Subgroups — #28371862

Order 13952.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812727	C₂	1	1	yes	maximal, characteristic
2	#20295507	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295500	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295137	C₂	1	2	yes	maximal, complemented
5	#20295125	C₂	1	2	yes	maximal, complemented
6	#20295113	C₂	1	2	yes	maximal, complemented
7	#20295108	C₂	1	2	yes	maximal, complemented
8	#20295020	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812619	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294984	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
14	#16812727	C₄	1	8	yes	complemented
15	C₂×#13182018	C₂²	1	2	yes	—
16	C₂×D₈₇₂	C₂²	1	2	yes	—
17	C₂²×#11724784	C₂²	1	1	yes	characteristic
18	#16812664	C₂²	1	2	yes	complemented
19	C₄×D₄₃₆	—	2	1	no	—
20	C₄×D₄₃₆	—	2	1	no	—
21	#16812661	C₂²	1	2	yes	—
22	#16812661	C₂²	1	2	yes	—
23	#16812661	C₂²	1	2	yes	complemented
24	C₄×C₂²⋊D₁₀₉	—	2	2	no	—
25	#16812658	C₂²	1	1	yes	characteristic
26	#16812657	—	2	1	no	—
27	C₄⋊C₄×D₁₀₉	—	2	1	no	—
28	C₂²⋊C₄×D₁₀₉	—	2	2	no	—
29	#16812622	C₂²	1	1	yes	characteristic, complemented
30	#16812622	C₂²	1	1	yes	characteristic, complemented
31	#16812619	C₂²	1	2	yes	—
32	#16812619	C₂²	1	2	yes	—
33	#16812619	C₂²	1	1	yes	characteristic, complemented
34	#16812619	C₂²	1	1	yes	characteristic, complemented
35	#16812609	C₂²	1	2	yes	complemented
36	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
37	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
38	#117×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#107×C₁₀₉	C₂²	1	2	yes	complemented
41	#106×C₁₀₉	C₂²	1	2	yes	complemented
42	C₂³×D₁₀₉	—	2	1	no	—
43	#13182029	—	2	2	no	—
44	#13182029	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
46	C₂×D₄₃₆	—	2	1	no	—
47	C₂×D₄₃₆	—	2	2	no	—
48	C₂×D₄₃₆	C₂×C₄	1	4	yes	complemented
49	C₂×D₄₃₆	C₂³	1	2	yes	—
50	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
52	C₂×C₄×D₁₀₉	—	2	1	no	—
53	C₂×C₄×D₁₀₉	—	2	2	no	—
54	C₂×C₄×D₁₀₉	—	2	2	no	—
55	C₂×C₄×D₁₀₉	—	2	2	no	—
56	#13182018	C₂×C₄	1	8	yes	complemented
57	D₈₇₂	C₂×C₄	1	8	yes	complemented
58	C₂×#11724784	—	2	2	no	—
59	C₂×#11724784	C₂³	1	1	yes	characteristic
60	C₂×#11724784	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
62	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
63	#13182008	—	2	2	no	—
64	C₂²⋊Dic₁₀₉	D₄	1	2	yes	—
65	#13182006	—	2	2	no	—
66	#13182006	—	2	2	no	—
67	#13182006	—	2	2	no	—
68	C₄×Dic₁₀₉	—	2	2	no	—
69	C₄×Dic₁₀₉	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₉	—	2	1	no	—
71	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₀₉	—	2	1	no	—
75	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
77	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	—	2	1	no	—
80	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₁₀₉	—	2	1	no	—
82	C₂²×D₁₀₉	—	2	2	no	—
83	C₂²×D₁₀₉	—	2	4	no	—
84	C₂²×D₁₀₉	—	8	1	no	—
85	C₂²×D₁₀₉	—	8	1	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	8	1	no	—
90	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	#41	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₁₀₉	—	2	4	no	—
