Subgroups — C₁₀₃×#33860 #39939135

Order 26368.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₃×#3417	C₂	1	2	yes	maximal, complemented
2	C₁₀₃×#3293	C₂	1	2	yes	maximal, complemented
3	C₁₀₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₃×#2222	C₂	1	2	yes	maximal, complemented
5	C₁₀₃×#2214	C₂	1	2	yes	maximal, complemented
6	C₁₀₃×#2195	C₂	1	4	yes	maximal, complemented
7	C₁₀₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	#578×C₁₀₃	—	2	2	no	—
10	#578×C₁₀₃	C₂²	1	4	yes	—
11	#510×C₁₀₃	—	2	4	no	—
12	#510×C₁₀₃	C₂²	1	4	yes	—
13	#508×C₁₀₃	—	2	2	no	—
14	#508×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₁₀₃	—	2	2	no	—
16	C₂×Q₃₂×C₁₀₃	C₂²	1	2	yes	—
17	C₂×Q₃₂×C₁₀₃	C₂²	1	2	yes	—
18	#504×C₁₀₃	C₂²	1	2	yes	—
19	#494×C₁₀₃	C₂²	1	2	yes	—
20	#473×C₁₀₃	—	2	2	no	—
21	#473×C₁₀₃	C₂²	1	4	yes	—
22	#456×C₁₀₃	—	2	4	no	—
23	#443×C₁₀₃	C₂²	1	2	yes	—
24	#443×C₁₀₃	C₂²	1	1	yes	characteristic
25	#368×C₁₀₃	C₂²	1	1	yes	characteristic
26	#364×C₁₀₃	C₂²	1	1	yes	characteristic
27	#362×C₁₀₃	C₂²	1	4	yes	—
28	#359×C₁₀₃	C₂²	1	2	yes	—
29	#350×C₁₀₃	C₂²	1	2	yes	—
30	C₄×C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
31	2₋¹⁺⁴×C₁₀₃	—	2	4	no	—
32	#137×C₁₀₃	—	2	4	no	—
33	#137×C₁₀₃	—	2	4	no	—
34	#137×C₁₀₃	—	4	4	no	—
35	#137×C₁₀₃	—	4	4	no	—
36	#135×C₁₀₃	—	2	4	no	—
37	#135×C₁₀₃	—	4	2	no	—
38	#135×C₁₀₃	D₄	1	1	yes	characteristic
39	#135×C₁₀₃	C₂³	1	2	yes	—
40	C₂×Q₁₆×C₁₀₃	—	2	4	no	—
41	C₂×Q₁₆×C₁₀₃	—	2	2	no	—
42	C₂×Q₁₆×C₁₀₃	—	4	2	no	—
43	C₂×Q₁₆×C₁₀₃	C₂³	1	4	yes	—
44	#131×C₁₀₃	—	2	2	no	—
45	#131×C₁₀₃	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₁₀₃	—	2	2	no	—
47	Q₃₂×C₁₀₃	—	2	4	no	—
48	Q₃₂×C₁₀₃	—	2	2	no	—
49	Q₃₂×C₁₀₃	—	2	2	no	—
50	Q₃₂×C₁₀₃	—	4	2	no	—
51	Q₃₂×C₁₀₃	D₄	1	1	yes	characteristic
52	SD₃₂×C₁₀₃	—	2	4	no	—
53	SD₃₂×C₁₀₃	D₄	1	2	yes	—
54	D₁₆×C₁₀₃	D₄	1	1	yes	characteristic
55	M₃₂×C₁₀₃	—	2	2	no	—
56	M₃₂×C₁₀₃	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₀₃	—	2	2	no	—
58	C₂×C₁₆×C₁₀₃	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₁₀₃	C₂³	1	1	yes	characteristic
60	#108×C₁₀₃	D₄	1	1	yes	characteristic
61	#108×C₁₀₃	C₂³	1	2	yes	—
62	#104×C₁₀₃	—	2	2	no	—
63	#104×C₁₀₃	D₄	1	2	yes	—
64	#101×C₁₀₃	—	2	4	no	—
65	C₄×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
66	#41×C₁₀₃	—	2	2	no	—
