Subgroups — #27203942

Order 13184.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₄×#13110105	C₂	1	1	yes	maximal, characteristic, complemented
3	#20267901	C₂	1	2	yes	maximal, complemented
4	#20267896	C₂	1	2	yes	maximal, complemented
5	#20267891	C₂	1	2	yes	maximal, complemented
6	#20267889	C₂	1	2	yes	maximal, complemented
7	#20267827	C₂	1	1	yes	maximal, characteristic, complemented
8	#20267787	C₂	1	1	yes	maximal, characteristic, complemented
9	#20267785	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692932	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×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	C₂²×C₁₀₃⋊C₈	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	C₄×C₂²⋊D₁₀₃	—	2	2	no	—
23	#15692980	C₂²	1	1	yes	characteristic
24	#15692978	—	2	1	no	—
25	C₄²×D₁₀₃	—	2	1	no	—
26	#15692954	C₂²	1	2	yes	—
27	#15692952	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	#15692932	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×C₄²×C₁₀₃	—	2	1	no	—
41	C₂×C₄²×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	C₂×C₄×D₁₀₃	—	2	1	no	—
48	C₂×C₄×D₁₀₃	—	2	2	no	—
49	#13110095	C₂×C₄	1	4	yes	complemented
50	#13110093	C₂×C₄	1	4	yes	complemented
51	#13110092	C₂×C₄	1	4	yes	complemented
52	#13110090	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₀₃⋊C₈	—	2	2	no	—
54	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₀₃⋊C₈	C₂³	1	2	yes	—
56	C₂×C₁₀₃⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₀₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₃	C₂³	1	2	yes	—
59	Dic₁₀₃⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₁₀₃	—	2	1	no	—
61	(C₂×D₁₀₃)⋊C₄	—	2	2	no	—
62	C₄×Dic₁₀₃	—	2	1	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	2	no	—
71	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
72	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₃	—	2	2	no	—
74	C₄⋊C₄×C₁₀₃	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₀₃	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
77	C₄²×C₁₀₃	—	2	1	no	—
78	C₄²×C₁₀₃	—	2	2	no	—
79	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
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	—	4	2	no	—
94	C₂×C₄×C₁₀₃	—	4	4	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	D₄₁₂	—	2	1	no	—
105	D₄₁₂	C₂²×C₄	1	2	yes	—
106	C₄×D₁₀₃	—	2	4	no	—
107	C₄×D₁₀₃	—	4	4	no	—
108	Dic₂₀₆	—	2	1	no	—
109	Dic₂₀₆	C₂²×C₄	1	2	yes	—
110	C₁₀₃⋊C₈	—	2	2	no	—
111	C₁₀₃⋊C₈	C₂²×C₄	1	2	yes	—
112	C₁₀₃⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₁₀₃	—	2	2	no	—
114	C₂²×C₁₀₃	—	2	2	no	—
115	C₂²×C₁₀₃	—	4	2	no	—
116	C₂²×C₁₀₃	C₄×D₄	1	2	yes	—
117	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₃	—	2	2	no	—
119	C₂×D₁₀₃	—	4	2	no	—
120	C₄×C₁₀₃	—	2	2	no	—
121	C₄×C₁₀₃	—	2	2	no	—
122	C₄×C₁₀₃	—	2	2	no	—
123	C₄×C₁₀₃	—	4	2	no	—
124	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
125	C₄×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₀₃	C₂×#41	1	1	yes	characteristic
127	Dic₁₀₃	—	2	2	no	—
128	Dic₁₀₃	—	4	2	no	—
129	C₂×C₁₀₃	—	2	2	no	—
130	C₂×C₁₀₃	—	4	2	no	—
131	C₂×C₁₀₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₀₃	#575	1	2	yes	—
133	D₁₀₃	—	4	2	no	—
134	#2943	—	103	1	no	maximal, Sylow
135	C₁₀₃	#2943	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	103	1	no	—
137	C₄×#41	—	103	1	no	—
138	C₄×#41	D₁₀₃	1	1	yes	characteristic, complemented
139	#442	—	103	2	no	—
140	#441	—	103	2	no	—
141	#440	—	103	2	no	—
142	#440	—	103	2	no	—
143	#427	—	103	1	no	—
144	#416	—	103	1	no	—
145	#416	—	103	1	no	—
146	C₂×#97	—	103	1	no	—
147	C₂×#41	—	103	1	no	—
148	C₂×#41	C₂×D₁₀₃	1	1	yes	characteristic
149	#135	—	103	8	no	—
150	C₂×Q₁₆	—	103	1	no	—
151	C₂×SD₁₆	—	103	1	no	—
152	C₂×SD₁₆	—	103	1	no	—
153	C₂×D₈	—	103	1	no	—
154	C₂²×C₈	—	103	1	no	—
155	C₂²×C₈	—	103	1	no	—
156	C₄×Q₈	—	103	2	no	—
