Subgroups — #28371860

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×#16812724	C₂	1	1	yes	maximal, characteristic
2	#20295500	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295136	C₂	1	2	yes	maximal, complemented
4	#20295124	C₂	1	2	yes	maximal, complemented
5	C₄×#13182016	C₂	1	2	yes	maximal, complemented
6	C₄×#13182014	C₂	1	2	yes	maximal, complemented
7	#20295066	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295022	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812619	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294982	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
13	#16812724	C₄	1	8	yes	complemented
14	C₂×#13182016	C₂²	1	2	yes	—
15	C₂×#13182014	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	C₂²×#11724784	C₂²	1	1	yes	characteristic
18	C₄×D₄₃₆	C₂²	1	2	yes	—
19	C₄×D₄₃₆	C₂²	1	2	yes	—
20	#16812661	—	2	1	no	—
21	#16812661	—	2	1	no	—
22	#16812658	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×D₁₀₉	—	2	2	no	—
24	#16812632	C₂²	1	2	yes	complemented
25	#16812630	C₂²	1	2	yes	complemented
26	#16812623	C₂²	1	1	yes	characteristic, complemented
27	#16812623	C₂²	1	1	yes	characteristic, complemented
28	#16812620	C₂²	1	1	yes	characteristic, complemented
29	#16812620	C₂²	1	1	yes	characteristic, complemented
30	#16812619	C₂²	1	2	yes	—
31	#16812619	C₂²	1	2	yes	—
32	#16812610	C₂²	1	2	yes	complemented
33	C₄×#11724784	C₂²	1	2	yes	complemented
34	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
36	C₄×D₄×C₁₀₉	—	2	2	no	—
37	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
38	#117×C₁₀₉	—	2	1	no	—
39	#117×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂³×D₁₀₉	—	2	1	no	—
43	C₂×D₄₃₆	—	2	1	no	—
44	C₂×D₄₃₆	—	2	2	no	—
45	C₂×D₄₃₆	C₂×C₄	1	4	yes	complemented
46	C₂×D₄₃₆	C₂³	1	2	yes	—
47	C₂×C₄×D₁₀₉	—	2	2	no	—
48	C₂×C₄×D₁₀₉	—	2	2	no	—
49	#13182016	C₂×C₄	1	8	yes	complemented
50	#13182014	C₂×C₄	1	8	yes	complemented
51	#13182013	C₂×C₄	1	4	yes	complemented
52	C₂×#11724784	—	2	2	no	—
53	C₂×#11724784	C₂³	1	1	yes	characteristic
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₂×#11724784	C₂³	1	2	yes	—
56	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₁₀₉	—	2	1	no	—
59	#13182006	—	2	2	no	—
60	#13182006	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	1	no	—
62	#41×C₁₀₉	—	2	2	no	—
63	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
64	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₀₉	—	2	1	no	—
66	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	2	1	no	—
69	C₂²×C₄×C₁₀₉	—	2	2	no	—
70	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₉	—	2	1	no	—
72	C₄⋊C₄×C₁₀₉	—	2	2	no	—
73	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
77	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
78	C₄²×C₁₀₉	—	2	2	no	—
79	C₄²×C₁₀₉	C₂³	1	2	yes	—
80	C₂³×C₁₀₉	—	2	1	no	—
81	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
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	Q₈×C₁₀₉	—	2	1	no	—
87	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
88	D₄×C₁₀₉	—	2	4	no	—
89	D₄×C₁₀₉	—	2	1	no	—
90	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	4	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	8	2	no	—
95	C₂×C₄×C₁₀₉	—	8	1	no	—
96	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#41	1	2	yes	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₁₀₉	—	2	1	no	—
103	C₂×Dic₁₀₉	—	2	1	no	—
104	D₄₃₆	—	2	4	no	—
105	D₄₃₆	—	2	2	no	—
106	D₄₃₆	C₂²×C₄	1	4	yes	—
107	C₄×D₁₀₉	—	8	2	no	—
108	C₄×D₁₀₉	—	8	2	no	—
109	#11724784	—	2	1	no	—
110	#11724784	—	2	1	no	—
111	#11724784	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₀₉	—	2	1	no	—
113	C₂²×C₁₀₉	—	2	1	no	—
114	C₂²×C₁₀₉	—	2	2	no	—
115	C₂²×C₁₀₉	—	4	1	no	—
116	C₂²×C₁₀₉	—	4	1	no	—
117	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
118	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₀₉	—	2	4	no	—
120	C₂×D₁₀₉	—	4	2	no	—
121	C₂×D₁₀₉	—	4	2	no	—
122	C₂×D₁₀₉	—	8	4	no	—
123	C₄×C₁₀₉	—	2	2	no	—
124	C₄×C₁₀₉	—	2	2	no	—
125	C₄×C₁₀₉	—	2	2	no	—
126	C₄×C₁₀₉	—	4	2	no	—
127	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
128	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
130	Dic₁₀₉	—	4	1	no	—
131	Dic₁₀₉	—	4	1	no	—
132	C₂×C₁₀₉	—	2	2	no	—
133	C₂×C₁₀₉	—	4	2	no	—
