Subgroups — C₁₀₃×#33460 #39938735

Order 26368.

674 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₀₃	C₂	1	1	yes	maximal, characteristic
2	C₁₀₃×#2951	C₂	1	2	yes	maximal, complemented
3	C₁₀₃×#2177	C₂	1	2	yes	maximal, complemented
4	C₁₀₃×#2176	C₂	1	2	yes	maximal, complemented
5	C₄×SD₃₂×C₁₀₃	C₂	1	4	yes	maximal, complemented
6	C₂×#367×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₃×#2139	C₂	1	2	yes	maximal, complemented
8	C₁₀₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₀₃	C₂²	1	2	yes	—
10	#520×C₁₀₃	—	2	2	no	—
11	#508×C₁₀₃	C₄	1	8	yes	complemented
12	C₂×Q₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₁₀₃	C₂²	1	2	yes	—
14	C₂×D₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
17	#440×C₁₀₃	—	2	4	no	—
18	C₄×Q₁₆×C₁₀₃	C₂²	1	4	yes	—
19	C₄×D₈×C₁₀₃	C₂²	1	4	yes	—
20	C₂×#107×C₁₀₃	C₂²	1	2	yes	—
21	#417×C₁₀₃	—	2	2	no	—
22	#405×C₁₀₃	C₂²	1	1	yes	characteristic
23	#367×C₁₀₃	C₂²	1	2	yes	—
24	#367×C₁₀₃	C₂²	1	2	yes	—
25	#358×C₁₀₃	C₂²	1	4	yes	—
26	#357×C₁₀₃	C₂²	1	4	yes	—
27	#346×C₁₀₃	C₂²	1	2	yes	—
28	C₄×C₁₆×C₁₀₃	C₂²	1	2	yes	—
29	C₂×#41×C₁₀₃	—	2	2	no	—
30	#135×C₁₀₃	—	2	4	no	—
31	#135×C₁₀₃	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₁₀₃	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₁₀₃	—	2	2	no	—
34	C₂×D₈×C₁₀₃	C₂³	1	2	yes	—
35	C₂×M₁₆×C₁₀₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₀₃	—	2	4	no	—
38	C₄×D₄×C₁₀₃	—	2	4	no	—
39	C₄×D₄×C₁₀₃	—	4	4	no	—
40	#117×C₁₀₃	—	4	2	no	—
41	#117×C₁₀₃	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₃	—	4	2	no	—
44	Q₃₂×C₁₀₃	C₂×C₄	1	4	yes	complemented
45	SD₃₂×C₁₀₃	C₂×C₄	1	8	yes	complemented
46	D₁₆×C₁₀₃	C₂×C₄	1	4	yes	complemented
47	M₃₂×C₁₀₃	—	2	2	no	—
48	C₂×C₁₆×C₁₀₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₁₀₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₁₀₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₁₀₃	C₂³	1	1	yes	characteristic
52	#107×C₁₀₃	—	2	2	no	—
53	#107×C₁₀₃	C₂³	1	4	yes	—
54	#103×C₁₀₃	—	2	4	no	—
55	#102×C₁₀₃	—	2	4	no	—
56	#97×C₁₀₃	D₄	1	2	yes	—
57	C₄×C₈×C₁₀₃	C₂³	1	2	yes	—
58	#41×C₁₀₃	—	2	8	no	—
59	#41×C₁₀₃	—	4	4	no	—
60	C₂×Q₈×C₁₀₃	—	2	2	no	—
61	C₂×D₄×C₁₀₃	—	2	2	no	—
62	C₂×D₄×C₁₀₃	—	4	2	no	—
63	C₂²×C₄×C₁₀₃	—	4	2	no	—
64	C₂²×C₄×C₁₀₃	—	4	2	no	—
65	C₂²×C₄×C₁₀₃	—	4	4	no	—
66	C₂²×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₁₀₃	—	2	2	no	—
68	Q₁₆×C₁₀₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₁₀₃	—	2	8	no	—
70	D₈×C₁₀₃	—	2	2	no	—
71	D₈×C₁₀₃	C₂²×C₄	1	4	yes	—
72	M₁₆×C₁₀₃	—	2	2	no	—
73	C₂×C₈×C₁₀₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₀₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₀₃	#41	1	2	yes	—
77	C₂×C₈×C₁₀₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₃	—	2	4	no	—
79	C₄⋊C₄×C₁₀₃	—	4	2	no	—
80	C₄⋊C₄×C₁₀₃	—	4	4	no	—
81	C₄⋊C₄×C₁₀₃	D₈	1	2	yes	—
82	C₂²⋊C₄×C₁₀₃	—	4	4	no	—
83	C₂²⋊C₄×C₁₀₃	D₈	1	2	yes	—
84	C₄²×C₁₀₃	—	4	4	no	—
85	C₄²×C₁₀₃	C₂×D₄	1	2	yes	—
86	C₁₆×C₁₀₃	—	2	2	no	—
87	C₁₆×C₁₀₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₁₀₃	C₂²×C₄	1	2	yes	—
89	C₂³×C₁₀₃	—	4	2	no	—
90	C₂³×C₁₀₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₁₀₃	—	2	4	no	—
