Subgroups — C₁₀₃×#10824 #39916099

Order 26368.

722 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₁₀₃	C₂	1	1	yes	maximal, characteristic
2	C₁₀₃×#1742	C₂	1	1	yes	maximal, characteristic
3	C₁₀₃×#1740	C₂	1	1	yes	maximal, characteristic
4	C₂×#339×C₁₀₃	C₂	1	2	yes	maximal
5	C₂×H(3,ℤ/4)×C₁₀₃	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₀₃	C₂²	1	2	yes	—
7	C₂×#117×C₁₀₃	C₂²	1	2	yes	—
8	C₂×#117×C₁₀₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₀₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₁₀₃	—	2	2	no	—
12	C₄×C₂²⋊C₄×C₁₀₃	—	2	2	no	—
13	#376×C₁₀₃	C₄	1	2	yes	complemented
14	C₄³×C₁₀₃	C₄	1	2	yes	complemented
15	#343×C₁₀₃	—	2	2	no	—
16	#342×C₁₀₃	—	2	2	no	—
17	#339×C₁₀₃	C₄	1	8	yes	complemented
18	H(3,ℤ/4)×C₁₀₃	—	2	4	no	—
19	C₂³×C₄×C₁₀₃	—	2	1	no	—
20	C₂³×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₀₃	—	2	2	no	—
22	C₂×M₁₆×C₁₀₃	—	2	2	no	—
23	C₂×M₁₆×C₁₀₃	—	2	2	no	—
24	C₂×M₁₆×C₁₀₃	—	2	2	no	—
25	C₂×M₁₆×C₁₀₃	—	2	2	no	—
26	C₂×M₁₆×C₁₀₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₀₃	—	2	2	no	—
28	#117×C₁₀₃	—	2	2	no	—
29	#117×C₁₀₃	—	2	2	no	—
30	#117×C₁₀₃	—	2	1	no	—
31	#117×C₁₀₃	—	2	1	no	—
32	#117×C₁₀₃	—	2	2	no	—
33	#117×C₁₀₃	—	2	2	no	—
34	#117×C₁₀₃	—	2	2	no	—
35	#117×C₁₀₃	C₂×C₄	1	4	yes	—
36	#117×C₁₀₃	D₄	1	2	yes	—
37	#117×C₁₀₃	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₁₀₃	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₁₀₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₀₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₀₃	—	2	1	no	—
45	C₂×C₄²×C₁₀₃	—	2	2	no	—
46	C₂×C₄²×C₁₀₃	—	2	2	no	—
47	C₂×C₄²×C₁₀₃	—	2	4	no	—
48	C₂×C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₀₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
53	#95×C₁₀₃	—	2	4	no	—
54	C₂⁴×C₁₀₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₃	—	2	1	no	—
56	C₂²×C₄×C₁₀₃	—	2	1	no	—
57	C₂²×C₄×C₁₀₃	—	2	2	no	—
58	C₂²×C₄×C₁₀₃	—	2	4	no	—
59	C₂²×C₄×C₁₀₃	—	4	4	no	—
60	C₂²×C₄×C₁₀₃	—	4	2	no	—
61	C₂²×C₄×C₁₀₃	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₀₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₀₃	#41	1	2	yes	—
72	M₁₆×C₁₀₃	—	2	4	no	—
73	M₁₆×C₁₀₃	—	2	4	no	—
74	M₁₆×C₁₀₃	—	4	2	no	—
75	M₁₆×C₁₀₃	—	4	2	no	—
76	C₂×C₈×C₁₀₃	—	2	4	no	—
77	C₂×C₈×C₁₀₃	—	4	2	no	—
78	C₂×C₈×C₁₀₃	—	4	2	no	—
79	C₄⋊C₄×C₁₀₃	—	2	2	no	—
80	C₄⋊C₄×C₁₀₃	—	2	2	no	—
81	C₄⋊C₄×C₁₀₃	—	2	4	no	—
82	C₄⋊C₄×C₁₀₃	—	2	4	no	—
83	C₄⋊C₄×C₁₀₃	—	4	2	no	—
84	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
85	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
86	C₂²⋊C₄×C₁₀₃	—	2	4	no	—
87	C₂²⋊C₄×C₁₀₃	—	2	4	no	—
88	C₂²⋊C₄×C₁₀₃	—	4	2	no	—
89	C₄²×C₁₀₃	—	2	1	no	—
90	C₄²×C₁₀₃	—	2	1	no	—
91	C₄²×C₁₀₃	—	2	2	no	—
92	C₄²×C₁₀₃	—	2	2	no	—
93	C₄²×C₁₀₃	—	2	2	no	—
94	C₄²×C₁₀₃	—	2	4	no	—
95	C₄²×C₁₀₃	—	2	4	no	—
96	C₄²×C₁₀₃	—	4	8	no	—
97	C₄²×C₁₀₃	—	4	2	no	—
98	C₄²×C₁₀₃	C₄²	1	2	yes	—
99	C₄²×C₁₀₃	C₄²	1	2	yes	—
100	C₄²×C₁₀₃	C₄⋊C₄	1	4	yes	—
101	C₂³×C₁₀₃	—	4	2	no	—
102	C₂³×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₀₃	C₄×D₄	1	2	yes	—
104	C₂³×C₁₀₃	#121	1	1	yes	characteristic
105	C₂³×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₀₃	#123	1	2	yes	—
