Subgroups — C₁₁₃×#15201 #43283221

Order 28928.

1714 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#391×C₁₁₃	C₂	1	3	yes	maximal
3	C₁₁₃×#1446	C₂	1	2	yes	maximal
4	C₁₁₃×#1442	C₂	1	6	yes	maximal
5	C₁₁₃×C₄⋊#95	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₁₃	—	2	12	no	—
9	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₁₃	C₂²	1	3	yes	—
12	#391×C₁₁₃	—	2	12	no	—
13	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
14	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
15	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
16	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
17	C₂²×Q₈×C₁₁₃	—	2	6	no	—
18	C₂²×Q₈×C₁₁₃	—	2	6	no	—
19	C₂²×Q₈×C₁₁₃	—	2	6	no	—
20	C₂²×Q₈×C₁₁₃	—	2	6	no	—
21	C₂²×Q₈×C₁₁₃	Q₈	1	4	yes	complemented
22	C₂²×Q₈×C₁₁₃	Q₈	1	4	yes	complemented
23	C₂³×C₄×C₁₁₃	C₂³	1	6	yes	—
24	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
25	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₁₃	—	2	32	no	—
29	C₂×C₄⋊C₄×C₁₁₃	—	2	12	no	—
30	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
31	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
32	C₂×C₄⋊C₄×C₁₁₃	—	2	12	no	—
33	C₂×C₄²×C₁₁₃	—	2	12	no	—
34	C₂×C₄²×C₁₁₃	D₄	1	12	yes	—
35	#95×C₁₁₃	—	2	12	no	—
36	#95×C₁₁₃	—	2	12	no	—
37	#95×C₁₁₃	—	2	12	no	—
38	#95×C₁₁₃	—	4	6	no	—
39	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₁₁₃	—	2	8	no	—
41	C₂×Q₈×C₁₁₃	—	2	24	no	—
42	C₂×Q₈×C₁₁₃	—	2	24	no	—
43	C₂×Q₈×C₁₁₃	—	2	24	no	—
44	C₂×Q₈×C₁₁₃	—	4	4	no	—
45	C₂×Q₈×C₁₁₃	—	4	12	no	—
46	C₂²×C₄×C₁₁₃	—	2	6	no	—
47	C₂²×C₄×C₁₁₃	—	2	6	no	—
48	C₂²×C₄×C₁₁₃	—	2	6	no	—
49	C₂²×C₄×C₁₁₃	—	2	6	no	—
50	C₂²×C₄×C₁₁₃	—	2	6	no	—
51	C₂²×C₄×C₁₁₃	—	2	3	no	—
52	C₂²×C₄×C₁₁₃	—	2	6	no	—
53	C₂²×C₄×C₁₁₃	—	2	3	no	—
54	C₂²×C₄×C₁₁₃	—	2	6	no	—
55	C₂²×C₄×C₁₁₃	—	2	12	no	—
56	C₂²×C₄×C₁₁₃	—	2	12	no	—
57	C₂²×C₄×C₁₁₃	—	2	6	no	—
58	C₂²×C₄×C₁₁₃	—	2	6	no	—
59	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
60	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₁₁₃	C₂×Q₈	1	6	yes	—
64	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₁₃	#41	1	6	yes	—
66	C₂²×C₄×C₁₁₃	#41	1	6	yes	—
67	C₄⋊C₄×C₁₁₃	—	2	24	no	—
68	C₄⋊C₄×C₁₁₃	—	4	12	no	—
69	C₄²×C₁₁₃	—	2	24	no	—
70	C₂³×C₁₁₃	C₂²×D₄	1	3	yes	—
71	C₂³×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂³×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂³×C₁₁₃	C₂×#41	1	3	yes	—
74	C₂³×C₁₁₃	C₂×#41	1	3	yes	—
75	C₂³×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂³×C₁₁₃	2₋¹⁺⁴	1	3	yes	—
77	Q₈×C₁₁₃	—	2	32	no	—
78	Q₈×C₁₁₃	—	4	16	no	—
79	C₂×C₄×C₁₁₃	—	2	6	no	—
80	C₂×C₄×C₁₁₃	—	2	12	no	—
81	C₂×C₄×C₁₁₃	—	2	6	no	—
82	C₂×C₄×C₁₁₃	—	2	12	no	—
83	C₂×C₄×C₁₁₃	—	2	6	no	—
84	C₂×C₄×C₁₁₃	—	2	12	no	—
85	C₂×C₄×C₁₁₃	—	2	6	no	—
86	C₂×C₄×C₁₁₃	—	2	6	no	—
87	C₂×C₄×C₁₁₃	—	4	2	no	—
88	C₂×C₄×C₁₁₃	—	4	6	no	—
89	C₂×C₄×C₁₁₃	—	4	6	no	—
90	C₂×C₄×C₁₁₃	—	4	6	no	—
