Subgroups — #28371864

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×#16812725	C₂	1	1	yes	maximal, characteristic
2	#20295506	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295136	C₂	1	2	yes	maximal, complemented
4	#20295126	C₂	1	2	yes	maximal, complemented
5	C₄×#13182017	C₂	1	2	yes	maximal, complemented
6	C₄×#13182014	C₂	1	2	yes	maximal, complemented
7	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295020	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812620	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294982	C₂	1	1	yes	maximal, characteristic, complemented
11	#518×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812725	C₄	1	8	yes	complemented
14	C₂×#13182017	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×Dic₂₁₈	C₂²	1	2	yes	complemented
19	C₄×D₄₃₆	C₂²	1	2	yes	complemented
20	#16812659	—	2	2	no	—
21	#16812657	—	2	1	no	—
22	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₀₉	—	2	1	no	—
24	#16812632	C₂²	1	2	yes	complemented
25	#16812630	C₂²	1	2	yes	complemented
26	#16812622	C₂²	1	1	yes	characteristic, complemented
27	#16812622	C₂²	1	1	yes	characteristic, complemented
28	#16812620	C₂²	1	2	yes	—
29	#16812620	C₂²	1	2	yes	—
30	#16812619	C₂²	1	1	yes	characteristic, complemented
31	#16812619	C₂²	1	1	yes	characteristic, complemented
32	#16812610	C₂²	1	2	yes	complemented
33	C₄×#11724784	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₀₉	—	2	1	no	—
36	C₄×D₄×C₁₀₉	—	2	1	no	—
37	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
38	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
39	#117×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
42	#13182029	—	2	2	no	—
43	#13182029	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
45	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
47	C₂²×Dic₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	1	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	#13182017	C₂×C₄	1	8	yes	complemented
51	#13182014	C₂×C₄	1	8	yes	complemented
52	#13182013	C₂×C₄	1	4	yes	complemented
53	C₂×#11724784	—	2	2	no	—
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₂×#11724784	C₂³	1	1	yes	characteristic
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₄⋊Dic₁₀₉	—	2	1	no	—
58	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
59	#13182008	—	2	2	no	—
60	#13182006	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	2	no	—
62	C₂⁴×C₁₀₉	—	2	1	no	—
63	C₂×D₄×C₁₀₉	—	2	1	no	—
64	C₂×D₄×C₁₀₉	—	2	2	no	—
65	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
66	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	—	2	2	no	—
69	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
76	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
77	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
78	C₄²×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	C₂³	1	2	yes	—
80	C₂³×C₁₀₉	—	2	2	no	—
81	C₂³×C₁₀₉	—	2	4	no	—
82	C₂³×C₁₀₉	—	8	1	no	—
83	C₂³×C₁₀₉	—	8	1	no	—
84	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₀₉	—	2	1	no	—
86	D₄×C₁₀₉	—	2	4	no	—
87	D₄×C₁₀₉	—	2	2	no	—
88	D₄×C₁₀₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	8	2	no	—
92	C₂×C₄×C₁₀₉	—	8	2	no	—
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	#41	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₁₀₉	—	2	4	no	—
100	C₂×Dic₁₀₉	—	2	1	no	—
101	C₂×Dic₁₀₉	—	2	2	no	—
102	C₂×Dic₁₀₉	—	8	1	no	—
103	D₄₃₆	—	2	1	no	—
104	D₄₃₆	C₂²×C₄	1	2	yes	—
105	C₄×D₁₀₉	—	2	4	no	—
106	C₄×D₁₀₉	—	8	2	no	—
107	Dic₂₁₈	—	2	1	no	—
108	Dic₂₁₈	C₂²×C₄	1	2	yes	—
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	4	no	—
115	C₂²×C₁₀₉	—	4	2	no	—
116	C₂²×C₁₀₉	—	4	2	no	—
117	C₂²×C₁₀₉	—	8	4	no	—
118	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
119	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₀₉	—	2	2	no	—
121	C₂×D₁₀₉	—	4	1	no	—
122	C₂×D₁₀₉	—	4	1	no	—
123	C₄×C₁₀₉	—	2	2	no	—