92	Q₈×C₁₀₃	—	4	2	no	—
93	D₄×C₁₀₃	—	2	4	no	—
94	D₄×C₁₀₃	—	4	8	no	—
95	D₄×C₁₀₃	—	4	2	no	—
96	C₂×C₄×C₁₀₃	—	4	2	no	—
97	C₂×C₄×C₁₀₃	—	4	8	no	—
98	C₂×C₄×C₁₀₃	—	4	4	no	—
99	C₂×C₄×C₁₀₃	—	16	4	no	—
100	C₂×C₄×C₁₀₃	—	16	2	no	—
101	C₂×C₄×C₁₀₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₃	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₁₀₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₀₃	#135	1	2	yes	—
105	C₂×C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₁₀₃	—	2	2	no	—
107	C₈×C₁₀₃	C₄×D₄	1	2	yes	—
108	C₈×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₁₀₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₁₀₃	—	2	2	no	—
111	C₂²×C₁₀₃	—	4	4	no	—
112	C₂²×C₁₀₃	—	8	2	no	—
113	C₂²×C₁₀₃	—	8	2	no	—
114	C₂²×C₁₀₃	C₄×D₈	1	2	yes	—
115	C₂²×C₁₀₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₁₀₃	—	2	4	no	—
117	C₄×C₁₀₃	—	4	4	no	—
118	C₄×C₁₀₃	—	8	4	no	—
119	C₄×C₁₀₃	C₄×D₈	1	2	yes	—
120	C₄×C₁₀₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₁₀₃	C₂×#135	1	1	yes	characteristic
122	#33460	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₀₃	—	2	2	no	—
124	C₂×C₁₀₃	—	8	4	no	—
125	C₂×C₁₀₃	C₂×C₄×D₈	1	1	yes	characteristic
126	C₂×C₁₀₃	#3416	1	2	yes	—
127	C₂×#508	C₂×C₁₀₃	1	1	yes	characteristic
128	#2951	C₂×C₁₀₃	1	2	yes	complemented
129	#2177	C₂×C₁₀₃	1	2	yes	complemented
130	#2176	C₂×C₁₀₃	1	2	yes	complemented
131	C₄×SD₃₂	C₂×C₁₀₃	1	4	yes	complemented
132	C₂×#367	C₂×C₁₀₃	1	1	yes	characteristic, complemented
133	#2139	C₂×C₁₀₃	1	2	yes	complemented
134	#2108	C₂×C₁₀₃	1	1	yes	characteristic, complemented
135	C₁₀₃	#33460	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#135	C₂²×C₁₀₃	1	2	yes	—
137	#520	—	2	2	no	—
138	#508	C₄×C₁₀₃	1	8	yes	complemented
139	C₂×Q₃₂	C₂²×C₁₀₃	1	1	yes	characteristic
140	C₂×SD₃₂	C₂²×C₁₀₃	1	2	yes	—
141	C₂×D₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
142	C₂×M₃₂	C₂²×C₁₀₃	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
144	#440	—	2	4	no	—
145	C₄×Q₁₆	C₂²×C₁₀₃	1	4	yes	—
146	C₄×D₈	C₂²×C₁₀₃	1	4	yes	—
147	C₂×#107	C₂²×C₁₀₃	1	2	yes	—
148	#417	—	2	2	no	—
149	#405	C₂²×C₁₀₃	1	1	yes	characteristic
150	#367	C₂²×C₁₀₃	1	2	yes	—
151	#367	C₂²×C₁₀₃	1	2	yes	—
152	#358	C₂²×C₁₀₃	1	4	yes	—
153	#357	C₂²×C₁₀₃	1	4	yes	—
154	#346	C₂²×C₁₀₃	1	2	yes	—
155	C₄×C₁₆	C₂²×C₁₀₃	1	2	yes	—
156	C₂×#41	—	2	2	no	—
157	#135	—	2	4	no	—
158	#135	C₂×C₄×C₁₀₃	1	8	yes	complemented
159	C₂×Q₁₆	C₂³×C₁₀₃	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×D₈	C₂³×C₁₀₃	1	2	yes	—
162	C₂×M₁₆	D₄×C₁₀₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
164	C₄×Q₈	—	2	4	no	—
165	C₄×D₄	—	2	4	no	—
166	C₄×D₄	—	4	4	no	—
167	#117	—	4	2	no	—
168	#117	D₄×C₁₀₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	4	2	no	—
171	Q₃₂	C₂×C₄×C₁₀₃	1	4	yes	complemented
172	SD₃₂	C₂×C₄×C₁₀₃	1	8	yes	complemented
173	D₁₆	C₂×C₄×C₁₀₃	1	4	yes	complemented
174	M₃₂	—	2	2	no	—
175	C₂×C₁₆	C₂×C₄×C₁₀₃	1	4	yes	complemented
176	C₂×C₁₆	C₂³×C₁₀₃	1	2	yes	—
177	C₂×C₁₆	C₂³×C₁₀₃	1	1	yes	characteristic
178	C₂×C₁₆	C₂³×C₁₀₃	1	1	yes	characteristic
179	#107	—	2	2	no	—
180	#107	C₂³×C₁₀₃	1	4	yes	—
181	#103	—	2	4	no	—
182	#102	—	2	4	no	—