134	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₀₉	#576	1	1	yes	characteristic
136	C₂×C₁₀₉	#577	1	1	yes	characteristic
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	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
148	#416	—	109	1	no	—
149	C₂×#102	—	109	1	no	—
150	#405	—	109	1	no	—
151	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	1	no	—
157	C₂²×C₈	—	109	1	no	—
158	C₄×Q₈	C₂×D₁₀₉	1	2	yes	complemented
159	C₄×D₄	—	2	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	C₂×D₁₀₉	1	2	yes	complemented
165	#117	—	2	1	no	—
166	#117	—	109	1	no	—
167	#117	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	218	2	no	—
171	#107	—	109	2	no	—
172	#106	—	109	2	no	—
173	#103	—	109	1	no	—
174	#103	—	109	1	no	—
175	#102	—	109	1	no	—
176	#102	—	109	1	no	—
177	#102	—	109	2	no	—
178	#102	—	109	2	no	—
179	#97	—	109	2	no	—
180	C₄×C₈	—	109	2	no	—
181	C₂⁴	—	218	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₁₀₉	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	109	4	no	—
187	C₂×D₄	—	109	2	no	—
188	C₂×D₄	—	218	1	no	—
189	C₂×D₄	—	218	2	no	—
190	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	218	2	no	—
195	C₂²×C₄	—	218	2	no	—
196	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
197	SD₁₆	—	109	8	no	—
198	D₈	—	109	8	no	—
199	M₁₆	—	109	4	no	—
200	C₂×C₈	—	109	2	no	—
201	C₂×C₈	—	109	1	no	—
202	C₂×C₈	—	109	1	no	—
203	C₂×C₈	—	218	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	109	1	no	—
207	C₄⋊C₄	—	109	1	no	—
208	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
209	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	218	1	no	—
213	C₂²⋊C₄	—	218	2	no	—
214	C₂²⋊C₄	—	218	2	no	—
215	C₂²⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	218	1	no	—
218	C₄²	C₂²×D₁₀₉	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	872	1	no	—
221	C₂³	—	872	1	no	—
222	C₂³	—	218	4	no	—
223	C₂³	—	218	2	no	—
224	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	109	4	no	—
230	D₄	—	218	4	no	—
231	D₄	—	218	2	no	—
232	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	872	2	no	—
239	C₂×C₄	—	872	2	no	—
240	C₂×C₄	—	218	1	no	—
241	C₂×C₄	—	218	1	no	—
242	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
246	C₂×C₄	#13182029	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
248	C₈	—	109	4	no	—
249	C₈	—	218	1	no	—
250	C₈	—	218	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	872	4	no	—
257	C₂²	—	436	2	no	—
258	C₂²	—	436	2	no	—
259	C₂²	—	218	4	no	—
260	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	436	1	no	—
267	C₄	—	436	1	no	—
268	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
269	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
270	C₄	C₂×#13182029	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	436	4	no	—
274	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
275	C₂	#20295829	1	1	yes	characteristic
276	C₂	#20295834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#520×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812724	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#2946
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #20295500	C₂	semidirect
3. #20295136	C₂	semidirect
4. #20295124	C₂	semidirect
5. C₄×#13182016	C₂	semidirect
6. C₄×#13182014	C₂	semidirect
7. #20295066	C₂	semidirect
8. #20295022	C₂	semidirect
9. C₂×#16812619	C₂	semidirect
10. #20294982	C₂	semidirect
11. #520×C₁₀₉	C₂	semidirect
13. #16812724	C₄	semidirect
24. #16812632	C₂²	semidirect
25. #16812630	C₂²	semidirect
26. #16812623	C₂²	semidirect
27. #16812623	C₂²	semidirect
28. #16812620	C₂²	semidirect
29. #16812620	C₂²	semidirect
32. #16812610	C₂²	semidirect
33. C₄×#11724784	C₂²	semidirect
35. C₄×Q₈×C₁₀₉	C₂²	semidirect
37. C₄×D₄×C₁₀₉	C₂²	semidirect
45. C₂×D₄₃₆	C₂×C₄	semidirect
49. #13182016	C₂×C₄	semidirect
50. #13182014	C₂×C₄	semidirect
51. #13182013	C₂×C₄	semidirect
63. #41×C₁₀₉	C₂×C₄	semidirect
139. C₁₀₉	#2946	semidirect
141. #520	D₁₀₉	semidirect
158. C₄×Q₈	C₂×D₁₀₉	semidirect
164. C₄×D₄	C₂×D₁₀₉	semidirect
183. #41	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×#16812724	2
#20295500	2
#20295136	2
#20295124	2
C₄×#13182016	2
C₄×#13182014	2
#20295066	2
#20295022	2
C₂×#16812619	2
#20294982	2
#520×C₁₀₉	2
#2946	109