Subgroups — #27203979

Order 13184.

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×#15693046	C₂	1	1	yes	maximal, characteristic
2	#20268274	C₂	1	1	yes	maximal, characteristic, complemented
3	#20268265	C₂	1	1	yes	maximal, characteristic, complemented
4	#20267900	C₂	1	2	yes	maximal, complemented
5	#20267894	C₂	1	2	yes	maximal, complemented
6	#20267876	C₂	1	2	yes	maximal, complemented
7	#20267871	C₂	1	2	yes	maximal, complemented
8	#20267834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692943	C₂	1	1	yes	maximal, characteristic, complemented
10	#20267746	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₁₂	C₂²	1	1	yes	characteristic
13	#15693046	C₄	1	8	yes	complemented
14	C₂×#13110092	C₂²	1	2	yes	—
15	C₂×#13110090	C₂²	1	2	yes	—
16	C₂×#13110089	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₀₃	C₂²	1	2	yes	complemented
18	C₄×D₄₁₂	—	2	1	no	—
19	C₄×D₄₁₂	—	2	1	no	—
20	D₄×Dic₁₀₃	—	2	2	no	—
21	D₄×Dic₁₀₃	C₂²	1	2	yes	complemented
22	#15692983	C₂²	1	2	yes	—
23	#15692983	C₂²	1	2	yes	—
24	#15692980	—	2	1	no	—
25	#15692980	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₀₃	—	2	1	no	—
27	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×D₁₀₃	—	2	2	no	—
29	#15692954	C₂²	1	2	yes	complemented
30	#15692952	C₂²	1	2	yes	complemented
31	#15692943	C₂²	1	2	yes	—
32	#15692943	C₂²	1	2	yes	—
33	#15692931	C₂²	1	2	yes	complemented
34	C₈×Dic₁₀₃	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₀₃	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
37	#117×C₁₀₃	C₂²	1	1	yes	characteristic
38	#103×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
39	#103×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
40	#102×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
41	#102×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₀₃	—	2	1	no	—
43	C₂×D₄₁₂	—	2	1	no	—
44	C₂×D₄₁₂	—	2	2	no	—
45	C₂×D₄₁₂	C₂×C₄	1	4	yes	complemented
46	C₂×D₄₁₂	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₃	—	2	1	no	—
48	C₂²×Dic₁₀₃	—	2	2	no	—
49	C₂×C₄×D₁₀₃	—	2	2	no	—
50	C₂×C₄×D₁₀₃	—	2	2	no	—
51	#13110092	C₂×C₄	1	8	yes	complemented
52	#13110090	C₂×C₄	1	8	yes	complemented
53	#13110089	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₀₃⋊C₈	C₂³	1	2	yes	—
55	C₄⋊Dic₁₀₃	—	2	1	no	—
56	C₄⋊Dic₁₀₃	—	2	2	no	—
57	C₄⋊Dic₁₀₃	D₄	1	1	yes	characteristic
58	C₄⋊Dic₁₀₃	D₄	1	1	yes	characteristic
59	C₄⋊Dic₁₀₃	C₂³	1	2	yes	—
60	C₂²⋊Dic₁₀₃	—	2	2	no	—
61	C₂²⋊Dic₁₀₃	D₄	1	2	yes	—
62	(C₂×D₁₀₃)⋊C₄	—	2	2	no	—
63	(C₂×D₁₀₃)⋊C₄	—	2	2	no	—
64	C₄×Dic₁₀₃	—	2	2	no	—
65	C₄×Dic₁₀₃	C₂³	1	2	yes	—
66	#41×C₁₀₃	—	2	2	no	—
67	#41×C₁₀₃	C₂×C₄	1	4	yes	complemented
68	C₂×Q₈×C₁₀₃	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₁₀₃	—	2	1	no	—
70	C₂×D₄×C₁₀₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₃	—	2	1	no	—
72	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₀₃	—	2	2	no	—
74	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₃	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₃	—	2	1	no	—
79	C₄²×C₁₀₃	—	2	1	no	—
80	C₂³×C₁₀₃	—	2	1	no	—
81	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₃	—	2	2	no	—
83	C₂²×D₁₀₃	—	2	4	no	—
84	C₂²×D₁₀₃	—	8	1	no	—
85	C₂²×D₁₀₃	—	8	1	no	—
86	Q₈×C₁₀₃	—	2	1	no	—
87	Q₈×C₁₀₃	C₂²×C₄	1	2	yes	—
88	D₄×C₁₀₃	—	2	4	no	—
89	D₄×C₁₀₃	—	2	1	no	—
