Subgroups — C₁₁₃×#17975 #43285995

Order 28928.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2931	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1581	C₂	1	8	yes	maximal, complemented
4	C₁₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×#322×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#131×C₁₁₃	—	2	2	no	—
8	#532×C₁₁₃	C₂²	1	2	yes	—
9	#500×C₁₁₃	C₄	1	1	yes	characteristic
10	#493×C₁₁₃	C₄	1	1	yes	characteristic
11	#486×C₁₁₃	C₄	1	2	yes	—
12	#466×C₁₁₃	C₄	1	4	yes	—
13	C₄×D₈×C₁₁₃	C₂²	1	4	yes	—
14	#436×C₁₁₃	—	2	2	no	—
15	#436×C₁₁₃	C₂²	1	4	yes	—
16	C₈×D₄×C₁₁₃	—	2	2	no	—
17	C₈×D₄×C₁₁₃	C₂²	1	4	yes	—
18	#423×C₁₁₃	C₂²	1	2	yes	—
19	#408×C₁₁₃	—	2	2	no	—
20	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
23	#335×C₁₁₃	C₂²	1	4	yes	—
24	D₄⋊C₈×C₁₁₃	C₂²	1	8	yes	—
25	#322×C₁₁₃	C₂²	1	4	yes	—
26	C₂×#41×C₁₁₃	—	2	2	no	—
27	C₂×D₈×C₁₁₃	C₂×C₄	1	4	yes	—
28	#131×C₁₁₃	—	4	8	no	—
29	C₂×M₁₆×C₁₁₃	—	2	2	no	—
30	C₂×M₁₆×C₁₁₃	—	2	4	no	—
31	C₂²×C₈×C₁₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
37	#127×C₁₁₃	C₂×C₄	1	2	yes	—
38	#124×C₁₁₃	—	2	2	no	—
39	#121×C₁₁₃	—	2	2	no	—
40	#121×C₁₁₃	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₁₁₃	—	2	2	no	—
42	C₄×D₄×C₁₁₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
44	#107×C₁₁₃	C₂×C₄	1	4	yes	—
45	C₄⋊C₈×C₁₁₃	—	2	2	no	—
46	C₄⋊C₈×C₁₁₃	C₂³	1	4	yes	—
47	#102×C₁₁₃	C₂×C₄	1	8	yes	—
48	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
49	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
50	C₄×C₈×C₁₁₃	—	2	1	no	—
51	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁₃	D₄	1	2	yes	—
54	C₄×C₈×C₁₁₃	C₂³	1	2	yes	—
55	C₄×C₈×C₁₁₃	C₂³	1	2	yes	—
56	C₄×C₈×C₁₁₃	C₂³	1	2	yes	—
57	#41×C₁₁₃	—	4	4	no	—
58	#41×C₁₁₃	—	4	4	no	—
59	C₂×Q₈×C₁₁₃	—	2	2	no	—
60	C₂×D₄×C₁₁₃	—	2	4	no	—
61	C₂×D₄×C₁₁₃	—	2	2	no	—
62	C₂×D₄×C₁₁₃	—	2	2	no	—
63	C₂×D₄×C₁₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁₃	#41	1	1	yes	characteristic
68	D₈×C₁₁₃	—	4	4	no	—
69	M₁₆×C₁₁₃	—	4	4	no	—
70	M₁₆×C₁₁₃	—	4	8	no	—
71	C₂×C₈×C₁₁₃	—	2	2	no	—
72	C₂×C₈×C₁₁₃	—	2	4	no	—
73	C₂×C₈×C₁₁₃	—	2	2	no	—
74	C₂×C₈×C₁₁₃	—	4	2	no	—
75	C₂×C₈×C₁₁₃	—	4	2	no	—
76	C₂×C₈×C₁₁₃	—	4	8	no	—
77	C₂×C₈×C₁₁₃	D₈	1	4	yes	complemented
78	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₁₃	C₂×D₄	1	2	yes	—
82	C₄⋊C₄×C₁₁₃	—	2	2	no	—
83	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
84	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
85	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
86	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₁₃	#41	1	1	yes	characteristic
89	C₄²×C₁₁₃	C₂⁴	1	1	yes	characteristic
90	C₂³×C₁₁₃	—	2	4	no	—
91	C₂³×C₁₁₃	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₁₁₃	—	4	4	no	—
93	D₄×C₁₁₃	—	2	8	no	—
94	D₄×C₁₁₃	—	4	8	no	—
95	D₄×C₁₁₃	—	4	4	no	—
96	D₄×C₁₁₃	—	4	4	no	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	2	4	no	—
100	C₂×C₄×C₁₁₃	—	4	4	no	—
101	C₂×C₄×C₁₁₃	—	4	4	no	—
102	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₁₃	#135	1	2	yes	—
107	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
110	C₈×C₁₁₃	—	2	2	no	—
111	C₈×C₁₁₃	—	2	2	no	—
112	C₈×C₁₁₃	—	2	2	no	—
113	C₈×C₁₁₃	—	4	2	no	—
114	C₈×C₁₁₃	—	4	4	no	—
115	C₈×C₁₁₃	C₂×D₈	1	4	yes	—
116	C₂²×C₁₁₃	—	2	1	no	—
117	C₂²×C₁₁₃	—	2	1	no	—
118	C₂²×C₁₁₃	—	4	4	no	—
119	C₂²×C₁₁₃	—	4	4	no	—
120	C₂²×C₁₁₃	—	4	4	no	—
121	C₂²×C₁₁₃	C₄×D₈	1	2	yes	—
122	C₂²×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₁₁₃	—	2	1	no	—
124	C₄×C₁₁₃	—	2	1	no	—
125	C₄×C₁₁₃	—	4	4	no	—
126	C₄×C₁₁₃	C₄×D₈	1	2	yes	—
127	C₄×C₁₁₃	#444	1	2	yes	—
128	C₄×C₁₁₃	#568	1	2	yes	—
129	C₄×C₁₁₃	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₁₁₃	C₂×#135	1	1	yes	characteristic
131	#17975	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₁₃	—	2	2	no	—
133	C₂×C₁₁₃	—	8	4	no	—
134	C₂×C₁₁₃	#2932	1	1	yes	characteristic
135	C₂×C₁₁₃	C₂×C₄×D₈	1	1	yes	characteristic
