Subgroups — C₂×C₁₀₃×#2091 #39928997

Order 26368.

1102 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₁₀₃	C₂	1	3	yes	maximal, complemented
2	C₂×#407×C₁₀₃	C₂	1	3	yes	maximal
3	C₁₀₃×#2091	C₂	1	8	yes	maximal, direct product factor
4	C₂×#338×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₀₃	C₂²	1	3	yes	—
7	C₂²×M₁₆×C₁₀₃	C₂²	1	3	yes	—
8	#430×C₁₀₃	—	2	6	no	—
9	#430×C₁₀₃	C₂²	1	12	yes	complemented
10	C₂×#108×C₁₀₃	—	2	6	no	—
11	#407×C₁₀₃	—	2	6	no	—
12	#407×C₁₀₃	C₂²	1	12	yes	—
13	C₂×C₂²⋊C₈×C₁₀₃	—	2	6	no	—
14	#338×C₁₀₃	C₂²	1	4	yes	complemented
15	C₂⁵×C₁₀₃	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₃	—	2	3	no	—
17	C₂³×C₄×C₁₀₃	—	2	3	no	—
18	C₂³×C₄×C₁₀₃	—	2	3	no	—
19	C₂³×C₄×C₁₀₃	—	2	3	no	—
20	C₂³×C₄×C₁₀₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₀₃	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₀₃	—	2	6	no	—
25	C₂×M₁₆×C₁₀₃	—	2	6	no	—
26	C₂×M₁₆×C₁₀₃	—	2	3	no	—
27	C₂×M₁₆×C₁₀₃	—	2	6	no	—
28	C₂×M₁₆×C₁₀₃	—	2	6	no	—
29	C₂×M₁₆×C₁₀₃	D₄	1	6	yes	—
30	C₂×M₁₆×C₁₀₃	C₂³	1	6	yes	—
31	C₂×M₁₆×C₁₀₃	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₀₃	—	2	3	no	—
33	C₂²×C₈×C₁₀₃	D₄	1	6	yes	—
34	#108×C₁₀₃	—	2	24	no	—
35	C₂²⋊C₈×C₁₀₃	—	2	24	no	—
36	C₂⁴×C₁₀₃	—	2	3	no	—
37	C₂⁴×C₁₀₃	—	2	3	no	—
38	C₂⁴×C₁₀₃	—	4	4	no	—
39	C₂⁴×C₁₀₃	C₂×Q₈	1	2	yes	—
40	C₂⁴×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₃	—	2	3	no	—
42	C₂²×C₄×C₁₀₃	—	2	3	no	—
43	C₂²×C₄×C₁₀₃	—	2	6	no	—
44	C₂²×C₄×C₁₀₃	—	2	6	no	—
45	C₂²×C₄×C₁₀₃	—	2	6	no	—
46	C₂²×C₄×C₁₀₃	—	2	6	no	—
47	C₂²×C₄×C₁₀₃	—	4	3	no	—
48	C₂²×C₄×C₁₀₃	—	4	3	no	—
49	C₂²×C₄×C₁₀₃	—	4	6	no	—
50	C₂²×C₄×C₁₀₃	—	4	3	no	—
51	C₂²×C₄×C₁₀₃	—	4	1	no	—
52	C₂²×C₄×C₁₀₃	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₁₀₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₃	#41	1	3	yes	—
57	C₂²×C₄×C₁₀₃	#41	1	3	yes	—
58	C₂²×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₁₀₃	—	2	12	no	—
60	M₁₆×C₁₀₃	—	2	12	no	—
61	M₁₆×C₁₀₃	—	2	12	no	—
62	M₁₆×C₁₀₃	—	4	6	no	—
63	C₂×C₈×C₁₀₃	—	2	6	no	—
64	C₂×C₈×C₁₀₃	—	2	6	no	—
65	C₂×C₈×C₁₀₃	—	2	6	no	—
66	C₂×C₈×C₁₀₃	—	4	3	no	—
67	C₂×C₈×C₁₀₃	—	4	3	no	—
68	C₂×C₈×C₁₀₃	C₂×D₄	1	12	yes	—
69	C₂³×C₁₀₃	—	2	12	no	—
70	C₂³×C₁₀₃	—	2	6	no	—
71	C₂³×C₁₀₃	—	4	2	no	—
72	C₂³×C₁₀₃	—	4	4	no	—
73	C₂³×C₁₀₃	—	4	6	no	—
74	C₂³×C₁₀₃	—	4	12	no	—
75	C₂³×C₁₀₃	—	4	3	no	—
76	C₂³×C₁₀₃	—	4	1	no	—
77	C₂³×C₁₀₃	C₂²⋊Q₈	1	3	yes	—
78	C₂³×C₁₀₃	C₂²⋊Q₈	1	3	yes	—
79	C₂³×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₀₃	—	2	6	no	—
81	C₂×C₄×C₁₀₃	—	4	3	no	—
82	C₂×C₄×C₁₀₃	—	4	1	no	—
83	C₂×C₄×C₁₀₃	—	4	6	no	—
84	C₂×C₄×C₁₀₃	—	4	2	no	—
85	C₂×C₄×C₁₀₃	—	4	6	no	—
86	C₂×C₄×C₁₀₃	—	4	2	no	—
87	C₂×C₄×C₁₀₃	C₂²⋊Q₈	1	3	yes	—