95	C₂×Dic₁₀₉	—	2	1	no	—
96	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
97	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
98	D₄₃₆	—	2	4	no	—
99	D₄₃₆	—	2	2	no	—
100	D₄₃₆	—	2	1	no	—
101	D₄₃₆	C₂²×C₄	1	2	yes	—
102	D₄₃₆	C₂²×C₄	1	4	yes	—
103	C₄×D₁₀₉	—	2	4	no	—
104	C₄×D₁₀₉	—	8	2	no	—
105	C₄×D₁₀₉	—	8	2	no	—
106	C₄×D₁₀₉	—	8	2	no	—
107	Dic₂₁₈	—	2	1	no	—
108	Dic₂₁₈	C₂²×C₄	1	2	yes	—
109	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
110	#11724784	—	2	1	no	—
111	#11724784	—	2	1	no	—
112	C₂²×C₁₀₉	—	2	1	no	—
113	C₂²×C₁₀₉	—	2	1	no	—
114	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
115	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₀₉	—	2	2	no	—
117	C₂×D₁₀₉	—	2	4	no	—
118	C₂×D₁₀₉	—	4	1	no	—
119	C₂×D₁₀₉	—	4	2	no	—
120	C₂×D₁₀₉	—	4	2	no	—
121	C₂×D₁₀₉	—	4	1	no	—
122	C₂×D₁₀₉	—	8	4	no	—
123	C₄×C₁₀₉	—	4	1	no	—
124	C₄×C₁₀₉	—	4	2	no	—
125	C₄×C₁₀₉	—	4	1	no	—
126	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
127	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
129	Dic₁₀₉	—	2	2	no	—
130	Dic₁₀₉	—	2	2	no	—
131	Dic₁₀₉	—	2	2	no	—
132	C₂×C₁₀₉	—	2	2	no	—
133	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₀₉	#576	1	1	yes	characteristic
135	C₂×C₁₀₉	#577	1	1	yes	characteristic
136	D₁₀₉	—	4	2	no	—
137	D₁₀₉	—	4	4	no	—
138	#2946	—	109	1	no	maximal, Sylow
139	C₁₀₉	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	109	1	no	—
141	#520	—	109	1	no	—
142	#518	—	109	1	no	—
143	#442	—	109	2	no	—
144	#440	—	109	2	no	—
145	C₄×SD₁₆	—	109	2	no	—
146	C₄×D₈	—	109	2	no	—
147	#428	D₁₀₉	1	1	yes	characteristic, complemented
148	#416	—	109	1	no	—
149	C₂×#102	—	109	1	no	—
150	#405	—	109	1	no	—
151	C₂×#41	—	109	1	no	—
152	C₂²×D₄	—	109	1	no	—
153	AGL(1,ℤ/8)	—	109	8	no	—
154	C₂×SD₁₆	—	109	2	no	—
155	C₂×D₈	—	109	2	no	—
156	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
157	C₂²×C₈	—	109	1	no	—
158	C₄×Q₈	—	109	2	no	—
159	C₄×D₄	—	109	2	no	—
160	C₄×D₄	—	109	2	no	—
161	C₄×D₄	—	109	2	no	—
162	C₄×D₄	—	218	1	no	—
163	C₄×D₄	—	218	1	no	—
164	C₄×D₄	—	218	2	no	—
165	#117	—	109	1	no	—
166	#117	—	218	1	no	—
167	#117	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	218	1	no	—
169	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	218	2	no	—
171	#107	C₂×D₁₀₉	1	2	yes	complemented
172	#106	C₂×D₁₀₉	1	2	yes	complemented
173	#103	—	109	1	no	—
174	#103	—	109	1	no	—
175	#102	—	109	2	no	—
176	#102	—	109	2	no	—
177	#102	—	109	1	no	—
178	#102	—	109	1	no	—
179	#97	—	109	2	no	—
180	C₄×C₈	—	109	2	no	—
181	C₂⁴	—	218	1	no	—
182	#41	—	109	4	no	—
183	#41	—	218	2	no	—
184	C₂×Q₈	—	109	1	no	—
185	C₂×D₄	—	109	4	no	—
186	C₂×D₄	—	109	1	no	—
187	C₂×D₄	—	109	2	no	—
188	C₂×D₄	—	218	1	no	—
189	C₂×D₄	—	218	1	no	—
190	C₂×D₄	—	218	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	218	2	no	—
193	C₂²×C₄	—	218	2	no	—
194	C₂²×C₄	—	218	2	no	—
195	C₂²×C₄	—	218	1	no	—
196	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
197	SD₁₆	—	109	8	no	—
198	D₈	—	109	8	no	—
199	M₁₆	C₄×D₁₀₉	1	4	yes	complemented
200	C₂×C₈	—	109	1	no	—
201	C₂×C₈	—	109	1	no	—
202	C₂×C₈	—	218	2	no	—
203	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	109	1	no	—
