Subgroups — C₁₁₃×#15158 #43283178

Order 28928.

1778 conjugacy classes of subgroups.

The subgroup lattice has 258 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×#395×C₁₁₃	C₂	1	2	yes	maximal
3	C₂×#389×C₁₁₃	C₂	1	3	yes	maximal
4	C₁₁₃×#1441	C₂	1	6	yes	maximal
5	C₁₁₃×C₄⋊#95	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₁₃	—	2	12	no	—
8	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	6	yes	—
10	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
11	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
12	C₂²×C₄²×C₁₁₃	C₂²	1	3	yes	—
13	#395×C₁₁₃	—	2	8	no	—
14	#389×C₁₁₃	—	2	12	no	—
15	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
16	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
17	C₂×#95×C₁₁₃	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₁₃	—	2	6	no	—
19	C₂²×Q₈×C₁₁₃	—	2	6	no	—
20	C₂²×Q₈×C₁₁₃	D₄	1	4	yes	—
21	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
22	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
23	C₂³×C₄×C₁₁₃	C₂³	1	6	yes	—
24	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
25	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₁₁₃	—	2	32	no	—
27	C₂×C₄⋊C₄×C₁₁₃	—	2	12	no	—
28	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
29	C₂×C₄⋊C₄×C₁₁₃	—	2	6	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×C₄⋊C₄×C₁₁₃	—	2	12	no	—
34	C₂×C₄⋊C₄×C₁₁₃	—	2	12	no	—
35	C₂×C₄⋊C₄×C₁₁₃	Q₈	1	12	yes	complemented
36	C₂×C₄²×C₁₁₃	—	2	12	no	—
37	C₂×C₄²×C₁₁₃	D₄	1	12	yes	—
38	#95×C₁₁₃	—	2	12	no	—
39	#95×C₁₁₃	—	2	12	no	—
40	#95×C₁₁₃	—	4	6	no	—
41	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈×C₁₁₃	—	2	8	no	—
43	C₂×Q₈×C₁₁₃	—	2	24	no	—
44	C₂×Q₈×C₁₁₃	—	2	24	no	—
45	C₂²×C₄×C₁₁₃	—	2	6	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	6	no	—
52	C₂²×C₄×C₁₁₃	—	2	3	no	—
53	C₂²×C₄×C₁₁₃	—	2	6	no	—
54	C₂²×C₄×C₁₁₃	—	2	6	no	—
55	C₂²×C₄×C₁₁₃	—	2	3	no	—
56	C₂²×C₄×C₁₁₃	—	2	6	no	—
57	C₂²×C₄×C₁₁₃	—	2	6	no	—
58	C₂²×C₄×C₁₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₁₃	—	2	6	no	—
60	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
63	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₁₁₃	C₂×Q₈	1	12	yes	—
65	C₂²×C₄×C₁₁₃	C₂×Q₈	1	6	yes	—
66	C₂²×C₄×C₁₁₃	#41	1	6	yes	—
67	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
68	C₄⋊C₄×C₁₁₃	—	2	24	no	—
69	C₄⋊C₄×C₁₁₃	—	2	24	no	—
70	C₄⋊C₄×C₁₁₃	—	4	4	no	—
71	C₄⋊C₄×C₁₁₃	—	4	24	no	—
72	C₄²×C₁₁₃	—	2	24	no	—
73	C₂³×C₁₁₃	C₂²×D₄	1	3	yes	—
74	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
75	C₂³×C₁₁₃	C₂²×Q₈	1	3	yes	—
76	C₂³×C₁₁₃	C₂×#41	1	3	yes	—
77	C₂³×C₁₁₃	C₂×#41	1	1	yes	characteristic
78	C₂³×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
79	C₂³×C₁₁₃	2₋¹⁺⁴	1	3	yes	—
80	Q₈×C₁₁₃	—	2	32	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	12	no	—
87	C₂×C₄×C₁₁₃	—	2	6	no	—
88	C₂×C₄×C₁₁₃	—	2	6	no	—
89	C₂×C₄×C₁₁₃	—	2	12	no	—
90	C₂×C₄×C₁₁₃	—	2	12	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	6	no	—
