Subgroups — C₁₁₃×#3374 #28388170

Order 14464.

326 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	#557×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	#491×C₁₁₃	C₂	1	2	yes	maximal
4	#487×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	#479×C₁₁₃	C₂	1	2	yes	maximal
6	#475×C₁₁₃	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₁₃	C₂	1	2	yes	maximal
8	#462×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	#446×C₁₁₃	C₂	1	1	yes	maximal, characteristic
10	#441×C₁₁₃	C₂	1	2	yes	maximal
11	C₄×Q₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₁₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₁₃	—	2	1	no	—
15	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
18	#125×C₁₁₃	—	2	2	no	—
19	#125×C₁₁₃	—	2	1	no	—
20	#125×C₁₁₃	—	2	2	no	—
21	#125×C₁₁₃	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₁₃	—	2	1	no	—
23	C₄×Q₈×C₁₁₃	—	2	1	no	—
24	C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
28	C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
29	#107×C₁₁₃	C₂²	1	1	yes	characteristic
30	#106×C₁₁₃	C₂²	1	2	yes	—
31	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
33	#103×C₁₁₃	C₂²	1	1	yes	characteristic
34	#103×C₁₁₃	C₂²	1	1	yes	characteristic
35	#103×C₁₁₃	C₂²	1	2	yes	—
36	#103×C₁₁₃	C₂²	1	2	yes	—
37	#103×C₁₁₃	C₂²	1	2	yes	—
38	#103×C₁₁₃	C₂²	1	2	yes	—
39	#97×C₁₁₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₁₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
46	Q₁₆×C₁₁₃	—	2	2	no	—
47	Q₁₆×C₁₁₃	—	2	4	no	—
48	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₁₃	—	2	2	no	—
52	C₄⋊C₄×C₁₁₃	—	2	2	no	—
53	C₄⋊C₄×C₁₁₃	—	2	1	no	—
54	C₄⋊C₄×C₁₁₃	—	2	1	no	—
55	C₄⋊C₄×C₁₁₃	—	2	2	no	—
56	C₄⋊C₄×C₁₁₃	—	2	1	no	—
57	C₄⋊C₄×C₁₁₃	—	2	2	no	—
58	C₄⋊C₄×C₁₁₃	—	2	2	no	—
59	C₄⋊C₄×C₁₁₃	—	2	1	no	—
60	C₄⋊C₄×C₁₁₃	—	2	1	no	—
61	C₄⋊C₄×C₁₁₃	—	2	2	no	—
62	C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₁₃	D₄	1	2	yes	—
64	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
67	C₄²×C₁₁₃	—	2	2	no	—
68	C₄²×C₁₁₃	—	2	2	no	—
69	C₄²×C₁₁₃	—	2	1	no	—
70	C₄²×C₁₁₃	—	2	1	no	—
71	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₁₃	C₂³	1	2	yes	—
73	Q₈×C₁₁₃	—	2	2	no	—
74	Q₈×C₁₁₃	—	2	1	no	—
75	Q₈×C₁₁₃	—	2	1	no	—
76	Q₈×C₁₁₃	—	2	2	no	—
77	Q₈×C₁₁₃	—	2	1	no	—
78	Q₈×C₁₁₃	—	2	2	no	—
79	Q₈×C₁₁₃	—	4	2	no	—
80	Q₈×C₁₁₃	#41	1	2	yes	—
81	C₂×C₄×C₁₁₃	—	2	1	no	—
82	C₂×C₄×C₁₁₃	—	2	1	no	—
83	C₂×C₄×C₁₁₃	—	2	1	no	—
84	C₂×C₄×C₁₁₃	—	2	2	no	—
85	C₂×C₄×C₁₁₃	—	2	1	no	—
86	C₂×C₄×C₁₁₃	—	2	2	no	—
87	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₁₃	—	2	1	no	—
93	C₈×C₁₁₃	—	2	2	no	—
94	C₈×C₁₁₃	—	2	1	no	—
95	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₁₃	—	2	2	no	—
97	C₄×C₁₁₃	—	2	2	no	—
98	C₄×C₁₁₃	—	2	2	no	—
99	C₄×C₁₁₃	—	2	1	no	—
100	C₄×C₁₁₃	—	4	2	no	—
101	C₄×C₁₁₃	—	4	1	no	—
102	C₄×C₁₁₃	—	4	2	no	—
103	C₄×C₁₁₃	—	4	1	no	—
104	C₄×C₁₁₃	—	4	1	no	—
105	C₄×C₁₁₃	#137	1	2	yes	—
106	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
107	C₄×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₁₁₃	#548	1	1	yes	characteristic
109	C₂×C₁₁₃	C₂×#137	1	1	yes	characteristic
110	C₂×C₁₁₃	#578	1	1	yes	characteristic
