Subgroups — C₁₀₃×#13245 #39918520

Order 26368.

442 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₁₀₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#368×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₁₀₃	C₂	1	2	yes	maximal, complemented
4	C₁₀₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₃×#1978	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₀₃	C₂²	1	2	yes	—
7	C₂×D₁₆×C₁₀₃	—	2	1	no	—
8	C₂×D₁₆×C₁₀₃	—	2	1	no	—
9	C₂×D₁₆×C₁₀₃	—	2	2	no	—
10	C₂×D₁₆×C₁₀₃	C₄	1	4	yes	complemented
11	C₂×M₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₀₃	C₂²	1	2	yes	—
14	#418×C₁₀₃	—	2	2	no	—
15	C₂×#100×C₁₀₃	—	2	2	no	—
16	#405×C₁₀₃	C₂²	1	1	yes	characteristic
17	#368×C₁₀₃	—	2	2	no	—
18	#361×C₁₀₃	—	2	4	no	—
19	#346×C₁₀₃	—	2	1	no	—
20	C₄×C₁₆×C₁₀₃	—	2	1	no	—
21	C₂²×D₄×C₁₀₃	—	2	2	no	—
22	C₂×D₈×C₁₀₃	—	2	2	no	—
23	C₂×D₈×C₁₀₃	—	2	2	no	—
24	C₂×D₈×C₁₀₃	—	2	2	no	—
25	C₂×D₈×C₁₀₃	—	2	2	no	—
26	C₂×D₈×C₁₀₃	—	2	2	no	—
27	C₂×D₈×C₁₀₃	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₁₀₃	—	2	2	no	—
29	C₂×M₁₆×C₁₀₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
31	#117×C₁₀₃	D₄	1	1	yes	characteristic
32	D₁₆×C₁₀₃	—	2	4	no	—
33	D₁₆×C₁₀₃	—	2	4	no	—
34	M₃₂×C₁₀₃	—	2	1	no	—
35	M₃₂×C₁₀₃	D₄	1	2	yes	—
36	C₂×C₁₆×C₁₀₃	—	2	1	no	—
37	C₂×C₁₆×C₁₀₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₀₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₀₃	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₀₃	D₄	1	2	yes	—
41	#108×C₁₀₃	—	2	2	no	—
42	#108×C₁₀₃	—	2	2	no	—
43	#102×C₁₀₃	—	4	4	no	—
44	#100×C₁₀₃	—	4	4	no	—
45	#97×C₁₀₃	—	2	1	no	—
46	C₄×C₈×C₁₀₃	—	2	1	no	—
47	C₂⁴×C₁₀₃	—	4	2	no	—
48	C₂×D₄×C₁₀₃	—	2	4	no	—
49	C₂×D₄×C₁₀₃	—	4	2	no	—
50	C₂×D₄×C₁₀₃	—	4	2	no	—
51	C₂×D₄×C₁₀₃	—	4	2	no	—
52	C₂×D₄×C₁₀₃	—	4	2	no	—
53	C₂×D₄×C₁₀₃	—	4	2	no	—
54	C₂²×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
55	D₈×C₁₀₃	—	2	4	no	—
56	D₈×C₁₀₃	—	2	4	no	—
57	D₈×C₁₀₃	—	4	2	no	—
58	D₈×C₁₀₃	—	4	2	no	—
59	M₁₆×C₁₀₃	—	2	1	no	—
60	M₁₆×C₁₀₃	—	4	2	no	—
61	M₁₆×C₁₀₃	—	4	2	no	—
62	M₁₆×C₁₀₃	D₈	1	2	yes	—
63	C₂×C₈×C₁₀₃	—	2	1	no	—
64	C₂×C₈×C₁₀₃	—	4	2	no	—
65	C₂×C₈×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₀₃	#41	1	1	yes	characteristic
70	C₂×C₈×C₁₀₃	#41	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₃	D₈	1	2	yes	—
72	C₂²⋊C₄×C₁₀₃	—	2	1	no	—
73	C₄²×C₁₀₃	—	2	1	no	—
74	C₁₆×C₁₀₃	—	2	2	no	—
75	C₁₆×C₁₀₃	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₁₀₃	C₂²⋊C₄	1	2	yes	—
77	C₂³×C₁₀₃	—	4	4	no	—
78	C₂³×C₁₀₃	—	8	2	no	—
79	C₂³×C₁₀₃	—	8	2	no	—
80	C₂³×C₁₀₃	—	16	4	no	—
81	C₂³×C₁₀₃	#135	1	1	yes	characteristic
82	D₄×C₁₀₃	—	4	4	no	—
83	D₄×C₁₀₃	—	4	4	no	—
84	D₄×C₁₀₃	—	8	2	no	—
85	D₄×C₁₀₃	—	8	2	no	—
