Subgroups — C₁₀₃×#9064 #39914339

Order 26368.

738 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₀₃	C₂	1	1	yes	maximal, characteristic
2	C₁₀₃×#2366	C₂	1	1	yes	maximal, characteristic
3	C₁₀₃×#1987	C₂	1	2	yes	maximal, complemented
4	C₂×#331×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₁₀₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂²×SD₁₆×C₁₀₃	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
8	C₂×#127×C₁₀₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₀₃	—	2	2	no	—
10	#492×C₁₀₃	—	2	2	no	—
11	#492×C₁₀₃	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₁₀₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
14	#390×C₁₀₃	—	2	2	no	—
15	#386×C₁₀₃	—	2	2	no	—
16	C₄×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
17	#340×C₁₀₃	—	2	2	no	—
18	#340×C₁₀₃	—	2	2	no	—
19	#331×C₁₀₃	—	2	2	no	—
20	C₂²×Q₈×C₁₀₃	—	2	1	no	—
21	C₂²×D₄×C₁₀₃	—	2	2	no	—
22	C₂²×D₄×C₁₀₃	D₄	1	2	yes	—
23	C₂³×C₄×C₁₀₃	—	2	2	no	—
24	C₂×SD₁₆×C₁₀₃	—	2	8	no	—
25	C₂×SD₁₆×C₁₀₃	—	4	4	no	—
26	C₂×SD₁₆×C₁₀₃	—	4	4	no	—
27	C₂²×C₈×C₁₀₃	—	2	1	no	—
28	C₂²×C₈×C₁₀₃	—	2	2	no	—
29	C₄⋊Q₈×C₁₀₃	—	2	1	no	—
30	C₄⋊Q₈×C₁₀₃	—	2	2	no	—
31	C₄⋊Q₈×C₁₀₃	C₂×C₄	1	2	yes	complemented
32	#127×C₁₀₃	—	2	2	no	—
33	#127×C₁₀₃	C₂×C₄	1	2	yes	—
34	#127×C₁₀₃	D₄	1	2	yes	—
35	C₄×D₄×C₁₀₃	—	4	4	no	—
36	C₄×D₄×C₁₀₃	—	4	4	no	—
37	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₀₃	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₁₀₃	—	2	2	no	—
42	C₂×C₄²×C₁₀₃	—	2	2	no	—
43	C₂×C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₀₃	—	2	2	no	—
45	C₄⋊C₈×C₁₀₃	—	2	2	no	—
46	C₄×C₈×C₁₀₃	—	2	1	no	—
47	C₄×C₈×C₁₀₃	C₂×C₄	1	2	yes	complemented
48	#95×C₁₀₃	—	2	2	no	—
49	C₂⁴×C₁₀₃	—	2	2	no	—
50	C₂×Q₈×C₁₀₃	—	2	2	no	—
51	C₂×Q₈×C₁₀₃	—	4	2	no	—
52	C₂×Q₈×C₁₀₃	—	4	2	no	—
53	C₂×Q₈×C₁₀₃	—	4	1	no	—
54	C₂×D₄×C₁₀₃	—	2	4	no	—
55	C₂×D₄×C₁₀₃	—	2	4	no	—
56	C₂×D₄×C₁₀₃	—	2	2	no	—
57	C₂×D₄×C₁₀₃	—	4	2	no	—
58	C₂×D₄×C₁₀₃	—	4	2	no	—
59	C₂×D₄×C₁₀₃	—	4	2	no	—
60	C₂×D₄×C₁₀₃	—	4	2	no	—
61	C₂×D₄×C₁₀₃	—	4	2	no	—
62	C₂×D₄×C₁₀₃	—	4	2	no	—
63	C₂×D₄×C₁₀₃	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₀₃	—	2	2	no	—
65	C₂²×C₄×C₁₀₃	—	2	2	no	—
66	C₂²×C₄×C₁₀₃	—	4	8	no	—
67	C₂²×C₄×C₁₀₃	—	4	1	no	—
68	C₂²×C₄×C₁₀₃	—	8	2	no	—
69	C₂²×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
71	SD₁₆×C₁₀₃	—	4	16	no	—
72	C₂×C₈×C₁₀₃	—	2	2	no	—
73	C₂×C₈×C₁₀₃	—	2	2	no	—
74	C₂×C₈×C₁₀₃	—	2	2	no	—
75	C₂×C₈×C₁₀₃	—	4	2	no	—
76	C₂×C₈×C₁₀₃	—	4	2	no	—
77	C₂×C₈×C₁₀₃	—	4	2	no	—
78	C₄⋊C₄×C₁₀₃	—	2	2	no	—
79	C₄⋊C₄×C₁₀₃	—	2	4	no	—
80	C₄⋊C₄×C₁₀₃	—	2	2	no	—
81	C₄⋊C₄×C₁₀₃	—	2	2	no	—
82	C₄⋊C₄×C₁₀₃	—	4	2	no	—
83	C₄⋊C₄×C₁₀₃	SD₁₆	1	4	yes	complemented
84	C₂²⋊C₄×C₁₀₃	—	4	8	no	—
85	C₂²⋊C₄×C₁₀₃	—	8	2	no	—
86	C₄²×C₁₀₃	—	4	2	no	—
87	C₄²×C₁₀₃	—	4	2	no	—
88	C₄²×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₀₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₀₃	#41	1	2	yes	—
91	C₂³×C₁₀₃	—	2	4	no	—
92	C₂³×C₁₀₃	—	4	4	no	—
