Subgroups — C₄×#16812745 #28376239

Order 13952.

1660 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	1	yes	maximal, characteristic
2	#20295509	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295508	C₂	1	2	yes	maximal, complemented
4	#20295506	C₂	1	3	yes	maximal, complemented
5	#20295500	C₂	1	3	yes	maximal, complemented
6	#20295498	C₂	1	6	yes	maximal, complemented
7	C₄×#13182030	C₂	1	2	yes	maximal, complemented
8	C₄×#13182029	C₂	1	3	yes	maximal, complemented
9	C₂×C₄×D₄₃₆	C₂	1	3	yes	maximal, complemented
10	C₄×D₄×D₁₀₉	C₂	1	6	yes	maximal, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812745	C₄	1	16	yes	direct product factor
13	C₂×#13182030	C₂²	1	2	yes	—
14	C₂×#13182029	C₂²	1	3	yes	—
15	C₂²×D₄₃₆	C₂²	1	3	yes	—
16	C₂×D₄×D₁₀₉	C₂²	1	6	yes	—
17	C₂²×C₄×D₁₀₉	C₂²	1	6	yes	complemented
18	Q₈×Dic₁₀₉	C₂²	1	2	yes	complemented
19	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
20	C₄×Dic₂₁₈	C₂²	1	3	yes	complemented
21	C₄×D₄₃₆	C₂²	1	6	yes	complemented
22	C₄×D₄₃₆	C₂²	1	3	yes	complemented
23	C₄×D₄₃₆	C₂²	1	6	yes	complemented
24	C₄×D₄₃₆	C₂²	1	6	yes	complemented
25	C₄×D₄₃₆	C₂²	1	3	yes	complemented
26	D₄×Dic₁₀₉	C₂²	1	6	yes	complemented
27	#16812661	C₂²	1	6	yes	complemented
28	#16812661	C₂²	1	6	yes	complemented
29	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
30	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
31	#16812659	C₂²	1	12	yes	complemented
32	#16812658	C₂²	1	3	yes	complemented
33	#16812657	C₂²	1	6	yes	complemented
34	#16812656	C₂²	1	6	yes	complemented
35	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	complemented
36	C₄⋊C₄×D₁₀₉	C₂²	1	6	yes	complemented
37	C₂×#13182006	C₂²	1	6	yes	complemented
38	C₂²⋊C₄×D₁₀₉	C₂²	1	12	yes	complemented
39	C₄²×D₁₀₉	C₂²	1	6	yes	complemented
40	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	C₄×D₄×C₁₀₉	C₂²	1	3	yes	complemented
44	C₄×D₄×C₁₀₉	C₂²	1	3	yes	complemented
45	#117×C₁₀₉	C₂²	1	3	yes	complemented
46	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	complemented
47	C₂³×D₁₀₉	C₂³	1	6	yes	—
48	#13182030	C₂×C₄	1	16	yes	complemented
49	#13182029	C₂×C₄	1	24	yes	complemented
50	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
51	C₂×D₄₃₆	C₂×C₄	1	24	yes	complemented
52	C₂×D₄₃₆	C₂³	1	6	yes	—
53	C₂×D₄₃₆	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
55	D₄×D₁₀₉	C₂×C₄	1	48	yes	complemented
56	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
57	C₂×C₄×D₁₀₉	—	4	12	no	—
58	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
59	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
60	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
61	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
62	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
63	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
64	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
65	#13182008	C₂³	1	6	yes	—
66	#13182008	C₂³	1	2	yes	—
67	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
68	#13182006	C₂³	1	6	yes	—
69	#13182006	C₂³	1	12	yes	—
70	#13182006	C₂³	1	6	yes	—
71	C₄×Dic₁₀₉	C₂³	1	2	yes	—