90	D₄×C₁₀₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₃	—	2	1	no	—
92	C₂×C₄×C₁₀₃	—	2	4	no	—
93	C₂×C₄×C₁₀₃	—	2	1	no	—
94	C₂×C₄×C₁₀₃	—	2	1	no	—
95	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₃	#41	1	2	yes	—
98	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₀₃	—	2	2	no	—
100	C₂×Dic₁₀₃	—	8	2	no	—
101	C₂×Dic₁₀₃	—	8	1	no	—
102	C₂×Dic₁₀₃	C₂×D₄	1	2	yes	—
103	C₂×Dic₁₀₃	C₂×D₄	1	2	yes	—
104	D₄₁₂	—	2	4	no	—
105	D₄₁₂	—	2	2	no	—
106	D₄₁₂	C₂²×C₄	1	4	yes	—
107	C₄×D₁₀₃	—	8	2	no	—
108	C₄×D₁₀₃	—	8	2	no	—
109	C₈×C₁₀₃	—	2	1	no	—
110	C₈×C₁₀₃	—	2	1	no	—
111	C₁₀₃⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₀₃	—	2	1	no	—
113	C₂²×C₁₀₃	—	2	1	no	—
114	C₂²×C₁₀₃	—	2	2	no	—
115	C₂²×C₁₀₃	—	4	1	no	—
116	C₂²×C₁₀₃	—	4	1	no	—
117	C₂²×C₁₀₃	C₄×D₄	1	2	yes	—
118	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₀₃	—	2	4	no	—
120	C₂×D₁₀₃	—	4	2	no	—
121	C₂×D₁₀₃	—	4	2	no	—
122	C₂×D₁₀₃	—	8	4	no	—
123	C₄×C₁₀₃	—	2	2	no	—
124	C₄×C₁₀₃	—	4	1	no	—
125	C₄×C₁₀₃	—	4	1	no	—
126	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
127	C₄×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₀₃	C₂×#41	1	1	yes	characteristic
129	Dic₁₀₃	—	2	2	no	—
130	Dic₁₀₃	—	2	2	no	—
131	Dic₁₀₃	—	4	2	no	—
132	C₂×C₁₀₃	—	2	2	no	—
133	C₂×C₁₀₃	—	4	2	no	—
134	C₂×C₁₀₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₀₃	#576	1	1	yes	characteristic
136	C₂×C₁₀₃	#577	1	1	yes	characteristic
137	D₁₀₃	—	4	4	no	—
138	#2946	—	103	1	no	maximal, Sylow
139	C₁₀₃	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	103	1	no	—
141	#520	—	103	1	no	—
142	#518	—	103	1	no	—
143	#442	—	103	2	no	—
144	#440	—	103	2	no	—
145	C₄×SD₁₆	—	103	2	no	—
146	C₄×D₈	—	103	2	no	—
147	#428	—	103	1	no	—
148	#416	D₁₀₃	1	1	yes	characteristic, complemented
149	C₂×#102	—	103	1	no	—
150	#405	—	103	1	no	—
151	C₂×#41	C₂×D₁₀₃	1	1	yes	characteristic
152	C₂²×D₄	—	103	1	no	—
153	AGL(1,ℤ/8)	—	103	8	no	—
154	C₂×SD₁₆	—	103	2	no	—
155	C₂×D₈	—	103	2	no	—
156	C₂×M₁₆	—	103	1	no	—
157	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
158	C₄×Q₈	—	103	2	no	—
159	C₄×D₄	—	103	2	no	—
160	C₄×D₄	—	103	2	no	—
161	C₄×D₄	—	103	2	no	—
162	C₄×D₄	—	206	1	no	—
163	C₄×D₄	—	206	1	no	—
164	C₄×D₄	—	206	2	no	—
165	#117	—	103	1	no	—
166	#117	—	206	1	no	—
167	#117	C₂×D₁₀₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	103	1	no	—
169	C₂×C₄⋊C₄	—	206	1	no	—
170	C₂×C₂²⋊C₄	—	206	2	no	—
171	#107	—	103	2	no	—
172	#106	—	103	2	no	—
173	#103	C₂×D₁₀₃	1	1	yes	characteristic, complemented
174	#103	C₂×D₁₀₃	1	1	yes	characteristic, complemented
175	#102	—	103	2	no	—
176	#102	—	103	2	no	—
177	#102	C₂×D₁₀₃	1	1	yes	characteristic, complemented
178	#102	C₂×D₁₀₃	1	1	yes	characteristic, complemented
179	#97	—	103	2	no	—
180	C₄×C₈	—	103	2	no	—
181	C₂⁴	—	206	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₁₀₃	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₁₀₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	103	4	no	—
187	C₂×D₄	—	103	2	no	—
188	C₂×D₄	—	206	1	no	—
189	C₂×D₄	—	206	2	no	—
190	C₂×D₄	C₂²×D₁₀₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	206	2	no	—
193	C₂²×C₄	—	206	2	no	—
194	C₂²×C₄	—	206	2	no	—
195	C₂²×C₄	—	206	1	no	—
196	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
197	SD₁₆	—	103	8	no	—
198	D₈	—	103	8	no	—
199	M₁₆	—	103	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	103	2	no	—