183	#97	D₄×C₁₀₃	1	2	yes	—
184	C₄×C₈	C₂³×C₁₀₃	1	2	yes	—
185	#41	—	2	8	no	—
186	#41	—	4	4	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂×D₄×C₁₀₃	1	1	yes	characteristic
194	Q₁₆	—	2	2	no	—
195	Q₁₆	C₂²×C₄×C₁₀₃	1	4	yes	—
196	SD₁₆	—	2	8	no	—
197	D₈	—	2	2	no	—
198	D₈	C₂²×C₄×C₁₀₃	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	C₂×C₈	C₂²×C₄×C₁₀₃	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₁₀₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
203	C₂×C₈	#41×C₁₀₃	1	2	yes	—
204	C₂×C₈	C₂⁴×C₁₀₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	D₈×C₁₀₃	1	2	yes	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	D₈×C₁₀₃	1	2	yes	—
211	C₄²	—	4	4	no	—
212	C₄²	C₂×D₄×C₁₀₃	1	2	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂²×C₄×C₁₀₃	1	2	yes	—
215	C₁₆	C₂²×C₄×C₁₀₃	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	C₂×D₈×C₁₀₃	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	16	4	no	—
227	C₂×C₄	—	16	2	no	—
228	C₂×C₄	C₄×D₄×C₁₀₃	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₁₀₃	1	4	yes	—
230	C₂×C₄	C₂×D₈×C₁₀₃	1	1	yes	characteristic
231	C₂×C₄	#135×C₁₀₃	1	2	yes	—
232	C₂×C₄	C₂²×D₄×C₁₀₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	C₄×D₄×C₁₀₃	1	2	yes	—
235	C₈	C₂³×C₄×C₁₀₃	1	1	yes	characteristic
236	C₈	C₂×#41×C₁₀₃	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	4	4	no	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	C₄×D₈×C₁₀₃	1	2	yes	—
242	C₂²	C₂²×D₈×C₁₀₃	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	4	no	—
246	C₄	C₄×D₈×C₁₀₃	1	2	yes	—
247	C₄	C₂×C₄×D₄×C₁₀₃	1	1	yes	characteristic
248	C₄	C₂×#135×C₁₀₃	1	1	yes	characteristic
249	C₂	—	2	2	no	—
250	C₂	—	8	4	no	—
251	C₂	C₂×C₄×D₈×C₁₀₃	1	1	yes	characteristic
252	C₂	C₁₀₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₀₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₃
prime radical	C₂×#508×C₁₀₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ #117×C₁₀₃ ◁ #405×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₃×#2951	C₂	semidirect
3. C₁₀₃×#2177	C₂	semidirect
4. C₁₀₃×#2176	C₂	semidirect
5. C₄×SD₃₂×C₁₀₃	C₂	semidirect
6. C₂×#367×C₁₀₃	C₂	semidirect
7. C₁₀₃×#2139	C₂	semidirect
8. C₁₀₃×#2108	C₂	semidirect
11. #508×C₁₀₃	C₄	semidirect
31. #135×C₁₀₃	C₂×C₄	semidirect
44. Q₃₂×C₁₀₃	C₂×C₄	semidirect
45. SD₃₂×C₁₀₃	C₂×C₄	semidirect
46. D₁₆×C₁₀₃	C₂×C₄	semidirect
48. C₂×C₁₆×C₁₀₃	C₂×C₄	semidirect
122. #33460	C₁₀₃	direct (N × H)
128. #2951	C₂×C₁₀₃	semidirect
129. #2177	C₂×C₁₀₃	semidirect
130. #2176	C₂×C₁₀₃	semidirect
131. C₄×SD₃₂	C₂×C₁₀₃	semidirect
132. C₂×#367	C₂×C₁₀₃	semidirect
133. #2139	C₂×C₁₀₃	semidirect
134. #2108	C₂×C₁₀₃	semidirect
135. C₁₀₃	#33460	direct (N × H)
138. #508	C₄×C₁₀₃	semidirect
158. #135	C₂×C₄×C₁₀₃	semidirect
171. Q₃₂	C₂×C₄×C₁₀₃	semidirect
172. SD₃₂	C₂×C₄×C₁₀₃	semidirect
173. D₁₆	C₂×C₄×C₁₀₃	semidirect
175. C₂×C₁₆	C₂×C₄×C₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₀₃	2
C₁₀₃×#2951	2
C₁₀₃×#2177	2
C₁₀₃×#2176	2
C₄×SD₃₂×C₁₀₃	2
C₂×#367×C₁₀₃	2
C₁₀₃×#2139	2
C₁₀₃×#2108	2
#33460	103

Subgroups — C103×#33460 #39938735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#33460 #39938735