157	C₄×Q₈	C₂×D₁₀₃	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	103	2	no	—
160	C₄×D₄	—	206	2	no	—
161	C₄×D₄	C₂×D₁₀₃	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	103	1	no	—
164	#117	—	206	1	no	—
165	#117	C₂×D₁₀₃	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	206	1	no	—
168	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
169	#107	—	103	2	no	—
170	#106	—	103	2	no	—
171	#103	—	103	2	no	—
172	#103	—	103	2	no	—
173	#102	—	103	2	no	—
174	#102	—	103	2	no	—
175	#97	—	103	2	no	—
176	#97	—	103	2	no	—
177	#41	—	2	2	no	—
178	#41	—	103	4	no	—
179	#41	—	206	2	no	—
180	#41	C₄×D₁₀₃	1	4	yes	complemented
181	C₂×Q₈	—	103	1	no	—
182	C₂×Q₈	C₂²×D₁₀₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	103	1	no	—
185	C₂×D₄	—	206	1	no	—
186	C₂×D₄	C₂²×D₁₀₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	206	2	no	—
190	C₂²×C₄	—	206	1	no	—
191	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
193	Q₁₆	—	103	4	no	—
194	SD₁₆	—	103	4	no	—
195	SD₁₆	—	103	4	no	—
196	D₈	—	103	4	no	—
197	C₂×C₈	—	103	4	no	—
198	C₂×C₈	—	103	1	no	—
199	C₂×C₈	—	103	1	no	—
200	C₂×C₈	—	103	2	no	—
201	C₂×C₈	—	206	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	103	2	no	—
204	C₄⋊C₄	—	206	2	no	—
205	C₄⋊C₄	C₂²×D₁₀₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	206	1	no	—
209	C₂²⋊C₄	—	206	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	206	2	no	—
213	C₄²	—	206	1	no	—
214	C₄²	C₂²⋊D₁₀₃	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₀₃	1	1	yes	characteristic
216	C₄²	C₂²×D₁₀₃	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	206	1	no	—
219	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	103	2	no	—
222	Q₈	—	206	1	no	—
223	Q₈	C₂×C₄×D₁₀₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	103	2	no	—
227	D₄	—	206	4	no	—
228	D₄	—	206	1	no	—
229	D₄	C₂×C₄×D₁₀₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	412	4	no	—
236	C₂×C₄	—	206	2	no	—
237	C₂×C₄	—	206	1	no	—
238	C₂×C₄	—	206	4	no	—
239	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
243	C₂×C₄	#13110105	1	2	yes	—
244	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
245	C₈	—	103	2	no	—
246	C₈	—	103	2	no	—
247	C₈	—	206	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	412	2	no	—
252	C₂²	—	206	2	no	—
253	C₂²	C₄×C₂²⋊D₁₀₃	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₀₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	412	2	no	—
260	C₄	—	206	2	no	—
261	C₄	C₄×C₂²⋊D₁₀₃	1	2	yes	—
262	C₄	C₂²×C₄×D₁₀₃	1	1	yes	characteristic
263	C₄	C₂×#13110105	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	412	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₀₃	1	1	yes	characteristic
268	C₂	#20268589	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₃
commutator	C₄×C₁₀₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	C₄×#41×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
prime radical	C₂×#15693041	C₂
cosocle	C₂×C₄×C₁₀₃	C₂⁴
hypercenter	C₄×#41	D₁₀₃
hypocenter	C₁₀₃	#2943
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#13110105	C₂	semidirect
3. #20267901	C₂	semidirect
4. #20267896	C₂	semidirect
5. #20267891	C₂	semidirect
6. #20267889	C₂	semidirect
7. #20267827	C₂	semidirect
8. #20267787	C₂	semidirect
9. #20267785	C₂	semidirect
10. C₂×#15692932	C₂	semidirect
11. C₄×#41×C₁₀₃	C₂	semidirect
13. #15693041	C₄	semidirect
43. #13110105	C₂×C₄	semidirect
49. #13110095	C₂×C₄	semidirect
50. #13110093	C₂×C₄	semidirect
51. #13110092	C₂×C₄	semidirect
52. #13110090	C₂×C₄	semidirect
54. C₂×C₁₀₃⋊C₈	C₂×C₄	semidirect
65. #41×C₁₀₃	C₂×C₄	semidirect
135. C₁₀₃	#2943	semidirect
138. C₄×#41	D₁₀₃	semidirect
180. #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
C₄×#13110105	2
#20267901	2
#20267896	2
#20267891	2
#20267889	2
#20267827	2
#20267787	2
#20267785	2
C₂×#15692932	2
C₄×#41×C₁₀₃	2
#2943	103