136	C₂×C₁₁₃	C₂×#444	1	1	yes	characteristic
137	#3169	C₂×C₁₁₃	1	1	yes	characteristic
138	#2931	C₂×C₁₁₃	1	2	yes	complemented
139	#1581	C₂×C₁₁₃	1	8	yes	complemented
140	#1567	C₂×C₁₁₃	1	1	yes	characteristic, complemented
141	#1492	C₂×C₁₁₃	1	2	yes	complemented
142	C₂×#322	C₂×C₁₁₃	1	1	yes	characteristic, complemented
143	C₁₁₃	#17975	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#131	—	2	2	no	—
145	#532	C₂²×C₁₁₃	1	2	yes	—
146	#500	C₄×C₁₁₃	1	1	yes	characteristic
147	#493	C₄×C₁₁₃	1	1	yes	characteristic
148	#486	C₄×C₁₁₃	1	2	yes	—
149	#466	C₄×C₁₁₃	1	4	yes	—
150	C₄×D₈	C₂²×C₁₁₃	1	4	yes	—
151	#436	—	2	2	no	—
152	#436	C₂²×C₁₁₃	1	4	yes	—
153	C₈×D₄	—	2	2	no	—
154	C₈×D₄	C₂²×C₁₁₃	1	4	yes	—
155	#423	C₂²×C₁₁₃	1	2	yes	—
156	#408	—	2	2	no	—
157	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
160	#335	C₂²×C₁₁₃	1	4	yes	—
161	D₄⋊C₈	C₂²×C₁₁₃	1	8	yes	—
162	#322	C₂²×C₁₁₃	1	4	yes	—
163	C₂×#41	—	2	2	no	—
164	C₂×D₈	C₂×C₄×C₁₁₃	1	4	yes	—
165	#131	—	4	8	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	C₂×C₄×C₁₁₃	1	2	yes	—
171	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
174	#127	C₂×C₄×C₁₁₃	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×C₄×C₁₁₃	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂³×C₁₁₃	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
181	#107	C₂×C₄×C₁₁₃	1	4	yes	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂³×C₁₁₃	1	4	yes	—
184	#102	C₂×C₄×C₁₁₃	1	8	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
190	C₄×C₈	D₄×C₁₁₃	1	2	yes	—
191	C₄×C₈	C₂³×C₁₁₃	1	2	yes	—
192	C₄×C₈	C₂³×C₁₁₃	1	2	yes	—
193	C₄×C₈	C₂³×C₁₁₃	1	2	yes	—
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂²×C₄×C₁₁₃	1	4	yes	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
204	C₂²×C₄	#41×C₁₁₃	1	1	yes	characteristic
205	D₈	—	4	4	no	—
206	M₁₆	—	4	4	no	—
207	M₁₆	—	4	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	D₈×C₁₁₃	1	4	yes	complemented
215	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₁₃	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₁₁₃	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
225	C₄²	#41×C₁₁₃	1	1	yes	characteristic
226	C₄²	C₂⁴×C₁₁₃	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	C₄×D₄×C₁₁₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
240	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₁₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₈×C₁₁₃	1	2	yes	—
243	C₂×C₄	#135×C₁₁₃	1	2	yes	—
244	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₈	C₂×D₈×C₁₁₃	1	4	yes	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₄×D₈×C₁₁₃	1	2	yes	—
259	C₂²	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	C₄×D₈×C₁₁₃	1	2	yes	—
264	C₄	#444×C₁₁₃	1	2	yes	—
265	C₄	#568×C₁₁₃	1	2	yes	—
266	C₄	C₂²×D₈×C₁₁₃	1	1	yes	characteristic
267	C₄	C₂×#135×C₁₁₃	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₁₁₃×#2932	1	1	yes	characteristic
271	C₂	C₂×C₄×D₈×C₁₁₃	1	1	yes	characteristic
272	C₂	C₂×#444×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₁₃
socle	C₂²×C₁₁₃	C₂×C₄×D₄
prime radical	C₁₁₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ C₂×C₄×C₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2931	C₂	semidirect
3. C₁₁₃×#1581	C₂	semidirect
4. C₁₁₃×#1567	C₂	semidirect
5. C₁₁₃×#1492	C₂	semidirect
6. C₂×#322×C₁₁₃	C₂	semidirect
77. C₂×C₈×C₁₁₃	D₈	semidirect
131. #17975	C₁₁₃	direct (N × H)
138. #2931	C₂×C₁₁₃	semidirect
139. #1581	C₂×C₁₁₃	semidirect
140. #1567	C₂×C₁₁₃	semidirect
141. #1492	C₂×C₁₁₃	semidirect
142. C₂×#322	C₂×C₁₁₃	semidirect
143. C₁₁₃	#17975	direct (N × H)
214. C₂×C₈	D₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₁₃×#3169	2
C₁₁₃×#2931	2
C₁₁₃×#1581	2
C₁₁₃×#1567	2
C₁₁₃×#1492	2
C₂×#322×C₁₁₃	2
#17975	113

Subgroups — C113×#17975 #43285995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#17975 #43285995