72	C₄×Dic₁₀₉	C₂³	1	6	yes	—
73	C₄×Dic₁₀₉	C₂³	1	6	yes	—
74	#41×C₁₀₉	C₂×C₄	1	8	yes	complemented
75	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
77	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
78	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
79	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
80	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
81	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
82	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
83	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₄²×C₁₀₉	C₂³	1	3	yes	—
85	C₄²×C₁₀₉	C₂³	1	3	yes	—
86	C₄²×C₁₀₉	C₂³	1	3	yes	—
87	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
88	C₂²×D₁₀₉	—	4	6	no	—
89	C₂²×D₁₀₉	—	4	6	no	—
90	C₂²×D₁₀₉	C₂²×C₄	1	24	yes	complemented
91	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
92	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	complemented
93	D₄×C₁₀₉	C₂²×C₄	1	12	yes	complemented
94	C₂×C₄×C₁₀₉	—	2	6	no	—
95	C₂×C₄×C₁₀₉	—	4	3	no	—
96	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	complemented
97	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
99	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
100	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	complemented
104	C₂×Dic₁₀₉	—	4	6	no	—
105	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
106	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
107	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
108	D₄₃₆	C₂²×C₄	1	24	yes	complemented
109	D₄₃₆	C₂²×C₄	1	12	yes	complemented
110	C₄×D₁₀₉	—	2	12	no	—
111	C₄×D₁₀₉	—	4	12	no	—
112	C₄×D₁₀₉	—	4	6	no	—
113	C₄×D₁₀₉	C₂²×C₄	1	24	yes	complemented
114	Dic₂₁₈	C₂²×C₄	1	4	yes	complemented
115	C₂²×C₁₀₉	—	2	3	no	—
116	C₂²×C₁₀₉	—	2	3	no	—
117	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
118	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₀₉	—	2	6	no	—
120	C₂×D₁₀₉	—	2	6	no	—
121	C₂×D₁₀₉	—	4	24	no	—
122	C₂×D₁₀₉	C₂³×C₄	1	12	yes	—
123	C₄×C₁₀₉	—	2	3	no	—
124	C₄×C₁₀₉	—	2	3	no	—
125	C₄×C₁₀₉	—	2	1	no	—
126	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
127	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
128	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	complemented
129	Dic₁₀₉	—	2	6	no	—
130	Dic₁₀₉	—	2	2	no	—
131	Dic₁₀₉	C₂³×C₄	1	4	yes	—
132	C₂×C₁₀₉	—	2	6	no	—
133	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
134	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₁₀₉	#585	1	1	yes	characteristic
136	D₁₀₉	—	2	12	no	—
137	C₄×H(5,𝔽₂)	—	109	1	no	maximal, Sylow
138	C₁₀₉	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
139	C₂×H(5,𝔽₂)	—	109	1	no	—
140	#520	—	109	1	no	—
141	#520	—	109	2	no	—
142	#520	—	109	3	no	—
143	#518	—	109	6	no	—
144	#518	—	109	3	no	—
145	C₄×#41	—	109	2	no	—
146	C₄×#41	—	109	3	no	—
147	C₄×#41	D₁₀₉	1	1	yes	characteristic, complemented
148	C₂×C₄×D₄	—	109	6	no	—
149	C₂×C₄×D₄	—	109	3	no	—
150	H(5,𝔽₂)	—	109	16	no	—
151	C₂×#41	—	109	2	no	—
152	C₂×#41	—	109	3	no	—
153	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
154	C₂²×D₄	—	109	6	no	—
155	C₂²×D₄	—	109	3	no	—