88	C₂×C₄×C₁₀₃	C₂²⋊Q₈	1	3	yes	—
89	C₂×C₄×C₁₀₃	#123	1	3	yes	—
90	C₂×C₄×C₁₀₃	#123	1	3	yes	—
91	C₂×C₄×C₁₀₃	#123	1	6	yes	—
92	C₂×C₄×C₁₀₃	#127	1	3	yes	—
93	C₂×C₄×C₁₀₃	#127	1	1	yes	characteristic
94	C₂×C₄×C₁₀₃	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₁₀₃	C₂×#41	1	3	yes	—
96	C₈×C₁₀₃	—	2	12	no	—
97	C₈×C₁₀₃	—	4	6	no	—
98	C₂²×C₁₀₃	—	2	6	no	—
99	C₂²×C₁₀₃	—	2	12	no	—
100	C₂²×C₁₀₃	—	4	3	no	—
101	C₂²×C₁₀₃	—	4	1	no	—
102	C₂²×C₁₀₃	—	4	6	no	—
103	C₂²×C₁₀₃	—	4	12	no	—
104	C₂²×C₁₀₃	—	4	4	no	—
105	C₂²×C₁₀₃	—	4	2	no	—
106	C₂²×C₁₀₃	#399	1	3	yes	—
107	C₂²×C₁₀₃	#399	1	1	yes	characteristic
108	C₂²×C₁₀₃	C₂×C₂²⋊Q₈	1	3	yes	—
109	C₄×C₁₀₃	—	4	2	no	—
110	C₄×C₁₀₃	#399	1	3	yes	—
111	C₄×C₁₀₃	#399	1	1	yes	characteristic
112	C₄×C₁₀₃	C₂×#123	1	3	yes	—
113	C₄×C₁₀₃	C₂×#127	1	1	yes	characteristic
114	C₂×#2091	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₁₀₃	—	2	3	no	—
116	C₂×C₁₀₃	—	2	3	no	—
117	C₂×C₁₀₃	—	4	4	no	—
118	C₂×C₁₀₃	#2091	1	2	yes	direct product factor
119	C₂×C₁₀₃	C₂×#399	1	1	yes	characteristic
120	C₂×#430	C₂×C₁₀₃	1	3	yes	complemented
121	C₂×#407	C₂×C₁₀₃	1	3	yes	—
122	#2091	C₂×C₁₀₃	1	8	yes	direct product factor
123	C₂×#338	C₂×C₁₀₃	1	1	yes	characteristic, complemented
124	C₁₀₃	C₂×#2091	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴×C₄	C₂²×C₁₀₃	1	1	yes	characteristic
126	C₂²×M₁₆	C₂²×C₁₀₃	1	3	yes	—
127	C₂²×M₁₆	C₂²×C₁₀₃	1	3	yes	—
128	#430	—	2	6	no	—
129	#430	C₂²×C₁₀₃	1	12	yes	complemented
130	C₂×#108	—	2	6	no	—
131	#407	—	2	6	no	—
132	#407	C₂²×C₁₀₃	1	12	yes	—
133	C₂×C₂²⋊C₈	—	2	6	no	—
134	#338	C₂²×C₁₀₃	1	4	yes	complemented
135	C₂⁵	Q₈×C₁₀₃	1	1	yes	characteristic
136	C₂³×C₄	—	2	3	no	—
137	C₂³×C₄	—	2	3	no	—
138	C₂³×C₄	—	2	3	no	—
139	C₂³×C₄	—	2	3	no	—
140	C₂³×C₄	Q₈×C₁₀₃	1	1	yes	characteristic
141	C₂³×C₄	Q₈×C₁₀₃	1	2	yes	—
142	C₂³×C₄	C₂³×C₁₀₃	1	2	yes	—
143	C₂³×C₄	C₂³×C₁₀₃	1	1	yes	characteristic
144	C₂×M₁₆	—	2	6	no	—
145	C₂×M₁₆	—	2	6	no	—
146	C₂×M₁₆	—	2	3	no	—
147	C₂×M₁₆	—	2	6	no	—
148	C₂×M₁₆	—	2	6	no	—
149	C₂×M₁₆	D₄×C₁₀₃	1	6	yes	—
150	C₂×M₁₆	C₂³×C₁₀₃	1	6	yes	—
151	C₂×M₁₆	C₂³×C₁₀₃	1	6	yes	—
152	C₂²×C₈	—	2	3	no	—
153	C₂²×C₈	D₄×C₁₀₃	1	6	yes	—
154	#108	—	2	24	no	—
155	C₂²⋊C₈	—	2	24	no	—
156	C₂⁴	—	2	3	no	—
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	4	4	no	—
159	C₂⁴	C₂×Q₈×C₁₀₃	1	2	yes	—
160	C₂⁴	C₂×Q₈×C₁₀₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	4	3	no	—
168	C₂²×C₄	—	4	3	no	—
169	C₂²×C₄	—	4	6	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	3	no	—
172	C₂²×C₄	C₂×D₄×C₁₀₃	1	3	yes	—
173	C₂²×C₄	C₂×D₄×C₁₀₃	1	3	yes	—
174	C₂²×C₄	C₂×Q₈×C₁₀₃	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₁₀₃	1	1	yes	characteristic