86	C₂×C₄×C₁₀₃	—	2	2	no	—
87	C₂×C₄×C₁₀₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₀₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₀₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₀₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₀₃	#135	1	1	yes	characteristic
93	C₈×C₁₀₃	—	2	2	no	—
94	C₈×C₁₀₃	—	8	2	no	—
95	C₈×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₁₀₃	C₄×D₄	1	1	yes	characteristic
97	C₈×C₁₀₃	#121	1	1	yes	characteristic
98	C₈×C₁₀₃	#124	1	1	yes	characteristic
99	C₂²×C₁₀₃	—	2	2	no	—
100	C₂²×C₁₀₃	—	8	4	no	—
101	C₂²×C₁₀₃	—	16	4	no	—
102	C₂²×C₁₀₃	—	16	4	no	—
103	C₂²×C₁₀₃	—	16	4	no	—
104	C₂²×C₁₀₃	C₄×D₈	1	1	yes	characteristic
105	C₂²×C₁₀₃	#466	1	1	yes	characteristic
106	C₂²×C₁₀₃	#495	1	1	yes	characteristic
107	C₄×C₁₀₃	—	4	2	no	—
108	C₄×C₁₀₃	#390	1	1	yes	characteristic
109	C₄×C₁₀₃	C₄×D₈	1	1	yes	characteristic
110	C₄×C₁₀₃	#466	1	1	yes	characteristic
111	C₄×C₁₀₃	#493	1	1	yes	characteristic
112	#13245	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₃	—	2	1	no	—
114	C₂×C₁₀₃	—	2	1	no	—
115	C₂×C₁₀₃	—	16	4	no	—
116	C₂×C₁₀₃	#1970	1	1	yes	characteristic
117	C₂×C₁₀₃	#2221	1	2	yes	—
118	C₂²×D₁₆	C₂×C₁₀₃	1	1	yes	characteristic
119	C₂×#368	C₂×C₁₀₃	1	1	yes	characteristic, complemented
120	C₂×#361	C₂×C₁₀₃	1	2	yes	complemented
121	#2108	C₂×C₁₀₃	1	1	yes	characteristic, complemented
122	#1978	C₂×C₁₀₃	1	2	yes	complemented
123	C₁₀₃	#13245	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×D₈	C₂²×C₁₀₃	1	2	yes	—
125	C₂×D₁₆	—	2	1	no	—
126	C₂×D₁₆	—	2	1	no	—
127	C₂×D₁₆	—	2	2	no	—
128	C₂×D₁₆	C₄×C₁₀₃	1	4	yes	complemented
129	C₂×M₃₂	C₂²×C₁₀₃	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
131	C₂×#108	C₂²×C₁₀₃	1	2	yes	—
132	#418	—	2	2	no	—
133	C₂×#100	—	2	2	no	—
134	#405	C₂²×C₁₀₃	1	1	yes	characteristic
135	#368	—	2	2	no	—
136	#361	—	2	4	no	—
137	#346	—	2	1	no	—
138	C₄×C₁₆	—	2	1	no	—
139	C₂²×D₄	—	2	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	C₂×C₄×C₁₀₃	1	4	yes	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	D₄×C₁₀₃	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
149	#117	D₄×C₁₀₃	1	1	yes	characteristic
150	D₁₆	—	2	4	no	—
151	D₁₆	—	2	4	no	—
152	M₃₂	—	2	1	no	—
153	M₃₂	D₄×C₁₀₃	1	2	yes	—
154	C₂×C₁₆	—	2	1	no	—
155	C₂×C₁₆	C₂×C₄×C₁₀₃	1	2	yes	—
156	C₂×C₁₆	D₄×C₁₀₃	1	1	yes	characteristic
157	C₂×C₁₆	D₄×C₁₀₃	1	1	yes	characteristic
158	C₂×C₁₆	D₄×C₁₀₃	1	2	yes	—
159	#108	—	2	2	no	—
160	#108	—	2	2	no	—
161	#102	—	4	4	no	—
162	#100	—	4	4	no	—
163	#97	—	2	1	no	—
164	C₄×C₈	—	2	1	no	—
165	C₂⁴	—	4	2	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂²×C₄	C₂×D₄×C₁₀₃	1	1	yes	characteristic
173	D₈	—	2	4	no	—
174	D₈	—	2	4	no	—