156	C₂³×C₄	—	109	6	no	—
157	C₄×Q₈	—	109	2	no	—
158	C₄×Q₈	—	109	1	no	—
159	C₄×Q₈	—	109	3	no	—
160	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
161	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
162	C₄×D₄	—	109	6	no	—
163	C₄×D₄	—	109	6	no	—
164	C₄×D₄	—	109	6	no	—
165	C₄×D₄	—	109	6	no	—
166	C₄×D₄	—	109	6	no	—
167	C₄×D₄	—	109	12	no	—
168	C₄×D₄	—	109	6	no	—
169	C₄×D₄	—	109	3	no	—
170	C₄×D₄	—	109	6	no	—
171	C₄×D₄	—	109	6	no	—
172	C₄×D₄	—	109	3	no	—
173	C₄×D₄	C₂×D₁₀₉	1	3	yes	complemented
174	C₄×D₄	C₂×D₁₀₉	1	3	yes	complemented
175	#117	—	109	3	no	—
176	#117	—	109	6	no	—
177	#117	—	109	6	no	—
178	#117	C₂×D₁₀₉	1	3	yes	complemented
179	C₂×C₄⋊C₄	—	109	3	no	—
180	C₂×C₄⋊C₄	—	109	6	no	—
181	C₂×C₂²⋊C₄	—	109	6	no	—
182	C₂×C₂²⋊C₄	—	109	12	no	—
183	C₂×C₄²	—	109	6	no	—
184	C₂×C₄²	C₂×D₁₀₉	1	3	yes	complemented
185	C₂⁴	—	109	6	no	—
186	#41	—	109	16	no	—
187	#41	—	109	24	no	—
188	#41	C₄×D₁₀₉	1	8	yes	complemented
189	C₂×Q₈	—	109	1	no	—
190	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₂×D₄	—	109	6	no	—
192	C₂×D₄	—	109	48	no	—
193	C₂×D₄	—	109	24	no	—
194	C₂×D₄	—	109	6	no	—
195	C₂×D₄	—	109	3	no	—
196	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
197	C₂²×C₄	—	109	6	no	—
198	C₂²×C₄	—	109	6	no	—
199	C₂²×C₄	—	109	6	no	—
200	C₂²×C₄	—	109	12	no	—
201	C₂²×C₄	—	109	6	no	—
202	C₂²×C₄	—	436	12	no	—
203	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
204	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
205	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
206	C₄⋊C₄	—	109	6	no	—
207	C₄⋊C₄	—	109	2	no	—
208	C₄⋊C₄	—	109	6	no	—
209	C₄⋊C₄	—	109	3	no	—
210	C₄⋊C₄	—	109	1	no	—
211	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
212	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
213	C₂²⋊C₄	—	109	6	no	—
214	C₂²⋊C₄	—	109	6	no	—
215	C₂²⋊C₄	—	109	12	no	—
216	C₂²⋊C₄	—	109	6	no	—
217	C₂²⋊C₄	C₂²×D₁₀₉	1	6	yes	—
218	C₄²	—	109	6	no	—
219	C₄²	—	109	6	no	—
220	C₄²	—	109	2	no	—
221	C₄²	C₂²×D₁₀₉	1	3	yes	—
222	C₄²	C₂²×D₁₀₉	1	3	yes	—
223	C₄²	C₂²×D₁₀₉	1	3	yes	—
224	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
225	C₂³	—	109	24	no	—
226	C₂³	—	109	6	no	—
227	C₂³	—	436	6	no	—
228	C₂³	—	436	6	no	—
229	C₂³	C₂³×D₁₀₉	1	3	yes	—
230	Q₈	—	109	4	no	—
231	Q₈	C₂×C₄×D₁₀₉	1	4	yes	complemented
232	D₄	—	109	24	no	—
233	D₄	—	109	24	no	—
234	D₄	—	109	12	no	—
235	D₄	C₂×C₄×D₁₀₉	1	12	yes	complemented
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	4	3	no	—
238	C₂×C₄	—	109	2	no	—
239	C₂×C₄	—	109	24	no	—
240	C₂×C₄	—	109	2	no	—
241	C₂×C₄	—	109	6	no	—
242	C₂×C₄	—	436	12	no	—
243	C₂×C₄	—	436	6	no	—
244	C₂×C₄	—	436	6	no	—
245	C₂×C₄	—	218	12	no	—
246	C₂×C₄	C₂×C₄×D₁₀₉	1	12	yes	complemented
247	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
248	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
249	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
250	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
251	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	109	12	no	—
256	C₂²	—	436	24	no	—
257	C₂²	—	218	6	no	—
258	C₂²	—	218	6	no	—
259	C₂²	C₂²×C₄×D₁₀₉	1	6	yes	—