91	C₂×C₄×C₁₁₃	—	4	6	no	—
92	C₂×C₄×C₁₁₃	—	4	6	no	—
93	C₂×C₄×C₁₁₃	—	4	6	no	—
94	C₂×C₄×C₁₁₃	—	4	12	no	—
95	C₂×C₄×C₁₁₃	—	4	12	no	—
96	C₂×C₄×C₁₁₃	#120	1	4	yes	—
97	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	12	yes	—
98	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	12	yes	—
99	C₂×C₄×C₁₁₃	#124	1	12	yes	—
100	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
101	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	3	yes	—
102	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	3	yes	—
103	C₂²×C₁₁₃	#525	1	3	yes	—
104	C₂²×C₁₁₃	C₂×#124	1	3	yes	—
105	C₂²×C₁₁₃	#536	1	3	yes	—
106	C₂²×C₁₁₃	#537	1	3	yes	—
107	C₂²×C₁₁₃	#556	1	6	yes	—
108	C₂²×C₁₁₃	#557	1	3	yes	—
109	C₂²×C₁₁₃	#557	1	3	yes	—
110	C₂²×C₁₁₃	Q₈²	1	1	yes	characteristic
111	C₂²×C₁₁₃	#559	1	3	yes	—
112	C₄×C₁₁₃	—	2	12	no	—
113	C₄×C₁₁₃	—	4	6	no	—
114	C₄×C₁₁₃	—	4	4	no	—
115	C₄×C₁₁₃	—	8	6	no	—
116	C₄×C₁₁₃	#393	1	8	yes	—
117	#15201	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₁₁₃	C₂×#393	1	1	yes	characteristic
119	C₂×C₁₁₃	#2568	1	3	yes	—
120	C₂×C₁₁₃	#2581	1	3	yes	—
121	C₂×C₁₁₃	#2582	1	3	yes	—
122	C₂×C₁₁₃	#2592	1	3	yes	—
123	C₂×C₁₁₃	#2616	1	1	yes	characteristic
124	C₂×C₁₁₃	#2848	1	1	yes	characteristic
125	C₂²×C₄×Q₈	C₂×C₁₁₃	1	1	yes	characteristic
126	C₂×#391	C₂×C₁₁₃	1	3	yes	—
127	#1446	C₂×C₁₁₃	1	2	yes	—
128	#1442	C₂×C₁₁₃	1	6	yes	—
129	C₄⋊#95	C₂×C₁₁₃	1	3	yes	—
130	C₁₁₃	#15201	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
132	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
133	C₂×C₄×Q₈	—	2	12	no	—
134	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	3	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	3	yes	—
136	C₂²×C₄²	C₂²×C₁₁₃	1	3	yes	—
137	#391	—	2	12	no	—
138	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
139	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
140	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
141	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
142	C₂²×Q₈	—	2	6	no	—
143	C₂²×Q₈	—	2	6	no	—
144	C₂²×Q₈	—	2	6	no	—
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	Q₈×C₁₁₃	1	4	yes	complemented
147	C₂²×Q₈	Q₈×C₁₁₃	1	4	yes	complemented
148	C₂³×C₄	C₂³×C₁₁₃	1	6	yes	—
149	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
150	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
151	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
152	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
153	C₄×Q₈	—	2	32	no	—
154	C₂×C₄⋊C₄	—	2	12	no	—
155	C₂×C₄⋊C₄	—	2	6	no	—
156	C₂×C₄⋊C₄	—	2	6	no	—
157	C₂×C₄⋊C₄	—	2	12	no	—
158	C₂×C₄²	—	2	12	no	—
159	C₂×C₄²	D₄×C₁₁₃	1	12	yes	—
160	#95	—	2	12	no	—
161	#95	—	2	12	no	—
162	#95	—	2	12	no	—
163	#95	—	4	6	no	—
164	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	8	no	—
166	C₂×Q₈	—	2	24	no	—
167	C₂×Q₈	—	2	24	no	—