107	C₂×C₄×C₁₀₃	—	2	4	no	—
108	C₂×C₄×C₁₀₃	—	2	2	no	—
109	C₂×C₄×C₁₀₃	—	2	2	no	—
110	C₂×C₄×C₁₀₃	—	2	4	no	—
111	C₂×C₄×C₁₀₃	—	2	4	no	—
112	C₂×C₄×C₁₀₃	—	4	8	no	—
113	C₂×C₄×C₁₀₃	—	4	4	no	—
114	C₂×C₄×C₁₀₃	—	4	2	no	—
115	C₂×C₄×C₁₀₃	—	4	4	no	—
116	C₂×C₄×C₁₀₃	—	4	4	no	—
117	C₂×C₄×C₁₀₃	—	8	4	no	—
118	C₂×C₄×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₀₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₀₃	#117	1	2	yes	—
123	C₂×C₄×C₁₀₃	#117	1	2	yes	—
124	C₂×C₄×C₁₀₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₀₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₀₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₀₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₀₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₀₃	#123	1	2	yes	—
132	C₂×C₄×C₁₀₃	#124	1	2	yes	—
133	C₂×C₄×C₁₀₃	#125	1	2	yes	—
134	C₂×C₄×C₁₀₃	#126	1	2	yes	—
135	C₂×C₄×C₁₀₃	#126	1	2	yes	—
136	C₈×C₁₀₃	—	4	4	no	—
137	C₂²×C₁₀₃	—	2	2	no	—
138	C₂²×C₁₀₃	—	2	4	no	—
139	C₂²×C₁₀₃	—	4	2	no	—
140	C₂²×C₁₀₃	—	4	1	no	—
141	C₂²×C₁₀₃	—	4	1	no	—
142	C₂²×C₁₀₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₀₃	#380	1	1	yes	characteristic
144	C₂²×C₁₀₃	#381	1	1	yes	characteristic
145	C₂²×C₁₀₃	#387	1	2	yes	—
146	C₂²×C₁₀₃	#388	1	2	yes	—
147	C₄×C₁₀₃	—	2	4	no	—
148	C₄×C₁₀₃	—	4	4	no	—
149	C₄×C₁₀₃	—	4	2	no	—
150	C₄×C₁₀₃	—	4	4	no	—
151	C₄×C₁₀₃	—	8	4	no	—
152	C₄×C₁₀₃	#376	1	1	yes	characteristic
153	C₄×C₁₀₃	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₁₀₃	#382	1	1	yes	characteristic
155	C₄×C₁₀₃	#383	1	1	yes	characteristic
156	C₄×C₁₀₃	#387	1	2	yes	—
157	C₄×C₁₀₃	#388	1	2	yes	—
158	#10824	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₁₀₃	—	2	1	no	—
160	C₂×C₁₀₃	—	2	1	no	—
161	C₂×C₁₀₃	—	2	2	no	—
162	C₂×C₁₀₃	—	2	2	no	—
163	C₂×C₁₀₃	#1434	1	1	yes	characteristic
164	C₂×C₁₀₃	#1827	1	2	yes	—
165	C₄×#117	C₂×C₁₀₃	1	1	yes	characteristic
166	#1742	C₂×C₁₀₃	1	1	yes	characteristic
167	#1740	C₂×C₁₀₃	1	1	yes	characteristic
168	C₂×#339	C₂×C₁₀₃	1	2	yes	—
169	C₂×H(3,ℤ/4)	C₂×C₁₀₃	1	2	yes	—
170	C₁₀₃	#10824	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×M₁₆	C₂²×C₁₀₃	1	2	yes	—
172	C₂×#117	C₂²×C₁₀₃	1	2	yes	—
173	C₂×#117	C₂²×C₁₀₃	1	1	yes	characteristic
174	C₂×#117	C₂²×C₁₀₃	1	1	yes	characteristic
175	C₂²×C₄²	C₂²×C₁₀₃	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	2	2	no	—
177	C₄×C₂²⋊C₄	—	2	2	no	—
178	#376	C₄×C₁₀₃	1	2	yes	complemented
179	C₄³	C₄×C₁₀₃	1	2	yes	complemented
180	#343	—	2	2	no	—
181	#342	—	2	2	no	—
182	#339	C₄×C₁₀₃	1	8	yes	complemented
183	H(3,ℤ/4)	—	2	4	no	—
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂³×C₁₀₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₁₀₃	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	C₂×C₄×C₁₀₃	1	4	yes	—
201	#117	D₄×C₁₀₃	1	2	yes	—
202	#117	Q₈×C₁₀₃	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	D₄×C₁₀₃	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	D₄×C₁₀₃	1	2	yes	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	C₂×C₄×C₁₀₃	1	2	yes	—
214	C₂×C₄²	C₂×C₄×C₁₀₃	1	4	yes	—
215	C₂×C₄²	C₂×C₄×C₁₀₃	1	1	yes	characteristic
216	C₂×C₄²	C₂×C₄×C₁₀₃	1	1	yes	characteristic
217	C₂×C₄²	C₂×C₄×C₁₀₃	1	2	yes	—
218	#95	—	2	4	no	—