95	C₂×C₄×C₁₁₃	—	4	6	no	—
96	C₂×C₄×C₁₁₃	—	4	6	no	—
97	C₂×C₄×C₁₁₃	—	4	2	no	—
98	C₂×C₄×C₁₁₃	—	4	6	no	—
99	C₂×C₄×C₁₁₃	—	4	6	no	—
100	C₂×C₄×C₁₁₃	#120	1	4	yes	—
101	C₂×C₄×C₁₁₃	#121	1	12	yes	—
102	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	12	yes	—
103	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	12	yes	—
104	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
105	C₂²×C₁₁₃	C₂×#121	1	3	yes	—
106	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	3	yes	—
107	C₂²×C₁₁₃	#525	1	3	yes	—
108	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	3	yes	—
109	C₂²×C₁₁₃	#536	1	3	yes	—
110	C₂²×C₁₁₃	#537	1	3	yes	—
111	C₂²×C₁₁₃	D₄×Q₈	1	3	yes	—
112	C₂²×C₁₁₃	#550	1	1	yes	characteristic
113	C₂²×C₁₁₃	#554	1	3	yes	—
114	C₂²×C₁₁₃	#556	1	3	yes	—
115	C₂²×C₁₁₃	#557	1	6	yes	—
116	C₄×C₁₁₃	—	2	12	no	—
117	C₄×C₁₁₃	—	4	6	no	—
118	C₄×C₁₁₃	—	4	12	no	—
119	C₄×C₁₁₃	—	8	2	no	—
120	C₄×C₁₁₃	#395	1	8	yes	complemented
121	#15158	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₁₁₃	C₂×#395	1	1	yes	characteristic
123	C₂×C₁₁₃	#2538	1	3	yes	—
124	C₂×C₁₁₃	#2546	1	3	yes	—
125	C₂×C₁₁₃	#2581	1	3	yes	—
126	C₂×C₁₁₃	#2584	1	3	yes	—
127	C₂×C₁₁₃	#2616	1	1	yes	characteristic
128	C₂×C₁₁₃	#2847	1	1	yes	characteristic
129	C₂²×C₄×Q₈	C₂×C₁₁₃	1	1	yes	characteristic
130	C₂×#395	C₂×C₁₁₃	1	2	yes	—
131	C₂×#389	C₂×C₁₁₃	1	3	yes	—
132	#1441	C₂×C₁₁₃	1	6	yes	—
133	C₄⋊#95	C₂×C₁₁₃	1	3	yes	—
134	C₁₁₃	#15158	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
136	C₂×C₄×Q₈	—	2	12	no	—
137	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	6	yes	—
138	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	3	yes	—
139	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	3	yes	—
141	C₂²×C₄²	C₂²×C₁₁₃	1	3	yes	—
142	#395	—	2	8	no	—
143	#389	—	2	12	no	—
144	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
145	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
146	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
147	C₂²×Q₈	—	2	6	no	—
148	C₂²×Q₈	—	2	6	no	—
149	C₂²×Q₈	D₄×C₁₁₃	1	4	yes	—
150	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
151	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
152	C₂³×C₄	C₂³×C₁₁₃	1	6	yes	—
153	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
154	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
155	C₄×Q₈	—	2	32	no	—
156	C₂×C₄⋊C₄	—	2	12	no	—
157	C₂×C₄⋊C₄	—	2	6	no	—
158	C₂×C₄⋊C₄	—	2	6	no	—
159	C₂×C₄⋊C₄	—	2	6	no	—
160	C₂×C₄⋊C₄	—	2	6	no	—
161	C₂×C₄⋊C₄	—	2	12	no	—
162	C₂×C₄⋊C₄	—	2	12	no	—
163	C₂×C₄⋊C₄	—	2	12	no	—
164	C₂×C₄⋊C₄	Q₈×C₁₁₃	1	12	yes	complemented
165	C₂×C₄²	—	2	12	no	—
166	C₂×C₄²	D₄×C₁₁₃	1	12	yes	—
167	#95	—	2	12	no	—
168	#95	—	2	12	no	—
169	#95	—	4	6	no	—
170	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	—	2	24	no	—
173	C₂×Q₈	—	2	24	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	6	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	—	2	6	no	—
