Subgroups — C₁₀₃×#12981 #39918256

Order 26368.

488 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₃×#2929	C₂	1	1	yes	maximal, characteristic
3	C₁₀₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₃×#2112	C₂	1	2	yes	maximal, complemented
5	C₁₀₃×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₀₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
8	#525×C₁₀₃	C₄	1	1	yes	characteristic
9	#504×C₁₀₃	—	2	4	no	—
10	C₂×M₃₂×C₁₀₃	—	2	2	no	—
11	C₂×M₃₂×C₁₀₃	C₂²	1	2	yes	—
12	C₂×M₃₂×C₁₀₃	C₂²	1	2	yes	—
13	#435×C₁₀₃	—	2	2	no	—
14	C₈×D₄×C₁₀₃	—	2	2	no	—
15	#424×C₁₀₃	C₄	1	1	yes	characteristic
16	#408×C₁₀₃	C₄	1	1	yes	characteristic
17	#407×C₁₀₃	C₄	1	1	yes	characteristic
18	#405×C₁₀₃	C₂²	1	1	yes	characteristic
19	#349×C₁₀₃	—	2	2	no	—
20	C₂²⋊C₁₆×C₁₀₃	—	2	4	no	—
21	C₂×#41×C₁₀₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
23	#131×C₁₀₃	—	2	2	no	—
24	#131×C₁₀₃	—	2	2	no	—
25	C₂×M₁₆×C₁₀₃	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₀₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₁₀₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₀₃	—	2	2	no	—
29	C₂²×C₈×C₁₀₃	—	2	1	no	—
30	C₂²×C₈×C₁₀₃	—	2	1	no	—
31	C₂²×C₈×C₁₀₃	—	2	2	no	—
32	C₂²×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₀₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
35	#123×C₁₀₃	C₈	1	2	yes	—
36	C₂²⋊Q₈×C₁₀₃	—	2	1	no	—
37	#121×C₁₀₃	—	2	1	no	—
38	#120×C₁₀₃	C₈	1	2	yes	—
39	C₄×D₄×C₁₀₃	—	2	2	no	—
40	#117×C₁₀₃	C₂×C₄	1	1	yes	characteristic
41	M₃₂×C₁₀₃	—	2	4	no	—
42	M₃₂×C₁₀₃	—	2	2	no	—
43	M₃₂×C₁₀₃	—	2	2	no	—
44	M₃₂×C₁₀₃	—	2	2	no	—
45	M₃₂×C₁₀₃	—	2	2	no	—
46	C₂×C₁₆×C₁₀₃	—	2	4	no	—
47	C₂×C₁₆×C₁₀₃	—	2	2	no	—
48	C₂×C₁₆×C₁₀₃	D₄	1	4	yes	complemented
49	C₄⋊C₈×C₁₀₃	—	2	2	no	—
50	C₂²⋊C₈×C₁₀₃	—	2	2	no	—
51	C₂²⋊C₈×C₁₀₃	—	2	2	no	—
52	#97×C₁₀₃	—	2	1	no	—
53	C₄×C₈×C₁₀₃	—	2	1	no	—
54	C₂⁴×C₁₀₃	C₂×C₈	1	1	yes	characteristic
55	#41×C₁₀₃	—	2	2	no	—
56	#41×C₁₀₃	—	2	2	no	—
57	C₂×Q₈×C₁₀₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₁₀₃	—	2	2	no	—
59	C₂×D₄×C₁₀₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₀₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₃	—	2	1	no	—
62	C₂²×C₄×C₁₀₃	—	2	2	no	—
63	C₂²×C₄×C₁₀₃	—	2	1	no	—
64	C₂²×C₄×C₁₀₃	—	2	2	no	—
65	C₂²×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₀₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₁₀₃	—	2	2	no	—
69	M₁₆×C₁₀₃	—	2	4	no	—
70	M₁₆×C₁₀₃	—	2	2	no	—
71	C₂×C₈×C₁₀₃	—	2	2	no	—
72	C₂×C₈×C₁₀₃	—	2	2	no	—
73	C₂×C₈×C₁₀₃	—	2	1	no	—
74	C₂×C₈×C₁₀₃	—	2	1	no	—
75	C₂×C₈×C₁₀₃	—	2	2	no	—
76	C₂×C₈×C₁₀₃	—	4	1	no	—
77	C₂×C₈×C₁₀₃	—	4	2	no	—
78	C₂×C₈×C₁₀₃	—	4	1	no	—
79	C₂×C₈×C₁₀₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₀₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₀₃	#41	1	2	yes	—
84	C₄⋊C₄×C₁₀₃	—	2	2	no	—
85	C₄⋊C₄×C₁₀₃	—	2	1	no	—