219	C₂⁴	C₂×D₄×C₁₀₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂²⋊C₄×C₁₀₃	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₀₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₁₀₃	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₀₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₀₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₁₀₃	1	1	yes	characteristic
232	C₂²×C₄	#41×C₁₀₃	1	1	yes	characteristic
233	C₂²×C₄	#41×C₁₀₃	1	1	yes	characteristic
234	C₂²×C₄	#41×C₁₀₃	1	2	yes	—
235	C₂²×C₄	#41×C₁₀₃	1	2	yes	—
236	C₂²×C₄	#41×C₁₀₃	1	2	yes	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	4	8	no	—
262	C₄²	—	4	2	no	—
263	C₄²	C₄²×C₁₀₃	1	2	yes	—
264	C₄²	C₄²×C₁₀₃	1	2	yes	—
265	C₄²	C₄⋊C₄×C₁₀₃	1	4	yes	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×C₂²⋊C₄×C₁₀₃	1	1	yes	characteristic
268	C₂³	C₄×D₄×C₁₀₃	1	2	yes	—
269	C₂³	#121×C₁₀₃	1	1	yes	characteristic
270	C₂³	C₂²⋊Q₈×C₁₀₃	1	1	yes	characteristic
271	C₂³	#123×C₁₀₃	1	2	yes	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	8	4	no	—
283	C₂×C₄	C₂×C₄²×C₁₀₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊C₄×C₁₀₃	1	1	yes	characteristic
285	C₂×C₄	#117×C₁₀₃	1	1	yes	characteristic
286	C₂×C₄	#117×C₁₀₃	1	1	yes	characteristic
287	C₂×C₄	#117×C₁₀₃	1	2	yes	—
288	C₂×C₄	#117×C₁₀₃	1	2	yes	—
289	C₂×C₄	#117×C₁₀₃	1	1	yes	characteristic
290	C₂×C₄	#117×C₁₀₃	1	1	yes	characteristic
291	C₂×C₄	C₄×D₄×C₁₀₃	1	2	yes	—
292	C₂×C₄	C₄×D₄×C₁₀₃	1	2	yes	—
293	C₂×C₄	C₄×Q₈×C₁₀₃	1	2	yes	—
294	C₂×C₄	#121×C₁₀₃	1	1	yes	characteristic
295	C₂×C₄	C₂²⋊Q₈×C₁₀₃	1	1	yes	characteristic
296	C₂×C₄	#123×C₁₀₃	1	2	yes	—
297	C₂×C₄	#124×C₁₀₃	1	2	yes	—
298	C₂×C₄	#125×C₁₀₃	1	2	yes	—
299	C₂×C₄	#126×C₁₀₃	1	2	yes	—
300	C₂×C₄	#126×C₁₀₃	1	2	yes	—
301	C₈	—	4	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₄×C₂²⋊C₄×C₁₀₃	1	1	yes	characteristic
308	C₂²	#380×C₁₀₃	1	1	yes	characteristic
309	C₂²	#381×C₁₀₃	1	1	yes	characteristic
310	C₂²	#387×C₁₀₃	1	2	yes	—
311	C₂²	#388×C₁₀₃	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	8	4	no	—
317	C₄	#376×C₁₀₃	1	1	yes	characteristic
318	C₄	C₄×C₄⋊C₄×C₁₀₃	1	1	yes	characteristic
319	C₄	#382×C₁₀₃	1	1	yes	characteristic
320	C₄	#383×C₁₀₃	1	1	yes	characteristic
321	C₄	#387×C₁₀₃	1	2	yes	—
322	C₄	#388×C₁₀₃	1	2	yes	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	2	2	no	—
327	C₂	C₁₀₃×#1434	1	1	yes	characteristic
328	C₂	C₁₀₃×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₃	#117
commutator	C₂×C₄	C₂×C₄²×C₁₀₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₃
socle	C₂²×C₁₀₃	#381
prime radical	C₂⁴×C₁₀₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₃ ◁ C₂×#117×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. #376×C₁₀₃	C₄	semidirect
14. C₄³×C₁₀₃	C₄	semidirect
17. #339×C₁₀₃	C₄	semidirect
158. #10824	C₁₀₃	direct (N × H)
170. C₁₀₃	#10824	direct (N × H)
178. #376	C₄×C₁₀₃	semidirect
179. C₄³	C₄×C₁₀₃	semidirect
182. #339	C₄×C₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#117×C₁₀₃	2
C₁₀₃×#1742	2
C₁₀₃×#1740	2
C₂×#339×C₁₀₃	2
C₂×H(3,ℤ/4)×C₁₀₃	2
#10824	103

Subgroups — C103×#10824 #39916099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#10824 #39916099