67	#41×C₁₀₃	—	4	4	no	—
68	#41×C₁₀₃	—	4	4	no	—
69	#41×C₁₀₃	—	8	4	no	—
70	#41×C₁₀₃	C₂×D₄	1	2	yes	—
71	C₂×Q₈×C₁₀₃	—	2	4	no	—
72	C₂×Q₈×C₁₀₃	—	4	4	no	—
73	C₂×Q₈×C₁₀₃	—	4	4	no	—
74	Q₁₆×C₁₀₃	—	2	4	no	—
75	Q₁₆×C₁₀₃	—	2	4	no	—
76	Q₁₆×C₁₀₃	—	4	4	no	—
77	Q₁₆×C₁₀₃	—	4	4	no	—
78	Q₁₆×C₁₀₃	—	4	2	no	—
79	Q₁₆×C₁₀₃	—	4	2	no	—
80	Q₁₆×C₁₀₃	D₈	1	1	yes	characteristic
81	Q₁₆×C₁₀₃	C₂×D₄	1	2	yes	—
82	SD₁₆×C₁₀₃	—	2	2	no	—
83	SD₁₆×C₁₀₃	—	4	4	no	—
84	SD₁₆×C₁₀₃	—	4	4	no	—
85	SD₁₆×C₁₀₃	D₈	1	2	yes	—
86	D₈×C₁₀₃	D₈	1	1	yes	characteristic
87	D₈×C₁₀₃	C₂×D₄	1	2	yes	—
88	M₁₆×C₁₀₃	—	2	2	no	—
89	M₁₆×C₁₀₃	—	2	2	no	—
90	M₁₆×C₁₀₃	—	4	2	no	—
91	M₁₆×C₁₀₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₀₃	—	2	2	no	—
93	C₂×C₈×C₁₀₃	—	4	2	no	—
94	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₀₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₁₀₃	—	8	2	no	—
97	C₄²×C₁₀₃	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₁₀₃	—	2	2	no	—
99	C₁₆×C₁₀₃	—	2	1	no	—
100	C₁₆×C₁₀₃	C₂×D₄	1	1	yes	characteristic
101	C₁₆×C₁₀₃	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₁₀₃	—	2	2	no	—
103	Q₈×C₁₀₃	—	4	4	no	—
104	Q₈×C₁₀₃	—	4	4	no	—
105	Q₈×C₁₀₃	—	8	4	no	—
106	Q₈×C₁₀₃	C₂×D₈	1	2	yes	—
107	D₄×C₁₀₃	—	2	2	no	—
108	D₄×C₁₀₃	—	2	2	no	—
109	D₄×C₁₀₃	—	4	2	no	—
110	D₄×C₁₀₃	—	8	4	no	—
111	D₄×C₁₀₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₁₀₃	—	2	1	no	—
113	C₂×C₄×C₁₀₃	—	2	2	no	—
114	C₂×C₄×C₁₀₃	—	4	2	no	—
115	C₂×C₄×C₁₀₃	—	4	4	no	—
116	C₂×C₄×C₁₀₃	—	8	4	no	—
117	C₂×C₄×C₁₀₃	—	8	4	no	—
118	C₂×C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₀₃	—	2	2	no	—
120	C₈×C₁₀₃	—	4	2	no	—
121	C₈×C₁₀₃	C₂×D₈	1	1	yes	characteristic
122	C₈×C₁₀₃	C₂×D₈	1	1	yes	characteristic
123	C₈×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₁₀₃	—	2	2	no	—
126	C₂²×C₁₀₃	—	4	2	no	—
127	C₂²×C₁₀₃	#576	1	1	yes	characteristic
128	C₄×C₁₀₃	—	2	2	no	—
129	C₄×C₁₀₃	—	2	2	no	—
130	C₄×C₁₀₃	—	4	2	no	—
131	C₄×C₁₀₃	—	8	4	no	—
132	C₄×C₁₀₃	D₄²	1	1	yes	characteristic
133	C₄×C₁₀₃	C₂²×D₈	1	1	yes	characteristic
134	#33860	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₁₀₃	—	2	1	no	—
136	C₂×C₁₀₃	—	4	2	no	—
137	C₂×C₁₀₃	—	8	2	no	—
138	C₂×C₁₀₃	D₄×D₈	1	1	yes	characteristic
139	#3417	C₂×C₁₀₃	1	2	yes	complemented
140	#3293	C₂×C₁₀₃	1	2	yes	complemented
