Subgroups — C₁₀₃×#19961 #39925236

Order 26368.

638 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₃×#3397	C₂	1	2	yes	maximal, complemented
2	C₁₀₃×#3304	C₂	1	2	yes	maximal, complemented
3	C₁₀₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₃×#1686	C₂	1	2	yes	maximal, complemented
5	C₁₀₃×#1681	C₂	1	2	yes	maximal, complemented
6	C₁₀₃×#1650	C₂	1	2	yes	maximal, complemented
7	C₁₀₃×#1621	C₂	1	2	yes	maximal, complemented
8	C₁₀₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#556×C₁₀₃	C₂²	1	2	yes	—
11	#550×C₁₀₃	C₂²	1	2	yes	—
12	#500×C₁₀₃	C₂²	1	1	yes	characteristic
13	#496×C₁₀₃	—	2	2	no	—
14	#493×C₁₀₃	C₂²	1	1	yes	characteristic
15	#492×C₁₀₃	C₂²	1	2	yes	—
16	#486×C₁₀₃	C₂²	1	2	yes	—
17	#482×C₁₀₃	—	2	2	no	—
18	#475×C₁₀₃	C₂²	1	2	yes	—
19	#465×C₁₀₃	—	2	2	no	—
20	#463×C₁₀₃	—	2	2	no	—
21	#461×C₁₀₃	C₂²	1	2	yes	—
22	#460×C₁₀₃	—	2	2	no	—
23	#460×C₁₀₃	C₂²	1	2	yes	—
24	#459×C₁₀₃	C₂²	1	2	yes	—
25	#452×C₁₀₃	—	2	2	no	—
26	#446×C₁₀₃	C₂²	1	2	yes	—
27	#440×C₁₀₃	—	2	2	no	—
28	C₄×Q₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₁₀₃	C₂²	1	1	yes	characteristic
30	C₈×D₄×C₁₀₃	C₂²	1	2	yes	—
31	#416×C₁₀₃	—	2	2	no	—
32	#335×C₁₀₃	C₂²	1	1	yes	characteristic
33	#335×C₁₀₃	C₂²	1	1	yes	characteristic
34	#332×C₁₀₃	C₂²	1	2	yes	—
35	#331×C₁₀₃	C₂²	1	2	yes	—
36	Q₈⋊C₈×C₁₀₃	C₂²	1	2	yes	—
37	D₄⋊C₈×C₁₀₃	C₂²	1	2	yes	—
38	C₈²×C₁₀₃	C₂²	1	1	yes	characteristic
39	C₂×#41×C₁₀₃	—	2	2	no	—
40	C₂×Q₁₆×C₁₀₃	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₁₀₃	—	2	2	no	—
42	C₂×SD₁₆×C₁₀₃	—	2	2	no	—
43	C₂×SD₁₆×C₁₀₃	—	2	2	no	—
44	C₂×SD₁₆×C₁₀₃	—	2	2	no	—
45	C₂×D₈×C₁₀₃	—	2	2	no	—
46	C₂×D₈×C₁₀₃	—	4	1	no	—
47	C₂×D₈×C₁₀₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₀₃	—	2	2	no	—
49	C₄⋊Q₈×C₁₀₃	C₂³	1	2	yes	—
50	#127×C₁₀₃	—	2	2	no	—
51	#127×C₁₀₃	C₂³	1	2	yes	—
52	#126×C₁₀₃	—	4	2	no	—
53	#124×C₁₀₃	—	2	2	no	—
54	C₂²⋊Q₈×C₁₀₃	—	2	2	no	—
55	#121×C₁₀₃	—	2	2	no	—
56	#121×C₁₀₃	—	4	2	no	—
57	C₄×Q₈×C₁₀₃	—	2	2	no	—
58	C₄×Q₈×C₁₀₃	C₂³	1	2	yes	—
59	C₄×D₄×C₁₀₃	—	2	2	no	—
60	C₄×D₄×C₁₀₃	C₂³	1	2	yes	—
61	#117×C₁₀₃	—	2	2	no	—
62	#107×C₁₀₃	—	4	1	no	—
63	#107×C₁₀₃	D₄	1	1	yes	characteristic
64	#107×C₁₀₃	D₄	1	1	yes	characteristic
65	#106×C₁₀₃	—	2	2	no	—
66	C₄⋊C₈×C₁₀₃	—	2	2	no	—
67	C₄⋊C₈×C₁₀₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₁₀₃	C₂³	1	2	yes	—
69	#103×C₁₀₃	—	2	2	no	—
70	#103×C₁₀₃	—	2	2	no	—
71	#103×C₁₀₃	D₄	1	2	yes	—
72	#102×C₁₀₃	—	2	2	no	—
73	#102×C₁₀₃	—	2	2	no	—
74	#102×C₁₀₃	—	2	2	no	—
75	#102×C₁₀₃	—	2	2	no	—
76	#102×C₁₀₃	D₄	1	2	yes	—
77	C₂²⋊C₈×C₁₀₃	—	2	2	no	—
