Subgroups — #27204017

Order 13184.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15693041	C₂	1	1	yes	maximal, characteristic
2	#20268271	C₂	1	1	yes	maximal, characteristic, complemented
3	#20267891	C₂	1	2	yes	maximal, complemented
4	#20267889	C₂	1	2	yes	maximal, complemented
5	C₄×#13110095	C₂	1	2	yes	maximal, complemented
6	C₄×#13110090	C₂	1	2	yes	maximal, complemented
7	C₂×#15692954	C₂	1	1	yes	maximal, characteristic, complemented
8	#20267792	C₂	1	1	yes	maximal, characteristic, complemented
9	#20267790	C₂	1	1	yes	maximal, characteristic, complemented
10	#20267747	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13110105	C₂²	1	1	yes	characteristic
13	#15693041	C₄	1	8	yes	complemented
14	C₂×#13110095	C₂²	1	1	yes	characteristic
15	C₂×#13110093	C₂²	1	1	yes	characteristic
16	C₂×#13110092	C₂²	1	1	yes	characteristic
17	C₂×#13110090	C₂²	1	1	yes	characteristic
18	C₂×#13110089	C₂²	1	1	yes	characteristic
19	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₀₆	C₂²	1	2	yes	—
21	C₄×D₄₁₂	C₂²	1	2	yes	—
22	(C₂²⋊D₁₀₃)⋊C₄	—	2	2	no	—
23	#15692979	—	2	1	no	—
24	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₀₃	—	2	1	no	—
26	#15692954	C₂²	1	2	yes	—
27	#15692954	C₂²	1	2	yes	—
28	#15692945	C₂²	1	2	yes	—
29	#15692944	C₂²	1	2	yes	—
30	#15692942	C₂²	1	2	yes	—
31	#15692941	C₂²	1	2	yes	—
32	#15692932	C₂²	1	2	yes	—
33	C₄×C₁₀₃⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₁₀₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₃	C₂²	1	2	yes	—
36	C₄×D₄×C₁₀₃	—	2	2	no	—
37	C₄×D₄×C₁₀₃	C₂²	1	2	yes	—
38	#117×C₁₀₃	—	2	1	no	—
39	#117×C₁₀₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
42	#13110105	—	2	2	no	—
43	#13110105	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₀₆	C₂³	1	1	yes	characteristic
45	C₂×D₄₁₂	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₃	—	2	1	no	—
47	C₂²×Dic₁₀₃	—	2	1	no	—
48	C₂×C₄×D₁₀₃	—	2	1	no	—
49	C₂×C₄×D₁₀₃	—	2	2	no	—
50	#13110095	C₂×C₄	1	4	yes	complemented
51	#13110093	C₂×C₄	1	4	yes	complemented
52	#13110092	C₂×C₄	1	4	yes	complemented
53	#13110090	C₂×C₄	1	4	yes	complemented
54	#13110089	—	2	2	no	—
55	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₀₃⋊C₈	C₂³	1	2	yes	—
57	C₂×C₁₀₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₀₃⋊C₈	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₀₃	—	2	1	no	—
60	C₄⋊Dic₁₀₃	C₂³	1	2	yes	—
61	Dic₁₀₃⋊C₄	—	2	2	no	—
62	(C₂×D₁₀₃)⋊C₄	—	2	2	no	—
63	C₄×Dic₁₀₃	—	2	2	no	—
64	#41×C₁₀₃	—	2	2	no	—
65	#41×C₁₀₃	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₀₃	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₀₃	—	2	1	no	—
68	C₂×D₄×C₁₀₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₃	—	2	1	no	—
70	C₂²×C₄×C₁₀₃	—	2	1	no	—
71	C₂²×C₄×C₁₀₃	—	2	2	no	—
72	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₃	—	2	1	no	—
74	C₄⋊C₄×C₁₀₃	—	2	2	no	—
75	C₄⋊C₄×C₁₀₃	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₃	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₃	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
79	C₂²⋊C₄×C₁₀₃	D₄	1	2	yes	—
80	C₄²×C₁₀₃	—	2	2	no	—
81	C₄²×C₁₀₃	C₂³	1	2	yes	—