206	C₄⋊C₄	—	109	1	no	—
207	C₄⋊C₄	—	218	2	no	—
208	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
209	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	109	2	no	—
213	C₂²⋊C₄	—	218	2	no	—
214	C₂²⋊C₄	—	218	2	no	—
215	C₂²⋊C₄	—	218	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	109	2	no	—
218	C₄²	—	218	2	no	—
219	C₂³	—	872	1	no	—
220	C₂³	—	872	1	no	—
221	C₂³	—	218	4	no	—
222	C₂³	—	218	1	no	—
223	C₂³	—	218	2	no	—
224	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
225	Q₈	—	109	2	no	—
226	Q₈	—	218	1	no	—
227	D₄	—	109	2	no	—
228	D₄	—	109	4	no	—
229	D₄	—	218	4	no	—
230	D₄	—	218	4	no	—
231	D₄	—	218	2	no	—
232	D₄	—	218	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	872	2	no	—
238	C₂×C₄	—	872	2	no	—
239	C₂×C₄	—	872	2	no	—
240	C₂×C₄	—	109	2	no	—
241	C₂×C₄	—	109	2	no	—
242	C₂×C₄	—	218	1	no	—
243	C₂×C₄	—	218	4	no	—
244	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
245	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
246	C₂×C₄	#13182030	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
248	C₈	—	218	1	no	—
249	C₈	—	218	1	no	—
250	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	872	4	no	—
254	C₂²	—	436	1	no	—
255	C₂²	—	436	2	no	—
256	C₂²	—	436	2	no	—
257	C₂²	—	436	1	no	—
258	C₂²	—	218	2	no	—
259	C₂²	—	218	4	no	—
260	C₂²	#16812661	1	2	yes	—
261	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	218	2	no	—
266	C₄	—	218	2	no	—
267	C₄	—	218	2	no	—
268	C₄	#16812661	1	2	yes	—
269	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
270	C₄	C₂×#13182030	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	436	4	no	—
273	C₂	—	436	2	no	—
274	C₂	C₂×#16812661	1	1	yes	characteristic
275	C₂	#20295830	1	1	yes	characteristic
276	C₂	#20295835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#428×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812727	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#428	D₁₀₉
hypocenter	C₁₀₉	#2946
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #20295507	C₂	semidirect
3. #20295500	C₂	semidirect
4. #20295137	C₂	semidirect
5. #20295125	C₂	semidirect
6. #20295113	C₂	semidirect
7. #20295108	C₂	semidirect
8. #20295020	C₂	semidirect
9. C₂×#16812619	C₂	semidirect
10. #20294984	C₂	semidirect
11. #428×C₁₀₉	C₂	semidirect
14. #16812727	C₄	semidirect
18. #16812664	C₂²	semidirect
23. #16812661	C₂²	semidirect
29. #16812622	C₂²	semidirect
30. #16812622	C₂²	semidirect
33. #16812619	C₂²	semidirect
34. #16812619	C₂²	semidirect
35. #16812609	C₂²	semidirect
36. C₈×Dic₁₀₉	C₂²	semidirect
40. #107×C₁₀₉	C₂²	semidirect
41. #106×C₁₀₉	C₂²	semidirect
44. #13182029	C₂×C₄	semidirect
48. C₂×D₄₃₆	C₂×C₄	semidirect
56. #13182018	C₂×C₄	semidirect
57. D₈₇₂	C₂×C₄	semidirect
72. M₁₆×C₁₀₉	C₂×C₄	semidirect
139. C₁₀₉	#2946	semidirect
147. #428	D₁₀₉	semidirect
171. #107	C₂×D₁₀₉	semidirect
172. #106	C₂×D₁₀₉	semidirect
199. M₁₆	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812727	2
#20295507	2
#20295500	2
#20295137	2
#20295125	2
#20295113	2
#20295108	2
#20295020	2
C₂×#16812619	2
#20294984	2
#428×C₁₀₉	2
#2946	109