260	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	3	no	—
264	C₄	—	109	4	no	—
265	C₄	—	218	6	no	—
266	C₄	—	218	2	no	—
267	C₄	C₂²×C₄×D₁₀₉	1	6	yes	—
268	C₄	C₂²×C₄×D₁₀₉	1	2	yes	—
269	C₄	#16812745	1	2	yes	direct product factor
270	C₂	—	2	6	no	—
271	C₂	—	218	12	no	—
272	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
273	C₂	C₂×#16812745	1	1	yes	characteristic
274	C₂	#20295862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₄×#41×C₁₀₉	C₂
socle, cosocle	C₂²×C₁₀₉	C₂⁵
prime radical	C₂×#16812745	C₂
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #20295509	C₂	semidirect
3. #20295508	C₂	semidirect
4. #20295506	C₂	semidirect
5. #20295500	C₂	semidirect
6. #20295498	C₂	semidirect
7. C₄×#13182030	C₂	semidirect
8. C₄×#13182029	C₂	semidirect
9. C₂×C₄×D₄₃₆	C₂	semidirect
10. C₄×D₄×D₁₀₉	C₂	semidirect
11. C₄×#41×C₁₀₉	C₂	semidirect
12. #16812745	C₄	direct (N × H)
17. C₂²×C₄×D₁₀₉	C₂²	semidirect
18. Q₈×Dic₁₀₉	C₂²	semidirect
19. C₄×Dic₂₁₈	C₂²	semidirect
20. C₄×Dic₂₁₈	C₂²	semidirect
21. C₄×D₄₃₆	C₂²	semidirect
22. C₄×D₄₃₆	C₂²	semidirect
23. C₄×D₄₃₆	C₂²	semidirect
24. C₄×D₄₃₆	C₂²	semidirect
25. C₄×D₄₃₆	C₂²	semidirect
26. D₄×Dic₁₀₉	C₂²	semidirect
27. #16812661	C₂²	semidirect
28. #16812661	C₂²	semidirect
29. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
30. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
31. #16812659	C₂²	semidirect
32. #16812658	C₂²	semidirect
33. #16812657	C₂²	semidirect
34. #16812656	C₂²	semidirect
35. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
36. C₄⋊C₄×D₁₀₉	C₂²	semidirect
37. C₂×#13182006	C₂²	semidirect
38. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
39. C₄²×D₁₀₉	C₂²	semidirect
41. C₄×Q₈×C₁₀₉	C₂²	semidirect
42. C₄×Q₈×C₁₀₉	C₂²	semidirect
43. C₄×D₄×C₁₀₉	C₂²	semidirect
44. C₄×D₄×C₁₀₉	C₂²	semidirect
45. #117×C₁₀₉	C₂²	semidirect
46. C₂×C₄²×C₁₀₉	C₂²	semidirect
48. #13182030	C₂×C₄	semidirect
49. #13182029	C₂×C₄	semidirect
51. C₂×D₄₃₆	C₂×C₄	semidirect
55. D₄×D₁₀₉	C₂×C₄	semidirect
74. #41×C₁₀₉	C₂×C₄	semidirect
90. C₂²×D₁₀₉	C₂²×C₄	semidirect
92. Q₈×C₁₀₉	C₂²×C₄	semidirect
93. D₄×C₁₀₉	C₂²×C₄	semidirect
96. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
103. C₂²⋊D₁₀₉	C₂²×C₄	semidirect
108. D₄₃₆	C₂²×C₄	semidirect
109. D₄₃₆	C₂²×C₄	semidirect
113. C₄×D₁₀₉	C₂²×C₄	semidirect
114. Dic₂₁₈	C₂²×C₄	semidirect
128. C₄×C₁₀₉	H(5,𝔽₂)	semidirect
138. C₁₀₉	C₄×H(5,𝔽₂)	semidirect
147. C₄×#41	D₁₀₉	semidirect
160. C₄×Q₈	C₂×D₁₀₉	semidirect
161. C₄×Q₈	C₂×D₁₀₉	semidirect
173. C₄×D₄	C₂×D₁₀₉	semidirect
174. C₄×D₄	C₂×D₁₀₉	semidirect
178. #117	C₂×D₁₀₉	semidirect
184. C₂×C₄²	C₂×D₁₀₉	semidirect
188. #41	C₄×D₁₀₉	semidirect
231. Q₈	C₂×C₄×D₁₀₉	semidirect
235. D₄	C₂×C₄×D₁₀₉	semidirect
246. C₂×C₄	C₂×C₄×D₁₀₉	semidirect
269. C₄	#16812745	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
#20295509	2
#20295508	2
#20295506	2
#20295500	2
#20295498	2
C₄×#13182030	2
C₄×#13182029	2
C₂×C₄×D₄₃₆	2
C₄×D₄×D₁₀₉	2
C₄×#41×C₁₀₉	2
C₄×H(5,𝔽₂)	109

Subgroups — C4×#16812745 #28376239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812745 #28376239