82	C₂³×C₁₀₃	—	2	1	no	—
83	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₀₃	—	2	1	no	—
85	Q₈×C₁₀₃	—	2	1	no	—
86	Q₈×C₁₀₃	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₃	—	2	4	no	—
88	D₄×C₁₀₃	—	2	1	no	—
89	D₄×C₁₀₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₃	—	2	1	no	—
91	C₂×C₄×C₁₀₃	—	2	4	no	—
92	C₂×C₄×C₁₀₃	—	2	2	no	—
93	C₂×C₄×C₁₀₃	—	8	2	no	—
94	C₂×C₄×C₁₀₃	—	8	1	no	—
95	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₃	#41	1	2	yes	—
100	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₀₃	—	2	4	no	—
102	C₂×Dic₁₀₃	—	2	1	no	—
103	C₂×Dic₁₀₃	—	2	2	no	—
104	C₂×Dic₁₀₃	—	8	1	no	—
105	D₄₁₂	—	2	1	no	—
106	D₄₁₂	C₂²×C₄	1	2	yes	—
107	C₄×D₁₀₃	—	2	4	no	—
108	C₄×D₁₀₃	—	8	2	no	—
109	Dic₂₀₆	—	2	1	no	—
110	Dic₂₀₆	C₂²×C₄	1	2	yes	—
111	C₁₀₃⋊C₈	—	2	2	no	—
112	C₁₀₃⋊C₈	C₂²×C₄	1	2	yes	—
113	C₁₀₃⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂²×C₁₀₃	—	2	1	no	—
115	C₂²×C₁₀₃	—	2	1	no	—
116	C₂²×C₁₀₃	—	2	2	no	—
117	C₂²×C₁₀₃	—	4	1	no	—
118	C₂²×C₁₀₃	—	4	1	no	—
119	C₂²×C₁₀₃	C₄×D₄	1	2	yes	—
120	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₀₃	—	2	2	no	—
122	C₂×D₁₀₃	—	4	1	no	—
123	C₂×D₁₀₃	—	4	1	no	—
124	C₄×C₁₀₃	—	2	2	no	—
125	C₄×C₁₀₃	—	2	2	no	—
126	C₄×C₁₀₃	—	2	2	no	—
127	C₄×C₁₀₃	—	4	2	no	—
128	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
129	C₄×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₁₀₃	C₂×#41	1	1	yes	characteristic
131	Dic₁₀₃	—	2	2	no	—
132	Dic₁₀₃	—	4	2	no	—
133	C₂×C₁₀₃	—	2	2	no	—
134	C₂×C₁₀₃	—	4	2	no	—
135	C₂×C₁₀₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₀₃	#576	1	1	yes	characteristic
137	C₂×C₁₀₃	#578	1	1	yes	characteristic
138	D₁₀₃	—	4	2	no	—
139	#2951	—	103	1	no	maximal, Sylow
140	C₁₀₃	#2951	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	103	1	no	—
142	#520	—	103	1	no	—
143	#520	D₁₀₃	1	1	yes	characteristic, complemented
144	#440	—	103	2	no	—
145	#440	—	103	2	no	—
146	C₄×Q₁₆	—	103	2	no	—
147	C₄×D₈	—	103	2	no	—
148	C₂×#107	—	103	1	no	—
149	#417	—	103	1	no	—
150	#417	—	103	1	no	—
151	#405	—	103	1	no	—
152	C₂×#41	—	103	1	no	—
153	C₂×#41	C₂×D₁₀₃	1	1	yes	characteristic
154	#135	—	103	8	no	—
155	C₂×Q₁₆	—	103	1	no	—
156	C₂×SD₁₆	—	103	1	no	—
157	C₂×SD₁₆	—	103	1	no	—
158	C₂×D₈	—	103	1	no	—
159	C₂×M₁₆	—	103	1	no	—
160	C₂²×C₈	—	103	1	no	—
161	C₄×Q₈	—	103	2	no	—
162	C₄×Q₈	C₂×D₁₀₃	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	103	2	no	—
165	C₄×D₄	—	206	2	no	—
166	C₄×D₄	C₂×D₁₀₃	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	206	1	no	—
169	#117	C₂×D₁₀₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	103	1	no	—
172	C₂×C₄⋊C₄	—	206	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
174	#107	—	103	2	no	—
175	#107	—	103	2	no	—
176	#103	—	103	2	no	—
177	#103	—	103	2	no	—
178	#102	—	103	2	no	—
179	#102	—	103	2	no	—
180	#97	—	103	2	no	—
181	C₄×C₈	—	103	2	no	—
182	#41	—	2	2	no	—
183	#41	—	103	4	no	—
184	#41	—	206	2	no	—
185	#41	C₄×D₁₀₃	1	4	yes	complemented
186	C₂×Q₈	—	103	1	no	—
187	C₂×Q₈	C₂²×D₁₀₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	103	1	no	—
190	C₂×D₄	—	206	1	no	—
