	D₄	1	4	yes	—
44	#13182029	C₂³	1	4	yes	—
45	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
46	C₂×D₄₃₆	—	2	4	no	—
47	C₂×D₄₃₆	—	2	2	no	—
48	C₂×D₄₃₆	—	2	4	no	—
49	C₂×D₄₃₆	—	2	16	no	—
50	C₂×D₄₃₆	D₄	1	1	yes	characteristic
51	C₂×D₄₃₆	C₂³	1	4	yes	—
52	C₂×D₄₃₆	C₂³	1	8	yes	complemented
53	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
54	C₂×C₂²⋊D₁₀₉	D₄	1	2	yes	—
55	D₄×D₁₀₉	—	2	32	no	—
56	D₄×D₁₀₉	—	2	8	no	—
57	D₄×D₁₀₉	C₂³	1	16	yes	complemented
58	C₂×C₄×D₁₀₉	—	2	4	no	—
59	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
60	#13182016	C₂³	1	16	yes	complemented
61	D₈₇₂	C₂³	1	8	yes	complemented
62	D₈₇₂	C₂³	1	8	yes	complemented
63	#13182014	C₂³	1	16	yes	complemented
64	#13182013	C₂³	1	4	yes	complemented
65	#13182012	C₂³	1	8	yes	complemented
66	C₂×#11724784	C₂³	1	2	yes	—
67	C₈×D₁₀₉	C₂³	1	8	yes	complemented
68	#41×C₁₀₉	—	2	4	no	—
69	#41×C₁₀₉	C₂³	1	8	yes	complemented
70	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
71	C₂×D₄×C₁₀₉	—	2	2	no	—
72	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₉	—	2	2	no	—
74	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
75	Q₁₆×C₁₀₉	C₂³	1	4	yes	complemented
76	SD₁₆×C₁₀₉	C₂³	1	8	yes	complemented
77	D₈×C₁₀₉	C₂³	1	4	yes	complemented
78	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₀₉	C₂³	1	4	yes	complemented
81	C₂³×C₁₀₉	—	2	2	no	—
82	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₁₀₉	—	2	4	no	—
84	C₂²×D₁₀₉	—	2	8	no	—
85	C₂²×D₁₀₉	—	2	16	no	—
86	C₂²×D₁₀₉	—	8	8	no	—
87	C₂²×D₁₀₉	—	8	4	no	—
88	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
89	Q₈×C₁₀₉	—	2	2	no	—
90	Q₈×C₁₀₉	C₂⁴	1	4	yes	—
91	D₄×C₁₀₉	—	2	8	no	—
92	D₄×C₁₀₉	—	2	2	no	—
93	D₄×C₁₀₉	C₂⁴	1	4	yes	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	8	no	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
100	C₂²⋊D₁₀₉	—	2	16	no	—
101	C₂²⋊D₁₀₉	C₂×D₄	1	8	yes	—
102	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
103	D₄₃₆	—	2	16	no	—
104	D₄₃₆	—	2	4	no	—
105	D₄₃₆	—	2	8	no	—
106	D₄₃₆	C₂×D₄	1	4	yes	—
107	D₄₃₆	C₂⁴	1	8	yes	—
108	C₄×D₁₀₉	—	2	16	no	—
109	C₄×D₁₀₉	C₂×D₄	1	4	yes	—
110	C₄×D₁₀₉	C₂⁴	1	4	yes	—
111	Dic₂₁₈	C₂×D₄	1	4	yes	—
112	C₈×C₁₀₉	C₂⁴	1	2	yes	—
113	C₈×C₁₀₉	C₂⁴	1	2	yes	—
114	#11724784	C₂⁴	1	4	yes	—
115	C₂²×C₁₀₉	—	2	2	no	—
116	C₂²×C₁₀₉	—	2	4	no	—
117	C₂²×C₁₀₉	—	4	4	no	—
118	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
119	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₀₉	—	2	4	no	—
121	C₂×D₁₀₉	—	2	8	no	—
122	C₂×D₁₀₉	—	4	8	no	—
123	C₂×D₁₀₉	—	8	16	no	—
124	C₂×D₁₀₉	—	8	8	no	—
125	C₂×D₁₀₉	C₂²×D₄	1	4	yes	—
126	C₄×C₁₀₉	—	2	4	no	—
127	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
128	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₁₀₉	C₂⁵	1	1	yes	characteristic
130	Dic₁₀₉	C₂²×D₄	1	4	yes	—
131	C₂×C₁₀₉	—	2	2	no	—
132	C₂×C₁₀₉	—	4	4	no	—
133	C₂×C₁₀₉	#576	1	2	yes	complemented
134	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
135	D₁₀₉	—	2	4	no	—
136	D₁₀₉	—	4	8	no	—
137	C₂×#576	—	109	1	no	maximal, Sylow
138	C₁₀₉	C₂×#576	1	1	yes	characteristic, complemented, Sylow