78	#97×C₁₀₃	—	2	2	no	—
79	C₄×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
82	#41×C₁₀₃	—	4	4	no	—
83	#41×C₁₀₃	—	8	2	no	—
84	C₂×Q₈×C₁₀₃	—	2	2	no	—
85	C₂×Q₈×C₁₀₃	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₁₀₃	—	2	2	no	—
87	C₂×D₄×C₁₀₃	—	2	2	no	—
88	C₂×D₄×C₁₀₃	—	2	2	no	—
89	C₂×D₄×C₁₀₃	—	4	2	no	—
90	C₂×D₄×C₁₀₃	—	4	2	no	—
91	C₂×D₄×C₁₀₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₀₃	—	2	2	no	—
93	C₂²×C₄×C₁₀₃	—	4	2	no	—
94	Q₁₆×C₁₀₃	—	4	1	no	—
95	SD₁₆×C₁₀₃	—	4	4	no	—
96	SD₁₆×C₁₀₃	—	8	2	no	—
97	SD₁₆×C₁₀₃	—	8	2	no	—
98	SD₁₆×C₁₀₃	—	8	2	no	—
99	D₈×C₁₀₃	—	2	2	no	—
100	D₈×C₁₀₃	—	8	2	no	—
101	D₈×C₁₀₃	—	8	2	no	—
102	C₂×C₈×C₁₀₃	—	2	1	no	—
103	C₂×C₈×C₁₀₃	—	2	2	no	—
104	C₂×C₈×C₁₀₃	—	2	2	no	—
105	C₂×C₈×C₁₀₃	—	8	2	no	—
106	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₁₀₃	—	2	2	no	—
111	C₄⋊C₄×C₁₀₃	—	2	2	no	—
112	C₄⋊C₄×C₁₀₃	—	2	2	no	—
113	C₄⋊C₄×C₁₀₃	—	4	2	no	—
114	C₄⋊C₄×C₁₀₃	C₂×D₄	1	2	yes	—
115	C₄⋊C₄×C₁₀₃	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
117	C₂²⋊C₄×C₁₀₃	—	4	2	no	—
118	C₂²⋊C₄×C₁₀₃	—	4	2	no	—
119	C₂²⋊C₄×C₁₀₃	—	4	2	no	—
120	C₄²×C₁₀₃	—	2	2	no	—
121	C₄²×C₁₀₃	—	4	2	no	—
122	C₄²×C₁₀₃	C₂⁴	1	1	yes	characteristic
123	C₂³×C₁₀₃	—	2	2	no	—
124	C₂³×C₁₀₃	—	4	2	no	—
125	Q₈×C₁₀₃	—	2	2	no	—
126	Q₈×C₁₀₃	—	2	2	no	—
127	Q₈×C₁₀₃	—	4	2	no	—
128	Q₈×C₁₀₃	—	4	2	no	—
129	D₄×C₁₀₃	—	2	2	no	—
130	D₄×C₁₀₃	—	2	2	no	—
131	D₄×C₁₀₃	—	4	2	no	—
132	D₄×C₁₀₃	—	4	2	no	—
133	D₄×C₁₀₃	—	4	2	no	—
134	D₄×C₁₀₃	—	4	2	no	—
135	D₄×C₁₀₃	—	8	2	no	—
136	D₄×C₁₀₃	—	8	4	no	—
137	D₄×C₁₀₃	—	16	2	no	—
138	C₂×C₄×C₁₀₃	—	2	2	no	—
139	C₂×C₄×C₁₀₃	—	2	2	no	—
140	C₂×C₄×C₁₀₃	—	2	2	no	—
141	C₂×C₄×C₁₀₃	—	4	2	no	—
142	C₂×C₄×C₁₀₃	—	4	2	no	—
143	C₂×C₄×C₁₀₃	—	4	2	no	—
144	C₂×C₄×C₁₀₃	—	8	4	no	—
145	C₂×C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₈×C₁₀₃	—	2	2	no	—
149	C₈×C₁₀₃	—	2	1	no	—
150	C₈×C₁₀₃	—	2	1	no	—
151	C₈×C₁₀₃	—	2	1	no	—
152	C₈×C₁₀₃	—	4	2	no	—
153	C₈×C₁₀₃	—	4	2	no	—
154	C₈×C₁₀₃	#135	1	2	yes	—
155	C₂²×C₁₀₃	—	4	2	no	—
156	C₂²×C₁₀₃	—	4	2	no	—
157	C₂²×C₁₀₃	—	4	2	no	—
158	C₂²×C₁₀₃	—	8	2	no	—
159	C₂²×C₁₀₃	—	8	2	no	—
160	C₂²×C₁₀₃	—	8	2	no	—
161	C₂²×C₁₀₃	D₄²	1	1	yes	characteristic
162	C₄×C₁₀₃	—	2	1	no	—
163	C₄×C₁₀₃	—	4	2	no	—
164	C₄×C₁₀₃	—	4	2	no	—
165	C₄×C₁₀₃	—	4	2	no	—
166	C₄×C₁₀₃	—	8	2	no	—
167	C₄×C₁₀₃	C₂×#135	1	1	yes	characteristic
168	C₄×C₁₀₃	#575	1	1	yes	characteristic
169	C₄×C₁₀₃	#576	1	1	yes	characteristic
170	C₄×C₁₀₃	#576	1	1	yes	characteristic
171	#19961	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₁₀₃	—	8	2	no	—
173	C₂×C₁₀₃	—	16	2	no	—