86	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
87	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
88	C₂²⋊C₄×C₁₀₃	C₂×C₈	1	2	yes	—
89	C₄²×C₁₀₃	—	2	1	no	—
90	C₁₆×C₁₀₃	—	2	4	no	—
91	C₁₆×C₁₀₃	—	4	2	no	—
92	C₂³×C₁₀₃	—	2	1	no	—
93	C₂³×C₁₀₃	—	2	2	no	—
94	C₂³×C₁₀₃	—	4	2	no	—
95	C₂³×C₁₀₃	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₁₀₃	C₂²×C₈	1	1	yes	characteristic
97	Q₈×C₁₀₃	—	2	2	no	—
98	D₄×C₁₀₃	—	2	4	no	—
99	D₄×C₁₀₃	—	2	2	no	—
100	D₄×C₁₀₃	—	4	4	no	—
101	C₂×C₄×C₁₀₃	—	2	2	no	—
102	C₂×C₄×C₁₀₃	—	2	2	no	—
103	C₂×C₄×C₁₀₃	—	2	2	no	—
104	C₂×C₄×C₁₀₃	—	2	1	no	—
105	C₂×C₄×C₁₀₃	—	2	2	no	—
106	C₂×C₄×C₁₀₃	—	4	1	no	—
107	C₂×C₄×C₁₀₃	—	4	1	no	—
108	C₂×C₄×C₁₀₃	—	4	2	no	—
109	C₂×C₄×C₁₀₃	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₀₃	#131	1	2	yes	—
114	C₈×C₁₀₃	—	2	2	no	—
115	C₈×C₁₀₃	—	4	1	no	—
116	C₈×C₁₀₃	—	4	1	no	—
117	C₈×C₁₀₃	#120	1	1	yes	characteristic
118	C₈×C₁₀₃	#121	1	2	yes	—
119	C₈×C₁₀₃	#123	1	1	yes	characteristic
120	C₂²×C₁₀₃	—	2	2	no	—
121	C₂²×C₁₀₃	—	2	4	no	—
122	C₂²×C₁₀₃	—	2	2	no	—
123	C₂²×C₁₀₃	—	4	1	no	—
124	C₂²×C₁₀₃	—	4	2	no	—
125	C₂²×C₁₀₃	—	4	1	no	—
126	C₂²×C₁₀₃	—	4	2	no	—
127	C₂²×C₁₀₃	—	4	1	no	—
128	C₂²×C₁₀₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₁₀₃	C₈×D₄	1	2	yes	—
130	C₄×C₁₀₃	—	2	2	no	—
131	C₄×C₁₀₃	—	4	2	no	—
132	C₄×C₁₀₃	—	4	1	no	—
133	C₄×C₁₀₃	#386	1	1	yes	characteristic
134	C₄×C₁₀₃	#408	1	1	yes	characteristic
135	C₄×C₁₀₃	#435	1	2	yes	—
136	#12981	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₁₀₃	—	2	1	no	—
138	C₂×C₁₀₃	—	2	2	no	—
139	C₂×C₁₀₃	—	4	2	no	—
140	C₂×C₁₀₃	—	4	2	no	—
141	C₂×C₁₀₃	#1869	1	1	yes	characteristic
142	#3407	C₂×C₁₀₃	1	1	yes	characteristic, complemented
143	#2929	C₂×C₁₀₃	1	1	yes	characteristic
144	#2115	C₂×C₁₀₃	1	1	yes	characteristic, complemented
145	#2112	C₂×C₁₀₃	1	2	yes	complemented
146	#1378	C₂×C₁₀₃	1	2	yes	complemented
147	C₁₀₃	#12981	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×#131	C₂²×C₁₀₃	1	1	yes	characteristic
149	C₂³×C₈	C₂²×C₁₀₃	1	1	yes	characteristic
150	#525	C₄×C₁₀₃	1	1	yes	characteristic
151	#504	—	2	4	no	—
152	C₂×M₃₂	—	2	2	no	—
153	C₂×M₃₂	C₂²×C₁₀₃	1	2	yes	—
154	C₂×M₃₂	C₂²×C₁₀₃	1	2	yes	—
155	#435	—	2	2	no	—
156	C₈×D₄	—	2	2	no	—
157	#424	C₄×C₁₀₃	1	1	yes	characteristic
158	#408	C₄×C₁₀₃	1	1	yes	characteristic
159	#407	C₄×C₁₀₃	1	1	yes	characteristic
160	#405	C₂²×C₁₀₃	1	1	yes	characteristic
161	#349	—	2	2	no	—
162	C₂²⋊C₁₆	—	2	4	no	—
163	C₂×#41	C₂×C₄×C₁₀₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₁₀₃	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	C₂×M₁₆	C₂×C₄×C₁₀₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₁₀₃	1	1	yes	characteristic
169	C₂×M₁₆	D₄×C₁₀₃	1	2	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₂×C₄×C₁₀₃	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₁₀₃	1	2	yes	—