168	C₂×Q₈	—	2	24	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	12	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	3	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	3	no	—
179	C₂²×C₄	—	2	6	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	6	no	—
183	C₂²×C₄	—	2	6	no	—
184	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
185	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
186	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
187	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
188	C₂²×C₄	C₂×Q₈×C₁₁₃	1	6	yes	—
189	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
190	C₂²×C₄	#41×C₁₁₃	1	6	yes	—
191	C₂²×C₄	#41×C₁₁₃	1	6	yes	—
192	C₄⋊C₄	—	2	24	no	—
193	C₄⋊C₄	—	4	12	no	—
194	C₄²	—	2	24	no	—
195	C₂³	C₂²×D₄×C₁₁₃	1	3	yes	—
196	C₂³	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
197	C₂³	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
198	C₂³	C₂×#41×C₁₁₃	1	3	yes	—
199	C₂³	C₂×#41×C₁₁₃	1	3	yes	—
200	C₂³	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
201	C₂³	2₋¹⁺⁴×C₁₁₃	1	3	yes	—
202	Q₈	—	2	32	no	—
203	Q₈	—	4	16	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	12	no	—
208	C₂×C₄	—	2	6	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	6	no	—
214	C₂×C₄	—	4	6	no	—
215	C₂×C₄	—	4	6	no	—
216	C₂×C₄	—	4	6	no	—
217	C₂×C₄	—	4	6	no	—
218	C₂×C₄	—	4	6	no	—
219	C₂×C₄	—	4	12	no	—
220	C₂×C₄	—	4	12	no	—
221	C₂×C₄	#120×C₁₁₃	1	4	yes	—
222	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	12	yes	—
223	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	12	yes	—
224	C₂×C₄	#124×C₁₁₃	1	12	yes	—
225	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
226	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	3	yes	—
227	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	3	yes	—
228	C₂²	#525×C₁₁₃	1	3	yes	—
229	C₂²	C₂×#124×C₁₁₃	1	3	yes	—
230	C₂²	#536×C₁₁₃	1	3	yes	—
231	C₂²	#537×C₁₁₃	1	3	yes	—
232	C₂²	#556×C₁₁₃	1	6	yes	—
233	C₂²	#557×C₁₁₃	1	3	yes	—
234	C₂²	#557×C₁₁₃	1	3	yes	—
235	C₂²	Q₈²×C₁₁₃	1	1	yes	characteristic
236	C₂²	#559×C₁₁₃	1	3	yes	—
237	C₄	—	2	12	no	—
238	C₄	—	4	6	no	—
239	C₄	—	4	4	no	—
240	C₄	—	8	6	no	—
241	C₄	#393×C₁₁₃	1	8	yes	—
242	C₂	C₂×#393×C₁₁₃	1	1	yes	characteristic
243	C₂	C₁₁₃×#2568	1	3	yes	—
244	C₂	C₁₁₃×#2581	1	3	yes	—
245	C₂	C₁₁₃×#2582	1	3	yes	—
246	C₂	C₁₁₃×#2592	1	3	yes	—
247	C₂	C₁₁₃×#2616	1	1	yes	characteristic
248	C₂	C₁₁₃×#2848	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
21. C₂²×Q₈×C₁₁₃	Q₈	semidirect
22. C₂²×Q₈×C₁₁₃	Q₈	semidirect
117. #15201	C₁₁₃	direct (N × H)
130. C₁₁₃	#15201	direct (N × H)
146. C₂²×Q₈	Q₈×C₁₁₃	semidirect
147. C₂²×Q₈	Q₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₁₁₃	2
C₂×#391×C₁₁₃	2
C₁₁₃×#1446	2
C₁₁₃×#1442	2
C₁₁₃×C₄⋊#95	2
#15201	113

Subgroups — C113×#15201 #43283221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#15201 #43283221