174	C₂×C₁₀₃	#3273	1	1	yes	characteristic
175	C₂×C₁₀₃	#3287	1	1	yes	characteristic
176	C₂×C₁₀₃	#3291	1	1	yes	characteristic
177	#3397	C₂×C₁₀₃	1	2	yes	complemented
178	#3304	C₂×C₁₀₃	1	2	yes	complemented
179	#1706	C₂×C₁₀₃	1	1	yes	characteristic, complemented
180	#1686	C₂×C₁₀₃	1	2	yes	complemented
181	#1681	C₂×C₁₀₃	1	2	yes	complemented
182	#1650	C₂×C₁₀₃	1	2	yes	complemented
183	#1621	C₂×C₁₀₃	1	2	yes	complemented
184	#1591	C₂×C₁₀₃	1	1	yes	characteristic, complemented
185	C₈×D₈	C₂×C₁₀₃	1	1	yes	characteristic, complemented
186	C₁₀₃	#19961	1	1	yes	characteristic, direct product factor, Sylow
187	#556	C₂²×C₁₀₃	1	2	yes	—
188	#550	C₂²×C₁₀₃	1	2	yes	—
189	#500	C₂²×C₁₀₃	1	1	yes	characteristic
190	#496	—	2	2	no	—
191	#493	C₂²×C₁₀₃	1	1	yes	characteristic
192	#492	C₂²×C₁₀₃	1	2	yes	—
193	#486	C₂²×C₁₀₃	1	2	yes	—
194	#482	—	2	2	no	—
195	#475	C₂²×C₁₀₃	1	2	yes	—
196	#465	—	2	2	no	—
197	#463	—	2	2	no	—
198	#461	C₂²×C₁₀₃	1	2	yes	—
199	#460	—	2	2	no	—
200	#460	C₂²×C₁₀₃	1	2	yes	—
201	#459	C₂²×C₁₀₃	1	2	yes	—
202	#452	—	2	2	no	—
203	#446	C₂²×C₁₀₃	1	2	yes	—
204	#440	—	2	2	no	—
205	C₄×Q₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
206	C₄×D₈	C₂²×C₁₀₃	1	1	yes	characteristic
207	C₈×D₄	C₂²×C₁₀₃	1	2	yes	—
208	#416	—	2	2	no	—
209	#335	C₂²×C₁₀₃	1	1	yes	characteristic
210	#335	C₂²×C₁₀₃	1	1	yes	characteristic
211	#332	C₂²×C₁₀₃	1	2	yes	—
212	#331	C₂²×C₁₀₃	1	2	yes	—
213	Q₈⋊C₈	C₂²×C₁₀₃	1	2	yes	—
214	D₄⋊C₈	C₂²×C₁₀₃	1	2	yes	—
215	C₈²	C₂²×C₁₀₃	1	1	yes	characteristic
216	C₂×#41	—	2	2	no	—
217	C₂×Q₁₆	D₄×C₁₀₃	1	1	yes	characteristic
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	2	2	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂×D₈	—	4	1	no	—
224	C₂×D₈	D₄×C₁₀₃	1	1	yes	characteristic
225	C₂²×C₈	—	2	2	no	—
226	C₄⋊Q₈	C₂³×C₁₀₃	1	2	yes	—
227	#127	—	2	2	no	—
228	#127	C₂³×C₁₀₃	1	2	yes	—
229	#126	—	4	2	no	—
230	#124	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	#121	—	2	2	no	—
233	#121	—	4	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	C₂³×C₁₀₃	1	2	yes	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	C₂³×C₁₀₃	1	2	yes	—
238	#117	—	2	2	no	—
239	#107	—	4	1	no	—
240	#107	D₄×C₁₀₃	1	1	yes	characteristic
241	#107	D₄×C₁₀₃	1	1	yes	characteristic
242	#106	—	2	2	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	C₂³×C₁₀₃	1	2	yes	—
245	C₄⋊C₈	C₂³×C₁₀₃	1	2	yes	—
246	#103	—	2	2	no	—
247	#103	—	2	2	no	—
248	#103	D₄×C₁₀₃	1	2	yes	—
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#102	—	2	2	no	—
253	#102	D₄×C₁₀₃	1	2	yes	—
254	C₂²⋊C₈	—	2	2	no	—