176	C₂²×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
177	#123	C₈×C₁₀₃	1	2	yes	—
178	C₂²⋊Q₈	—	2	1	no	—
179	#121	—	2	1	no	—
180	#120	C₈×C₁₀₃	1	2	yes	—
181	C₄×D₄	—	2	2	no	—
182	#117	C₂×C₄×C₁₀₃	1	1	yes	characteristic
183	M₃₂	—	2	4	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	M₃₂	—	2	2	no	—
188	C₂×C₁₆	—	2	4	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	D₄×C₁₀₃	1	4	yes	complemented
191	C₄⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₂⁴	C₂×C₈×C₁₀₃	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	C₂×Q₈	M₁₆×C₁₀₃	1	1	yes	characteristic
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂×C₈×C₁₀₃	1	2	yes	—
202	C₂×D₄	M₁₆×C₁₀₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂²⋊C₄×C₁₀₃	1	2	yes	—
208	C₂²×C₄	C₂×C₈×C₁₀₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₀₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²⋊C₄×C₁₀₃	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₁₀₃	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₀₃	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	C₂×C₈×C₁₀₃	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	2	4	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂²⋊C₈×C₁₀₃	1	2	yes	—
238	C₂³	C₂²×C₈×C₁₀₃	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₂²⋊C₈×C₁₀₃	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₃	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₁₀₃	1	2	yes	—
254	C₂×C₄	C₂×M₁₆×C₁₀₃	1	1	yes	characteristic
255	C₂×C₄	#131×C₁₀₃	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	#120×C₁₀₃	1	1	yes	characteristic
260	C₈	#121×C₁₀₃	1	2	yes	—
261	C₈	#123×C₁₀₃	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₂²⋊C₈×C₁₀₃	1	1	yes	characteristic
271	C₂²	C₈×D₄×C₁₀₃	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	#386×C₁₀₃	1	1	yes	characteristic
276	C₄	#408×C₁₀₃	1	1	yes	characteristic
277	C₄	#435×C₁₀₃	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	C₁₀₃×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₃	#120
commutator	C₂³	C₂²×C₈×C₁₀₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₃
socle	C₂×C₁₀₃	#1869
prime radical	#525×C₁₀₃	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₃ ◁ C₂²×C₈×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₃×#3407	C₂	semidirect
3. C₁₀₃×#2115	C₂	semidirect
4. C₁₀₃×#2112	C₂	semidirect
5. C₁₀₃×#1378	C₂	semidirect
48. C₂×C₁₆×C₁₀₃	D₄	semidirect
136. #12981	C₁₀₃	direct (N × H)
142. #3407	C₂×C₁₀₃	semidirect
144. #2115	C₂×C₁₀₃	semidirect
145. #2112	C₂×C₁₀₃	semidirect
146. #1378	C₂×C₁₀₃	semidirect
147. C₁₀₃	#12981	direct (N × H)
190. C₂×C₁₆	D₄×C₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₃×#3407	2
C₁₀₃×#2929	2
C₁₀₃×#2115	2
C₁₀₃×#2112	2
C₁₀₃×#1378	2
#12981	103

Subgroups — C103×#12981 #39918256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#12981 #39918256