191	C₂×D₄	C₂²×D₁₀₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	206	1	no	—
196	C₂²×C₄	—	206	2	no	—
197	C₂²×C₄	—	206	1	no	—
198	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
199	Q₁₆	—	103	4	no	—
200	SD₁₆	—	103	4	no	—
201	SD₁₆	—	103	4	no	—
202	D₈	—	103	4	no	—
203	M₁₆	—	206	2	no	—
204	C₂×C₈	—	103	4	no	—
205	C₂×C₈	—	103	1	no	—
206	C₂×C₈	—	103	1	no	—
207	C₂×C₈	—	103	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	103	2	no	—
211	C₄⋊C₄	—	206	2	no	—
212	C₄⋊C₄	—	206	1	no	—
213	C₄⋊C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₁₀₃	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	206	2	no	—
218	C₂²⋊C₄	C₂²⋊D₁₀₃	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	206	2	no	—
221	C₄²	C₂²×D₁₀₃	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	206	1	no	—
224	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	103	2	no	—
227	Q₈	—	206	1	no	—
228	Q₈	C₂×C₄×D₁₀₃	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	103	2	no	—
232	D₄	—	206	4	no	—
233	D₄	—	206	1	no	—
234	D₄	C₂×C₄×D₁₀₃	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	824	2	no	—
241	C₂×C₄	—	824	1	no	—
242	C₂×C₄	—	206	2	no	—
243	C₂×C₄	—	206	1	no	—
244	C₂×C₄	—	206	4	no	—
245	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
249	C₂×C₄	#13110105	1	2	yes	—
250	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
251	C₈	—	103	2	no	—
252	C₈	—	103	2	no	—
253	C₈	—	206	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	412	1	no	—
260	C₂²	—	412	1	no	—
261	C₂²	—	206	2	no	—
262	C₂²	C₄×C₂²⋊D₁₀₃	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₁₀₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	412	2	no	—
269	C₄	—	206	2	no	—
270	C₄	C₄×C₂²⋊D₁₀₃	1	2	yes	—
271	C₄	C₂²×C₄×D₁₀₃	1	1	yes	characteristic
272	C₄	C₂×#13110105	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	412	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₁₀₃	1	1	yes	characteristic
277	C₂	#20268594	1	1	yes	characteristic
278	C₂	#20268606	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₃
commutator	C₄×C₁₀₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	#520×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
prime radical	C₂×#15693041	C₂
cosocle	C₂×C₄×C₁₀₃	C₂⁴
hypercenter	#520	D₁₀₃
hypocenter	C₁₀₃	#2951
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #20268271	C₂	semidirect
3. #20267891	C₂	semidirect
4. #20267889	C₂	semidirect
5. C₄×#13110095	C₂	semidirect
6. C₄×#13110090	C₂	semidirect
7. C₂×#15692954	C₂	semidirect
8. #20267792	C₂	semidirect
9. #20267790	C₂	semidirect
10. #20267747	C₂	semidirect
11. #520×C₁₀₃	C₂	semidirect
13. #15693041	C₄	semidirect
43. #13110105	C₂×C₄	semidirect
50. #13110095	C₂×C₄	semidirect
51. #13110093	C₂×C₄	semidirect
52. #13110092	C₂×C₄	semidirect
53. #13110090	C₂×C₄	semidirect
55. C₂×C₁₀₃⋊C₈	C₂×C₄	semidirect
65. #41×C₁₀₃	C₂×C₄	semidirect
140. C₁₀₃	#2951	semidirect
143. #520	D₁₀₃	semidirect
185. #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×#15693041	2
#20268271	2
#20267891	2
#20267889	2
C₄×#13110095	2
C₄×#13110090	2
C₂×#15692954	2
#20267792	2
#20267790	2
#20267747	2
#520×C₁₀₃	2
#2951	103