202	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
204	C₄⋊C₄	—	103	1	no	—
205	C₄⋊C₄	—	103	1	no	—
206	C₄⋊C₄	—	103	2	no	—
207	C₄⋊C₄	—	206	2	no	—
208	C₄⋊C₄	—	206	1	no	—
209	C₄⋊C₄	C₂²×D₁₀₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₀₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	103	2	no	—
213	C₂²⋊C₄	—	206	2	no	—
214	C₂²⋊C₄	—	206	2	no	—
215	C₂²⋊C₄	—	206	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	103	2	no	—
218	C₄²	—	206	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	824	1	no	—
221	C₂³	—	824	1	no	—
222	C₂³	—	206	4	no	—
223	C₂³	—	206	2	no	—
224	C₂³	D₄×D₁₀₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₁₀₃	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	103	4	no	—
230	D₄	—	206	2	no	—
231	D₄	—	206	4	no	—
232	D₄	C₂×C₄×D₁₀₃	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	824	2	no	—
238	C₂×C₄	—	824	2	no	—
239	C₂×C₄	—	824	1	no	—
240	C₂×C₄	—	824	2	no	—
241	C₂×C₄	—	103	2	no	—
242	C₂×C₄	—	103	2	no	—
243	C₂×C₄	—	206	2	no	—
244	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
245	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
246	C₂×C₄	#13110104	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	103	4	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	—	824	4	no	—
257	C₂²	—	412	2	no	—
258	C₂²	—	412	2	no	—
259	C₂²	—	206	4	no	—
260	C₂²	(C₂²⋊D₁₀₃)⋊C₄	1	2	yes	—
261	C₂²	C₂×D₄×D₁₀₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	412	2	no	—
266	C₄	—	206	2	no	—
267	C₄	—	206	2	no	—
268	C₄	(C₂²⋊D₁₀₃)⋊C₄	1	2	yes	—
269	C₄	C₂²×C₄×D₁₀₃	1	1	yes	characteristic
270	C₄	C₂×#13110104	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	412	4	no	—
274	C₂	C₂×(C₂²⋊D₁₀₃)⋊C₄	1	1	yes	characteristic
275	C₂	#20268593	1	1	yes	characteristic
276	C₂	#20268598	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₃
commutator	C₄×C₁₀₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	#416×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
prime radical	C₂×#15693046	C₂
cosocle	C₂×C₄×C₁₀₃	C₂⁴
hypercenter	#416	D₁₀₃
hypocenter	C₁₀₃	#2946
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #20268274	C₂	semidirect
3. #20268265	C₂	semidirect
4. #20267900	C₂	semidirect
5. #20267894	C₂	semidirect
6. #20267876	C₂	semidirect
7. #20267871	C₂	semidirect
8. #20267834	C₂	semidirect
9. C₂×#15692943	C₂	semidirect
10. #20267746	C₂	semidirect
11. #416×C₁₀₃	C₂	semidirect
13. #15693046	C₄	semidirect
17. Q₈×Dic₁₀₃	C₂²	semidirect
21. D₄×Dic₁₀₃	C₂²	semidirect
29. #15692954	C₂²	semidirect
30. #15692952	C₂²	semidirect
33. #15692931	C₂²	semidirect
34. C₈×Dic₁₀₃	C₂²	semidirect
38. #103×C₁₀₃	C₂²	semidirect
39. #103×C₁₀₃	C₂²	semidirect
40. #102×C₁₀₃	C₂²	semidirect
41. #102×C₁₀₃	C₂²	semidirect
45. C₂×D₄₁₂	C₂×C₄	semidirect
51. #13110092	C₂×C₄	semidirect
52. #13110090	C₂×C₄	semidirect
53. #13110089	C₂×C₄	semidirect
67. #41×C₁₀₃	C₂×C₄	semidirect
139. C₁₀₃	#2946	semidirect
148. #416	D₁₀₃	semidirect
173. #103	C₂×D₁₀₃	semidirect
174. #103	C₂×D₁₀₃	semidirect
177. #102	C₂×D₁₀₃	semidirect
178. #102	C₂×D₁₀₃	semidirect
183. #41	C₄×D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15693046	2
#20268274	2
#20268265	2
#20267900	2
#20267894	2
#20267876	2
#20267871	2
#20267834	2
C₂×#15692943	2
#20267746	2
#416×C₁₀₃	2
#2946	103