175	D₈	—	4	2	no	—
176	D₈	—	4	2	no	—
177	M₁₆	—	2	1	no	—
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	2	no	—
180	M₁₆	D₈×C₁₀₃	1	2	yes	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₂²×C₄×C₁₀₃	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
187	C₂×C₈	#41×C₁₀₃	1	1	yes	characteristic
188	C₂×C₈	#41×C₁₀₃	1	1	yes	characteristic
189	C₄⋊C₄	D₈×C₁₀₃	1	2	yes	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	C₂²⋊C₄×C₁₀₃	1	2	yes	—
194	C₁₆	C₂²⋊C₄×C₁₀₃	1	2	yes	—
195	C₂³	—	4	4	no	—
196	C₂³	—	8	2	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	16	4	no	—
199	C₂³	#135×C₁₀₃	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	8	2	no	—
203	D₄	—	8	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	C₄×D₄×C₁₀₃	1	1	yes	characteristic
206	C₂×C₄	#121×C₁₀₃	1	1	yes	characteristic
207	C₂×C₄	#127×C₁₀₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₁₀₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₈×C₁₀₃	1	1	yes	characteristic
210	C₂×C₄	#135×C₁₀₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	8	2	no	—
213	C₈	C₂×C₂²⋊C₄×C₁₀₃	1	1	yes	characteristic
214	C₈	C₄×D₄×C₁₀₃	1	1	yes	characteristic
215	C₈	#121×C₁₀₃	1	1	yes	characteristic
216	C₈	#124×C₁₀₃	1	1	yes	characteristic
217	C₂²	—	2	2	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	4	no	—
220	C₂²	—	16	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	C₄×D₈×C₁₀₃	1	1	yes	characteristic
223	C₂²	#466×C₁₀₃	1	1	yes	characteristic
224	C₂²	#495×C₁₀₃	1	1	yes	characteristic
225	C₄	—	4	2	no	—
226	C₄	#390×C₁₀₃	1	1	yes	characteristic
227	C₄	C₄×D₈×C₁₀₃	1	1	yes	characteristic
228	C₄	#466×C₁₀₃	1	1	yes	characteristic
229	C₄	#493×C₁₀₃	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	16	4	no	—
233	C₂	C₁₀₃×#1970	1	1	yes	characteristic
234	C₂	C₁₀₃×#2221	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#466
commutator	C₂×C₈	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₃
prime radical	C₂²×D₁₆×C₁₀₃	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂²×C₄×C₁₀₃ ◁ #405×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#368×C₁₀₃	C₂	semidirect
3. C₂×#361×C₁₀₃	C₂	semidirect
4. C₁₀₃×#2108	C₂	semidirect
5. C₁₀₃×#1978	C₂	semidirect
10. C₂×D₁₆×C₁₀₃	C₄	semidirect
112. #13245	C₁₀₃	direct (N × H)
119. C₂×#368	C₂×C₁₀₃	semidirect
120. C₂×#361	C₂×C₁₀₃	semidirect
121. #2108	C₂×C₁₀₃	semidirect
122. #1978	C₂×C₁₀₃	semidirect
123. C₁₀₃	#13245	direct (N × H)
128. C₂×D₁₆	C₄×C₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₁₀₃	2
C₂×#368×C₁₀₃	2
C₂×#361×C₁₀₃	2
C₁₀₃×#2108	2
C₁₀₃×#1978	2
#13245	103

Subgroups — C103×#13245 #39918520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#13245 #39918520