176	C₂²×C₄	#41×C₁₀₃	1	3	yes	—
177	C₂²×C₄	#41×C₁₀₃	1	3	yes	—
178	C₂²×C₄	C₂⁴×C₁₀₃	1	1	yes	characteristic
179	M₁₆	—	2	12	no	—
180	M₁₆	—	2	12	no	—
181	M₁₆	—	2	12	no	—
182	M₁₆	—	4	6	no	—
183	C₂×C₈	—	2	6	no	—
184	C₂×C₈	—	2	6	no	—
185	C₂×C₈	—	2	6	no	—
186	C₂×C₈	—	4	3	no	—
187	C₂×C₈	—	4	3	no	—
188	C₂×C₈	C₂×D₄×C₁₀₃	1	12	yes	—
189	C₂³	—	2	12	no	—
190	C₂³	—	2	6	no	—
191	C₂³	—	4	2	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	4	6	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	4	3	no	—
196	C₂³	—	4	1	no	—
197	C₂³	C₂²⋊Q₈×C₁₀₃	1	3	yes	—
198	C₂³	C₂²⋊Q₈×C₁₀₃	1	3	yes	—
199	C₂³	C₂²×Q₈×C₁₀₃	1	1	yes	characteristic
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	6	no	—
205	C₂×C₄	—	4	6	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂²⋊Q₈×C₁₀₃	1	3	yes	—
208	C₂×C₄	C₂²⋊Q₈×C₁₀₃	1	3	yes	—
209	C₂×C₄	#123×C₁₀₃	1	3	yes	—
210	C₂×C₄	#123×C₁₀₃	1	3	yes	—
211	C₂×C₄	#123×C₁₀₃	1	6	yes	—
212	C₂×C₄	#127×C₁₀₃	1	1	yes	characteristic
213	C₂×C₄	#127×C₁₀₃	1	3	yes	—
214	C₂×C₄	C₂²×D₄×C₁₀₃	1	3	yes	—
215	C₂×C₄	C₂×#41×C₁₀₃	1	3	yes	—
216	C₈	—	2	12	no	—
217	C₈	—	4	6	no	—
218	C₂²	—	2	6	no	—
219	C₂²	—	2	12	no	—
220	C₂²	—	4	3	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	6	no	—
223	C₂²	—	4	12	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	#399×C₁₀₃	1	3	yes	—
227	C₂²	#399×C₁₀₃	1	1	yes	characteristic
228	C₂²	C₂×C₂²⋊Q₈×C₁₀₃	1	3	yes	—
229	C₄	—	4	2	no	—
230	C₄	#399×C₁₀₃	1	1	yes	characteristic
231	C₄	#399×C₁₀₃	1	3	yes	—
232	C₄	C₂×#123×C₁₀₃	1	3	yes	—
233	C₄	C₂×#127×C₁₀₃	1	1	yes	characteristic
234	C₂	—	2	3	no	—
235	C₂	—	2	3	no	—
236	C₂	—	4	4	no	—
237	C₂	C₁₀₃×#2091	1	2	yes	direct product factor
238	C₂	C₂×#399×C₁₀₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₃	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₃
socle	C₂²×C₁₀₃	#399
prime radical	C₂⁵×C₁₀₃	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₃ ◁ C₂³×C₄×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#430×C₁₀₃	C₂	semidirect
3. C₁₀₃×#2091	C₂	direct (N × H)
4. C₂×#338×C₁₀₃	C₂	semidirect
9. #430×C₁₀₃	C₂²	semidirect
14. #338×C₁₀₃	C₂²	semidirect
114. C₂×#2091	C₁₀₃	direct (N × H)
118. C₂×C₁₀₃	#2091	direct (N × H)
120. C₂×#430	C₂×C₁₀₃	semidirect
122. #2091	C₂×C₁₀₃	direct (N × H)
123. C₂×#338	C₂×C₁₀₃	semidirect
124. C₁₀₃	C₂×#2091	direct (N × H)
129. #430	C₂²×C₁₀₃	semidirect
134. #338	C₂²×C₁₀₃	semidirect
237. C₂	C₁₀₃×#2091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#430×C₁₀₃	2
C₂×#407×C₁₀₃	2
C₁₀₃×#2091	2
C₂×#338×C₁₀₃	2
C₂×#2091	103

Subgroups — C2×C103×#2091 #39928997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₃×#2091 #39928997