180	C₂²×C₄	—	2	6	no	—
181	C₂²×C₄	—	2	3	no	—
182	C₂²×C₄	—	2	6	no	—
183	C₂²×C₄	—	2	6	no	—
184	C₂²×C₄	—	2	3	no	—
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	—	2	6	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	6	no	—
189	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
190	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
191	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₁₃	1	12	yes	—
194	C₂²×C₄	C₂×Q₈×C₁₁₃	1	6	yes	—
195	C₂²×C₄	#41×C₁₁₃	1	6	yes	—
196	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
197	C₄⋊C₄	—	2	24	no	—
198	C₄⋊C₄	—	2	24	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	24	no	—
201	C₄²	—	2	24	no	—
202	C₂³	C₂²×D₄×C₁₁₃	1	3	yes	—
203	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
204	C₂³	C₂²×Q₈×C₁₁₃	1	3	yes	—
205	C₂³	C₂×#41×C₁₁₃	1	3	yes	—
206	C₂³	C₂×#41×C₁₁₃	1	1	yes	characteristic
207	C₂³	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
208	C₂³	2₋¹⁺⁴×C₁₁₃	1	3	yes	—
209	Q₈	—	2	32	no	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	2	6	no	—
215	C₂×C₄	—	2	12	no	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	—	2	6	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	12	no	—
220	C₂×C₄	—	4	6	no	—
221	C₂×C₄	—	4	6	no	—
222	C₂×C₄	—	4	6	no	—
223	C₂×C₄	—	4	6	no	—
224	C₂×C₄	—	4	6	no	—
225	C₂×C₄	—	4	6	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	6	no	—
228	C₂×C₄	—	4	6	no	—
229	C₂×C₄	#120×C₁₁₃	1	4	yes	—
230	C₂×C₄	#121×C₁₁₃	1	12	yes	—
231	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	12	yes	—
232	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	12	yes	—
233	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
234	C₂²	C₂×#121×C₁₁₃	1	3	yes	—
235	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	3	yes	—
236	C₂²	#525×C₁₁₃	1	3	yes	—
237	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	3	yes	—
238	C₂²	#536×C₁₁₃	1	3	yes	—
239	C₂²	#537×C₁₁₃	1	3	yes	—
240	C₂²	D₄×Q₈×C₁₁₃	1	3	yes	—
241	C₂²	#550×C₁₁₃	1	1	yes	characteristic
242	C₂²	#554×C₁₁₃	1	3	yes	—
243	C₂²	#556×C₁₁₃	1	3	yes	—
244	C₂²	#557×C₁₁₃	1	6	yes	—
245	C₄	—	2	12	no	—
246	C₄	—	4	12	no	—
247	C₄	—	4	6	no	—
248	C₄	—	8	2	no	—
249	C₄	#395×C₁₁₃	1	8	yes	complemented
250	C₂	C₂×#395×C₁₁₃	1	1	yes	characteristic
251	C₂	C₁₁₃×#2538	1	3	yes	—
252	C₂	C₁₁₃×#2546	1	3	yes	—
253	C₂	C₁₁₃×#2581	1	3	yes	—
254	C₂	C₁₁₃×#2584	1	3	yes	—
255	C₂	C₁₁₃×#2616	1	1	yes	characteristic
256	C₂	C₁₁₃×#2847	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
35. C₂×C₄⋊C₄×C₁₁₃	Q₈	semidirect
120. C₄×C₁₁₃	#395	semidirect
121. #15158	C₁₁₃	direct (N × H)
134. C₁₁₃	#15158	direct (N × H)
164. C₂×C₄⋊C₄	Q₈×C₁₁₃	semidirect
249. C₄	#395×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×#395×C₁₁₃	2
C₂×#389×C₁₁₃	2
C₁₁₃×#1441	2
C₁₁₃×C₄⋊#95	2
#15158	113

Subgroups — C113×#15158 #43283178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#15158 #43283178