141	#2247	C₂×C₁₀₃	1	1	yes	characteristic, complemented
142	#2222	C₂×C₁₀₃	1	2	yes	complemented
143	#2214	C₂×C₁₀₃	1	2	yes	complemented
144	#2195	C₂×C₁₀₃	1	4	yes	complemented
145	#2178	C₂×C₁₀₃	1	1	yes	characteristic, complemented
146	#2170	C₂×C₁₀₃	1	1	yes	characteristic, complemented
147	C₁₀₃	#33860	1	1	yes	characteristic, direct product factor, Sylow
148	#578	—	2	2	no	—
149	#578	C₂²×C₁₀₃	1	4	yes	—
150	#510	—	2	4	no	—
151	#510	C₂²×C₁₀₃	1	4	yes	—
152	#508	—	2	2	no	—
153	#508	C₂²×C₁₀₃	1	1	yes	characteristic
154	C₂×Q₃₂	—	2	2	no	—
155	C₂×Q₃₂	C₂²×C₁₀₃	1	2	yes	—
156	C₂×Q₃₂	C₂²×C₁₀₃	1	2	yes	—
157	#504	C₂²×C₁₀₃	1	2	yes	—
158	#494	C₂²×C₁₀₃	1	2	yes	—
159	#473	—	2	2	no	—
160	#473	C₂²×C₁₀₃	1	4	yes	—
161	#456	—	2	4	no	—
162	#443	C₂²×C₁₀₃	1	2	yes	—
163	#443	C₂²×C₁₀₃	1	1	yes	characteristic
164	#368	C₂²×C₁₀₃	1	1	yes	characteristic
165	#364	C₂²×C₁₀₃	1	1	yes	characteristic
166	#362	C₂²×C₁₀₃	1	4	yes	—
167	#359	C₂²×C₁₀₃	1	2	yes	—
168	#350	C₂²×C₁₀₃	1	2	yes	—
169	C₄×C₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
170	2₋¹⁺⁴	—	2	4	no	—
171	#137	—	2	4	no	—
172	#137	—	2	4	no	—
173	#137	—	4	4	no	—
174	#137	—	4	4	no	—
175	#135	—	2	4	no	—
176	#135	—	4	2	no	—
177	#135	D₄×C₁₀₃	1	1	yes	characteristic
178	#135	C₂³×C₁₀₃	1	2	yes	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	4	2	no	—
182	C₂×Q₁₆	C₂³×C₁₀₃	1	4	yes	—
183	#131	—	2	2	no	—
184	#131	C₂³×C₁₀₃	1	2	yes	—
185	C₄⋊Q₈	—	2	2	no	—
186	Q₃₂	—	2	4	no	—
187	Q₃₂	—	2	2	no	—
188	Q₃₂	—	2	2	no	—
189	Q₃₂	—	4	2	no	—
190	Q₃₂	D₄×C₁₀₃	1	1	yes	characteristic
191	SD₃₂	—	2	4	no	—
192	SD₃₂	D₄×C₁₀₃	1	2	yes	—
193	D₁₆	D₄×C₁₀₃	1	1	yes	characteristic
194	M₃₂	—	2	2	no	—
195	M₃₂	C₂³×C₁₀₃	1	2	yes	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	C₂³×C₁₀₃	1	1	yes	characteristic
198	C₂×C₁₆	C₂³×C₁₀₃	1	1	yes	characteristic
199	#108	D₄×C₁₀₃	1	1	yes	characteristic
200	#108	C₂³×C₁₀₃	1	2	yes	—
201	#104	—	2	2	no	—
202	#104	D₄×C₁₀₃	1	2	yes	—
203	#101	—	2	4	no	—
204	C₄×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
205	#41	—	2	2	no	—
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	#41	—	8	4	no	—
209	#41	C₂×D₄×C₁₀₃	1	2	yes	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	4	no	—
216	Q₁₆	—	4	4	no	—
217	Q₁₆	—	4	2	no	—
218	Q₁₆	—	4	2	no	—
219	Q₁₆	D₈×C₁₀₃	1	1	yes	characteristic