111	#3374	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₁₁₃	#3374	1	1	yes	characteristic, direct product factor, Sylow
113	#557	C₂×C₁₁₃	1	1	yes	characteristic
114	#557	C₂×C₁₁₃	1	1	yes	characteristic
115	#491	C₂×C₁₁₃	1	2	yes	—
116	#487	C₂×C₁₁₃	1	1	yes	characteristic
117	#479	C₂×C₁₁₃	1	2	yes	—
118	#475	C₂×C₁₁₃	1	1	yes	characteristic
119	#462	C₂×C₁₁₃	1	2	yes	—
120	#462	C₂×C₁₁₃	1	1	yes	characteristic
121	#446	C₂×C₁₁₃	1	1	yes	characteristic
122	#441	C₂×C₁₁₃	1	2	yes	—
123	C₄×Q₁₆	C₂×C₁₁₃	1	1	yes	characteristic
124	C₂×Q₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
125	C₂×Q₁₆	C₂²×C₁₁₃	1	2	yes	—
126	C₄⋊Q₈	—	2	1	no	—
127	C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
128	C₄⋊Q₈	C₂²×C₁₁₃	1	2	yes	—
129	C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
130	#125	—	2	2	no	—
131	#125	—	2	1	no	—
132	#125	—	2	2	no	—
133	#125	C₂²×C₁₁₃	1	2	yes	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	—	2	1	no	—
136	C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
137	C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
138	C₄×Q₈	C₂²×C₁₁₃	1	2	yes	—
139	C₄×Q₈	C₂²×C₁₁₃	1	2	yes	—
140	C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
141	#107	C₂²×C₁₁₃	1	1	yes	characteristic
142	#106	C₂²×C₁₁₃	1	2	yes	—
143	C₄⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
145	#103	C₂²×C₁₁₃	1	1	yes	characteristic
146	#103	C₂²×C₁₁₃	1	1	yes	characteristic
147	#103	C₂²×C₁₁₃	1	2	yes	—
148	#103	C₂²×C₁₁₃	1	2	yes	—
149	#103	C₂²×C₁₁₃	1	2	yes	—
150	#103	C₂²×C₁₁₃	1	2	yes	—
151	#97	C₂²×C₁₁₃	1	2	yes	—
152	C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	D₄×C₁₁₃	1	1	yes	characteristic
155	C₂×Q₈	C₂³×C₁₁₃	1	1	yes	characteristic
156	C₂×Q₈	C₂³×C₁₁₃	1	2	yes	—
157	C₂×Q₈	C₂³×C₁₁₃	1	1	yes	characteristic
158	Q₁₆	—	2	2	no	—
159	Q₁₆	—	2	4	no	—
160	C₂×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
162	C₂×C₈	C₂³×C₁₁₃	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
175	C₄⋊C₄	D₄×C₁₁₃	1	2	yes	—
176	C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
184	C₄²	C₂³×C₁₁₃	1	2	yes	—
185	Q₈	—	2	2	no	—
186	Q₈	—	2	1	no	—
187	Q₈	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	#41×C₁₁₃	1	2	yes	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
200	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
201	C₂×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
203	C₂×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₂²	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	4	1	no	—
217	C₄	#137×C₁₁₃	1	2	yes	—
218	C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
219	C₄	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
220	C₂	#548×C₁₁₃	1	1	yes	characteristic
221	C₂	C₂×#137×C₁₁₃	1	1	yes	characteristic
222	C₂	#578×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₄×Q₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
111. #3374	C₁₁₃	direct (N × H)
112. C₁₁₃	#3374	direct (N × H)

Maximal subgroups

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

subgroup	index
#557×C₁₁₃	2
#491×C₁₁₃	2
#487×C₁₁₃	2
#479×C₁₁₃	2
#475×C₁₁₃	2
#462×C₁₁₃	2
#446×C₁₁₃	2
#441×C₁₁₃	2
C₄×Q₁₆×C₁₁₃	2
#3374	113

Subgroups — C113×#3374 #28388170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#3374 #28388170