93	C₂³×C₁₀₃	—	4	4	no	—
94	C₂³×C₁₀₃	—	4	2	no	—
95	C₂³×C₁₀₃	—	4	2	no	—
96	C₂³×C₁₀₃	#120	1	1	yes	characteristic
97	Q₈×C₁₀₃	—	4	4	no	—
98	Q₈×C₁₀₃	—	8	2	no	—
99	D₄×C₁₀₃	—	4	4	no	—
100	D₄×C₁₀₃	—	4	4	no	—
101	D₄×C₁₀₃	—	4	4	no	—
102	D₄×C₁₀₃	—	4	4	no	—
103	D₄×C₁₀₃	—	4	4	no	—
104	D₄×C₁₀₃	—	8	2	no	—
105	C₂×C₄×C₁₀₃	—	2	1	no	—
106	C₂×C₄×C₁₀₃	—	2	1	no	—
107	C₂×C₄×C₁₀₃	—	2	4	no	—
108	C₂×C₄×C₁₀₃	—	4	2	no	—
109	C₂×C₄×C₁₀₃	—	4	2	no	—
110	C₂×C₄×C₁₀₃	—	4	2	no	—
111	C₂×C₄×C₁₀₃	—	4	2	no	—
112	C₂×C₄×C₁₀₃	—	4	2	no	—
113	C₂×C₄×C₁₀₃	—	4	2	no	—
114	C₂×C₄×C₁₀₃	—	8	8	no	—
115	C₂×C₄×C₁₀₃	—	8	1	no	—
116	C₂×C₄×C₁₀₃	—	8	1	no	—
117	C₂×C₄×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₃	#121	1	2	yes	—
120	C₂×C₄×C₁₀₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₀₃	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₁₀₃	#135	1	2	yes	—
123	C₂×C₄×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₀₃	#137	1	2	yes	—
125	C₈×C₁₀₃	—	2	4	no	—
126	C₈×C₁₀₃	—	8	2	no	—
127	C₂²×C₁₀₃	—	4	8	no	—
128	C₂²×C₁₀₃	—	4	4	no	—
129	C₂²×C₁₀₃	—	8	2	no	—
130	C₂²×C₁₀₃	—	8	2	no	—
131	C₂²×C₁₀₃	—	8	4	no	—
132	C₂²×C₁₀₃	#386	1	1	yes	characteristic
133	C₂²×C₁₀₃	#450	1	2	yes	—
134	C₂²×C₁₀₃	#452	1	2	yes	—
135	C₂²×C₁₀₃	#453	1	1	yes	characteristic
136	C₂²×C₁₀₃	#453	1	1	yes	characteristic
137	C₄×C₁₀₃	—	2	1	no	—
138	C₄×C₁₀₃	—	2	1	no	—
139	C₄×C₁₀₃	—	4	4	no	—
140	C₄×C₁₀₃	—	8	2	no	—
141	C₄×C₁₀₃	—	8	2	no	—
142	C₄×C₁₀₃	C₄×SD₁₆	1	2	yes	—
143	C₄×C₁₀₃	#440	1	2	yes	—
144	C₄×C₁₀₃	#461	1	2	yes	—
145	C₄×C₁₀₃	#463	1	2	yes	—
146	#9064	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₁₀₃	—	8	4	no	—
148	C₂×C₁₀₃	#1629	1	1	yes	characteristic
149	C₂×C₁₀₃	#1630	1	1	yes	characteristic
150	C₂×C₁₀₃	#1631	1	2	yes	—
151	C₂×C₁₀₃	#1877	1	2	yes	—
152	C₂×C₁₀₃	#1887	1	1	yes	characteristic
153	C₂×#492	C₂×C₁₀₃	1	1	yes	characteristic
154	#2366	C₂×C₁₀₃	1	1	yes	characteristic
155	#1987	C₂×C₁₀₃	1	2	yes	complemented
156	C₂×#331	C₂×C₁₀₃	1	1	yes	characteristic
157	#1279	C₂×C₁₀₃	1	2	yes	complemented
158	C₁₀₃	#9064	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×SD₁₆	—	2	2	no	—
160	C₂×C₄⋊Q₈	C₂²×C₁₀₃	1	1	yes	characteristic
161	C₂×#127	C₂²×C₁₀₃	1	1	yes	characteristic
162	C₂×C₄×D₄	—	2	2	no	—
163	#492	—	2	2	no	—
164	#492	C₄×C₁₀₃	1	4	yes	complemented
165	C₂×C₄⋊C₈	C₂²×C₁₀₃	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₁₀₃	1	1	yes	characteristic
167	#390	—	2	2	no	—
168	#386	—	2	2	no	—
169	C₄×C₄⋊C₄	C₂²×C₁₀₃	1	2	yes	—
170	#340	—	2	2	no	—
171	#340	—	2	2	no	—
172	#331	—	2	2	no	—
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×D₄	—	2	2	no	—
175	C₂²×D₄	D₄×C₁₀₃	1	2	yes	—
176	C₂³×C₄	—	2	2	no	—
177	C₂×SD₁₆	—	2	8	no	—
178	C₂×SD₁₆	—	4	4	no	—
179	C₂×SD₁₆	—	4	4	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	C₂×C₄×C₁₀₃	1	2	yes	complemented
185	#127	—	2	2	no	—
186	#127	C₂×C₄×C₁₀₃	1	2	yes	—
187	#127	D₄×C₁₀₃	1	2	yes	—
188	C₄×D₄	—	4	4	no	—