220	Q₁₆	C₂×D₄×C₁₀₃	1	2	yes	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	4	4	no	—
223	SD₁₆	—	4	4	no	—
224	SD₁₆	D₈×C₁₀₃	1	2	yes	—
225	D₈	D₈×C₁₀₃	1	1	yes	characteristic
226	D₈	C₂×D₄×C₁₀₃	1	2	yes	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	C₂×D₄×C₁₀₃	1	2	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
234	C₂×C₈	C₂⁴×C₁₀₃	1	1	yes	characteristic
235	C₄⋊C₄	—	8	2	no	—
236	C₄²	C₂×D₄×C₁₀₃	1	1	yes	characteristic
237	C₁₆	—	2	2	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	C₂×D₄×C₁₀₃	1	1	yes	characteristic
240	C₁₆	C₂×D₄×C₁₀₃	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	Q₈	C₂×D₈×C₁₀₃	1	2	yes	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	4	no	—
250	D₄	C₂×D₈×C₁₀₃	1	2	yes	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₂²×D₄×C₁₀₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×D₈×C₁₀₃	1	1	yes	characteristic
261	C₈	C₂×D₈×C₁₀₃	1	1	yes	characteristic
262	C₈	C₂²×D₄×C₁₀₃	1	1	yes	characteristic
263	C₈	H(5,𝔽₂)×C₁₀₃	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	#576×C₁₀₃	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	4	no	—
271	C₄	D₄²×C₁₀₃	1	1	yes	characteristic
272	C₄	C₂²×D₈×C₁₀₃	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	—	8	2	no	—
276	C₂	D₄×D₈×C₁₀₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₀₃	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₃ ◁ C₂×C₄×C₁₀₃ ◁ #443×C₁₀₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₃×#3417	C₂	semidirect
2. C₁₀₃×#3293	C₂	semidirect
3. C₁₀₃×#2247	C₂	semidirect
4. C₁₀₃×#2222	C₂	semidirect
5. C₁₀₃×#2214	C₂	semidirect
6. C₁₀₃×#2195	C₂	semidirect
7. C₁₀₃×#2178	C₂	semidirect
8. C₁₀₃×#2170	C₂	semidirect
134. #33860	C₁₀₃	direct (N × H)
139. #3417	C₂×C₁₀₃	semidirect
140. #3293	C₂×C₁₀₃	semidirect
141. #2247	C₂×C₁₀₃	semidirect
142. #2222	C₂×C₁₀₃	semidirect
143. #2214	C₂×C₁₀₃	semidirect
144. #2195	C₂×C₁₀₃	semidirect
145. #2178	C₂×C₁₀₃	semidirect
146. #2170	C₂×C₁₀₃	semidirect
147. C₁₀₃	#33860	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×#3417	2
C₁₀₃×#3293	2
C₁₀₃×#2247	2
C₁₀₃×#2222	2
C₁₀₃×#2214	2
C₁₀₃×#2195	2
C₁₀₃×#2178	2
C₁₀₃×#2170	2
#33860	103

Subgroups — C103×#33860 #39939135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#33860 #39939135