139	C₂×H(5,𝔽₂)	—	109	2	no	—
140	#576	—	109	16	no	—
141	C₂×AGL(1,ℤ/8)	—	109	2	no	—
142	C₂×AGL(1,ℤ/8)	—	109	4	no	—
143	C₂×#135	—	109	2	no	—
144	C₂×#135	D₁₀₉	1	1	yes	characteristic, complemented
145	C₂²×D₈	—	109	2	no	—
146	C₂²×D₈	—	109	1	no	—
147	C₂×#131	—	109	1	no	—
148	H(5,𝔽₂)	—	109	16	no	—
149	H(5,𝔽₂)	—	218	8	no	—
150	C₂×#41	—	109	1	no	—
151	C₂×#41	—	109	4	no	—
152	C₂×#41	—	218	2	no	—
153	C₂×#41	C₂×D₁₀₉	1	2	yes	complemented
154	C₂²×D₄	—	109	4	no	—
155	C₂²×D₄	—	109	2	no	—
156	C₂²×D₄	—	218	2	no	—
157	C₂²×D₄	—	218	4	no	—
158	AGL(1,ℤ/8)	—	109	16	no	—
159	AGL(1,ℤ/8)	—	109	32	no	—
160	#135	—	109	16	no	—
161	#135	C₂×D₁₀₉	1	8	yes	complemented
162	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic, complemented
163	C₂×SD₁₆	—	109	4	no	—
164	C₂×SD₁₆	C₂×D₁₀₉	1	2	yes	complemented
165	C₂×D₈	—	109	4	no	—
166	C₂×D₈	—	109	16	no	—
167	C₂×D₈	—	109	2	no	—
168	C₂×D₈	—	109	2	no	—
169	C₂×D₈	—	109	8	no	—
170	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
171	#131	—	109	8	no	—
172	C₂×M₁₆	—	109	1	no	—
173	C₂×M₁₆	—	109	2	no	—
174	C₂²×C₈	—	109	2	no	—
175	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
176	C₂⁴	—	218	4	no	—
177	C₂⁴	—	218	2	no	—
178	#41	—	2	4	no	—
179	#41	—	109	4	no	—
180	#41	—	109	4	no	—
181	#41	—	109	16	no	—
182	#41	—	218	16	no	—
183	#41	—	218	8	no	—
184	#41	C₂²×D₁₀₉	1	8	yes	complemented
185	C₂×Q₈	—	109	1	no	—
186	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	109	2	no	—
189	C₂×D₄	—	109	16	no	—
190	C₂×D₄	—	109	1	no	—
191	C₂×D₄	—	109	4	no	—
192	C₂×D₄	—	109	8	no	—
193	C₂×D₄	—	218	16	no	—
194	C₂×D₄	—	218	4	no	—
195	C₂×D₄	—	218	8	no	—
196	C₂×D₄	—	218	4	no	—
197	C₂×D₄	—	218	32	no	—
198	C₂×D₄	—	218	2	no	—
199	C₂×D₄	—	218	4	no	—
200	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	109	2	no	—
203	C₂²×C₄	—	218	4	no	—
204	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
205	Q₁₆	C₂²×D₁₀₉	1	4	yes	complemented
206	SD₁₆	—	109	16	no	—
207	SD₁₆	C₂²×D₁₀₉	1	8	yes	complemented
208	D₈	—	109	16	no	—
209	D₈	—	109	8	no	—
210	D₈	—	109	8	no	—
211	D₈	C₂²×D₁₀₉	1	4	yes	complemented
212	M₁₆	—	109	4	no	—
213	M₁₆	—	109	8	no	—
214	C₂×C₈	—	109	2	no	—
215	C₂×C₈	—	109	8	no	—
216	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₁₀₉	1	4	yes	complemented
219	C₂³	—	2	2	no	—
220	C₂³	—	872	4	no	—
221	C₂³	—	872	8	no	—
222	C₂³	—	109	2	no	—
223	C₂³	—	218	16	no	—
224	C₂³	—	218	8	no	—
225	C₂³	—	218	4	no	—
226	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	109	4	no	—
229	Q₈	C₂³×D₁₀₉	1	4	yes	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	109	8	no	—
233	D₄	—	109	4	no	—
234	D₄	—	109	8	no	—
235	D₄	—	218	16	no	—
236	D₄	—	218	16	no	—
237	D₄	—	218	8	no	—
238	D₄	—	218	4	no	—
239	D₄	C₂³×D₁₀₉	1	4	yes	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	109	2	no	—
243	C₂×C₄	—	109	4	no	—
244	C₂×C₄	—	109	4	no	—
245	C₂×C₄	—	218	16	no	—
246	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
247	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
248	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
250	C₈	—	109	4	no	—
251	C₈	C₂³×D₁₀₉	1	2	yes	—