189	C₄×D₄	—	4	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	D₄×C₁₀₃	1	2	yes	—
192	C₂×C₄⋊C₄	D₄×C₁₀₃	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂³×C₁₀₃	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	C₂×C₄×C₁₀₃	1	2	yes	complemented
201	#95	—	2	2	no	—
202	C₂⁴	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	C₂²⋊C₄×C₁₀₃	1	4	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	8	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	8	2	no	—
222	C₂²×C₄	C₂×D₄×C₁₀₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₀₃	1	2	yes	—
224	SD₁₆	—	4	16	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	SD₁₆×C₁₀₃	1	4	yes	complemented
237	C₂²⋊C₄	—	4	8	no	—
238	C₂²⋊C₄	—	8	2	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₂²×C₄×C₁₀₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₁₀₃	1	1	yes	characteristic
243	C₄²	#41×C₁₀₃	1	2	yes	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	#120×C₁₀₃	1	1	yes	characteristic
250	Q₈	—	4	4	no	—
251	Q₈	—	8	2	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	8	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₃	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₁₀₃	1	2	yes	—
272	C₂×C₄	#121×C₁₀₃	1	2	yes	—
273	C₂×C₄	#123×C₁₀₃	1	1	yes	characteristic
274	C₂×C₄	C₂×SD₁₆×C₁₀₃	1	2	yes	—
275	C₂×C₄	#135×C₁₀₃	1	2	yes	—
276	C₂×C₄	AGL(1,ℤ/8)×C₁₀₃	1	2	yes	—
277	C₂×C₄	#137×C₁₀₃	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	8	2	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	4	no	—
285	C₂²	#386×C₁₀₃	1	1	yes	characteristic
286	C₂²	#450×C₁₀₃	1	2	yes	—
287	C₂²	#452×C₁₀₃	1	2	yes	—
288	C₂²	#453×C₁₀₃	1	1	yes	characteristic
289	C₂²	#453×C₁₀₃	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	C₄×SD₁₆×C₁₀₃	1	2	yes	—
296	C₄	#440×C₁₀₃	1	2	yes	—
297	C₄	#461×C₁₀₃	1	2	yes	—
298	C₄	#463×C₁₀₃	1	2	yes	—
299	C₂	—	8	4	no	—
300	C₂	C₁₀₃×#1629	1	1	yes	characteristic
301	C₂	C₁₀₃×#1630	1	1	yes	characteristic
302	C₂	C₁₀₃×#1631	1	2	yes	—
303	C₂	C₁₀₃×#1877	1	2	yes	—
304	C₂	C₁₀₃×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₃	#120
commutator	C₄²	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₃
prime radical	C₂×#127×C₁₀₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₃ ◁ C₂×C₄²×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₃×#1987	C₂	semidirect
5. C₁₀₃×#1279	C₂	semidirect
11. #492×C₁₀₃	C₄	semidirect
31. C₄⋊Q₈×C₁₀₃	C₂×C₄	semidirect
47. C₄×C₈×C₁₀₃	C₂×C₄	semidirect
83. C₄⋊C₄×C₁₀₃	SD₁₆	semidirect
146. #9064	C₁₀₃	direct (N × H)
155. #1987	C₂×C₁₀₃	semidirect
157. #1279	C₂×C₁₀₃	semidirect
158. C₁₀₃	#9064	direct (N × H)
164. #492	C₄×C₁₀₃	semidirect
184. C₄⋊Q₈	C₂×C₄×C₁₀₃	semidirect
200. C₄×C₈	C₂×C₄×C₁₀₃	semidirect
236. C₄⋊C₄	SD₁₆×C₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₁₀₃	2
C₁₀₃×#2366	2
C₁₀₃×#1987	2
C₂×#331×C₁₀₃	2
C₁₀₃×#1279	2
#9064	103

Subgroups — C103×#9064 #39914339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#9064 #39914339