255	#97	—	2	2	no	—
256	C₄×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
259	#41	—	4	4	no	—
260	#41	—	8	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	C₂×D₄×C₁₀₃	1	2	yes	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	C₂×D₄×C₁₀₃	1	2	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	Q₁₆	—	4	1	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	8	2	no	—
274	SD₁₆	—	8	2	no	—
275	SD₁₆	—	8	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	8	2	no	—
278	D₈	—	8	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	8	2	no	—
283	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	C₂×D₄×C₁₀₃	1	2	yes	—
292	C₄⋊C₄	C₂×D₄×C₁₀₃	1	2	yes	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	C₂⁴×C₁₀₃	1	1	yes	characteristic
300	C₂³	—	2	2	no	—
301	C₂³	—	4	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	2	no	—
313	D₄	—	8	4	no	—
314	D₄	—	16	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	C₂²×D₄×C₁₀₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₁₀₃	1	1	yes	characteristic
324	C₂×C₄	H(5,𝔽₂)×C₁₀₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	2	no	—
331	C₈	#135×C₁₀₃	1	2	yes	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	D₄²×C₁₀₃	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₂×#135×C₁₀₃	1	1	yes	characteristic
345	C₄	#575×C₁₀₃	1	1	yes	characteristic
346	C₄	#576×C₁₀₃	1	1	yes	characteristic
347	C₄	#576×C₁₀₃	1	1	yes	characteristic
348	C₂	—	8	2	no	—
349	C₂	—	16	2	no	—
350	C₂	C₁₀₃×#3273	1	1	yes	characteristic
351	C₂	C₁₀₃×#3287	1	1	yes	characteristic
352	C₂	C₁₀₃×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₄²×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₃×#3397	C₂	semidirect
2. C₁₀₃×#3304	C₂	semidirect
3. C₁₀₃×#1706	C₂	semidirect
4. C₁₀₃×#1686	C₂	semidirect
5. C₁₀₃×#1681	C₂	semidirect
6. C₁₀₃×#1650	C₂	semidirect
7. C₁₀₃×#1621	C₂	semidirect
8. C₁₀₃×#1591	C₂	semidirect
9. C₈×D₈×C₁₀₃	C₂	semidirect
171. #19961	C₁₀₃	direct (N × H)
177. #3397	C₂×C₁₀₃	semidirect
178. #3304	C₂×C₁₀₃	semidirect
179. #1706	C₂×C₁₀₃	semidirect
180. #1686	C₂×C₁₀₃	semidirect
181. #1681	C₂×C₁₀₃	semidirect
182. #1650	C₂×C₁₀₃	semidirect
183. #1621	C₂×C₁₀₃	semidirect
184. #1591	C₂×C₁₀₃	semidirect
185. C₈×D₈	C₂×C₁₀₃	semidirect
186. C₁₀₃	#19961	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₃×#3397	2
C₁₀₃×#3304	2
C₁₀₃×#1706	2
C₁₀₃×#1686	2
C₁₀₃×#1681	2
C₁₀₃×#1650	2
C₁₀₃×#1621	2
C₁₀₃×#1591	2
C₈×D₈×C₁₀₃	2
#19961	103

Subgroups — C103×#19961 #39925236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#19961 #39925236