252	C₈	C₂³×D₁₀₉	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	872	8	no	—
257	C₂²	—	872	16	no	—
258	C₂²	—	109	4	no	—
259	C₂²	—	436	8	no	—
260	C₂²	—	218	8	no	—
261	C₂²	—	218	4	no	—
262	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
263	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	109	4	no	—
266	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
267	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
268	C₄	C₂⁴×D₁₀₉	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	436	8	no	—
272	C₂	—	218	4	no	—
273	C₂	#20295828	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₄×C₁₀₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₉
Fitting	C₂×#135×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	C₂×#135	D₁₀₉
hypocenter	C₁₀₉	C₂×#576
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#16812745	C₂	semidirect
2. #20295828	C₂	direct (N × H)
3. C₂×#16812726	C₂	semidirect
4. C₂×#16812724	C₂	semidirect
5. C₂×#16812722	C₂	semidirect
6. C₂²×D₈₇₂	C₂	semidirect
7. C₂×D₈×D₁₀₉	C₂	semidirect
8. C₂×#16812705	C₂	semidirect
9. C₂×#135×C₁₀₉	C₂	semidirect
11. #16812745	C₂²	semidirect
12. C₂×#13182030	C₂²	semidirect
16. C₂²×D₄₃₆	C₂²	semidirect
18. C₂×D₄×D₁₀₉	C₂²	semidirect
19. #16812726	C₂²	semidirect
20. #16812724	C₂²	semidirect
21. #16812722	C₂²	semidirect
22. C₂×#13182016	C₂²	semidirect
23. C₂×D₈₇₂	C₂²	semidirect
24. C₂×D₈₇₂	C₂²	semidirect
25. C₂×D₈₇₂	C₂²	semidirect
26. C₂×#13182014	C₂²	semidirect
27. D₈×D₁₀₉	C₂²	semidirect
28. #16812705	C₂²	semidirect
29. C₂×#13182013	C₂²	semidirect
30. C₂×#13182012	C₂²	semidirect
31. C₂×C₈×D₁₀₉	C₂²	semidirect
32. C₂×#41×C₁₀₉	C₂²	semidirect
33. #135×C₁₀₉	C₂²	semidirect
34. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
35. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
36. C₂×D₈×C₁₀₉	C₂²	semidirect
37. C₂²×C₈×C₁₀₉	C₂²	semidirect
41. #13182030	C₂³	semidirect
52. C₂×D₄₃₆	C₂³	semidirect
57. D₄×D₁₀₉	C₂³	semidirect
60. #13182016	C₂³	semidirect
61. D₈₇₂	C₂³	semidirect
62. D₈₇₂	C₂³	semidirect
63. #13182014	C₂³	semidirect
64. #13182013	C₂³	semidirect
65. #13182012	C₂³	semidirect
67. C₈×D₁₀₉	C₂³	semidirect
69. #41×C₁₀₉	C₂³	semidirect
75. Q₁₆×C₁₀₉	C₂³	semidirect
76. SD₁₆×C₁₀₉	C₂³	semidirect
77. D₈×C₁₀₉	C₂³	semidirect
80. C₂×C₈×C₁₀₉	C₂³	semidirect
133. C₂×C₁₀₉	#576	semidirect
138. C₁₀₉	C₂×#576	semidirect
144. C₂×#135	D₁₀₉	semidirect
153. C₂×#41	C₂×D₁₀₉	semidirect
161. #135	C₂×D₁₀₉	semidirect
162. C₂×Q₁₆	C₂×D₁₀₉	semidirect
164. C₂×SD₁₆	C₂×D₁₀₉	semidirect
170. C₂×D₈	C₂×D₁₀₉	semidirect
175. C₂²×C₈	C₂×D₁₀₉	semidirect
184. #41	C₂²×D₁₀₉	semidirect
205. Q₁₆	C₂²×D₁₀₉	semidirect
207. SD₁₆	C₂²×D₁₀₉	semidirect
211. D₈	C₂²×D₁₀₉	semidirect
218. C₂×C₈	C₂²×D₁₀₉	semidirect
273. C₂	#20295828	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×#16812745	2
#20295828	2
C₂×#16812726	2
C₂×#16812724	2
C₂×#16812722	2
C₂²×D₈₇₂	2
C₂×D₈×D₁₀₉	2
C₂×#16812705	2
C₂×#135×C₁₀₉	2
C₂×#576	109

Subgroups — C2×#20295828 #28377782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295828 #28377782