124	C₄×C₁₀₉	—	2	2	no	—
125	C₄×C₁₀₉	—	4	1	no	—
126	C₄×C₁₀₉	—	4	1	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	—	2	2	no	—
131	Dic₁₀₉	—	4	2	no	—
132	C₂×C₁₀₉	—	2	2	no	—
133	C₂×C₁₀₉	—	4	4	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	2	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	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
152	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	109	2	no	—
162	C₄×D₄	—	218	2	no	—
163	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
164	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
165	#117	—	218	1	no	—
166	#117	C₂×D₁₀₉	1	1	yes	characteristic
167	#117	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	109	1	no	—
169	C₂×C₄⋊C₄	—	218	1	no	—
170	C₂×C₂²⋊C₄	—	2	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	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	—	2	1	no	—
182	#41	—	109	4	no	—
183	#41	—	218	2	no	—
184	C₂×Q₈	—	109	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	109	1	no	—
188	C₂×D₄	—	218	1	no	—
189	C₂×D₄	C₄×D₁₀₉	1	4	yes	complemented
190	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	218	1	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	—	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	—	109	2	no	—
205	C₄⋊C₄	—	218	2	no	—
206	C₄⋊C₄	—	218	1	no	—
207	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
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	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	218	2	no	—
215	C₂²⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	218	2	no	—
218	C₄²	C₂²×D₁₀₉	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	218	1	no	—
224	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
225	Q₈	—	109	2	no	—
226	Q₈	—	218	1	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	109	2	no	—
230	D₄	—	218	4	no	—
231	D₄	—	218	1	no	—
232	D₄	C₂×C₄×D₁₀₉	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	872	2	no	—
238	C₂×C₄	—	872	1	no	—
239	C₂×C₄	—	218	2	no	—
240	C₂×C₄	—	218	1	no	—
241	C₂×C₄	—	218	4	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	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	436	1	no	—
258	C₂²	—	436	1	no	—
259	C₂²	—	218	2	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	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	436	2	no	—
267	C₄	—	218	2	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	4	no	—
273	C₂	—	436	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
275	C₂	#20295829	1	1	yes	characteristic
276	C₂	#20295837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#518×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812725	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#518	D₁₀₉
hypocenter	C₁₀₉	#2946
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #20295506	C₂	semidirect
3. #20295136	C₂	semidirect
4. #20295126	C₂	semidirect
5. C₄×#13182017	C₂	semidirect
6. C₄×#13182014	C₂	semidirect
7. #20295069	C₂	semidirect
8. #20295020	C₂	semidirect
9. C₂×#16812620	C₂	semidirect
10. #20294982	C₂	semidirect
11. #518×C₁₀₉	C₂	semidirect
13. #16812725	C₄	semidirect
18. C₄×Dic₂₁₈	C₂²	semidirect
19. C₄×D₄₃₆	C₂²	semidirect
24. #16812632	C₂²	semidirect
25. #16812630	C₂²	semidirect
26. #16812622	C₂²	semidirect
27. #16812622	C₂²	semidirect
30. #16812619	C₂²	semidirect
31. #16812619	C₂²	semidirect
32. #16812610	C₂²	semidirect
33. C₄×#11724784	C₂²	semidirect
43. #13182029	C₂×C₄	semidirect
50. #13182017	C₂×C₄	semidirect
51. #13182014	C₂×C₄	semidirect
52. #13182013	C₂×C₄	semidirect
65. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
139. C₁₀₉	#2946	semidirect
142. #518	D₁₀₉	semidirect
189. C₂×D₄	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×#16812725	2
#20295506	2
#20295136	2
#20295126	2
C₄×#13182017	2
C₄×#13182014	2
#20295069	2
#20295020	2
C₂×#16812620	2
#20294982	2
#518×C₁₀₉	2
#2946	109