Subgroups — C₁₁₃×#14274 #43282294

Order 28928.

1742 conjugacy classes of subgroups.

The subgroup lattice has 252 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	4	yes	maximal
3	C₂×#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#376×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₁₁₃×#1445	C₂	1	8	yes	maximal
6	C₂³×Q₈×C₁₁₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₁₁₃	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₁₁₃	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₁₁₃	C₄	1	8	yes	complemented
10	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₁₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
14	#391×C₁₁₃	—	2	16	no	—
15	#382×C₁₁₃	—	2	4	no	—
16	#376×C₁₁₃	—	2	4	no	—
17	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
18	C₂×#95×C₁₁₃	C₂²	1	16	yes	—
19	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
20	C₂²×Q₈×C₁₁₃	—	2	4	no	—
21	C₂²×Q₈×C₁₁₃	—	2	4	no	—
22	C₂²×Q₈×C₁₁₃	—	2	8	no	—
23	C₂²×Q₈×C₁₁₃	—	2	8	no	—
24	C₂²×Q₈×C₁₁₃	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₁₃	—	2	32	no	—
31	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
32	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	16	yes	—
36	C₂×C₄²×C₁₁₃	—	2	4	no	—
37	C₂×C₄²×C₁₁₃	—	2	4	no	—
38	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₁₃	D₄	1	8	yes	—
40	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
41	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
42	#95×C₁₁₃	—	2	8	no	—
43	#95×C₁₁₃	—	2	8	no	—
44	#95×C₁₁₃	—	2	32	no	—
45	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
46	C₂×Q₈×C₁₁₃	—	2	16	no	—
47	C₂×Q₈×C₁₁₃	—	2	32	no	—
48	C₂×Q₈×C₁₁₃	—	4	16	no	—
49	C₂×Q₈×C₁₁₃	—	4	8	no	—
50	C₂×Q₈×C₁₁₃	—	4	8	no	—
51	C₂²×C₄×C₁₁₃	—	2	8	no	—
52	C₂²×C₄×C₁₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₁₃	—	2	16	no	—
59	C₂²×C₄×C₁₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₁₃	—	2	8	no	—
61	C₂²×C₄×C₁₁₃	—	2	16	no	—
62	C₂²×C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
72	C₄⋊C₄×C₁₁₃	—	2	32	no	—
73	C₄⋊C₄×C₁₁₃	—	4	8	no	—
74	C₄²×C₁₁₃	—	2	16	no	—
75	C₄²×C₁₁₃	C₂²⋊C₄	1	16	yes	—
76	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
77	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
78	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
79	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
80	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
81	C₂³×C₁₁₃	2₋¹⁺⁴	1	8	yes	—
82	Q₈×C₁₁₃	—	4	32	no	—
83	C₂×C₄×C₁₁₃	—	2	8	no	—
84	C₂×C₄×C₁₁₃	—	2	8	no	—
85	C₂×C₄×C₁₁₃	—	2	8	no	—
86	C₂×C₄×C₁₁₃	—	2	8	no	—
87	C₂×C₄×C₁₁₃	—	2	16	no	—
88	C₂×C₄×C₁₁₃	—	2	8	no	—
89	C₂×C₄×C₁₁₃	—	2	16	no	—
90	C₂×C₄×C₁₁₃	—	4	8	no	—
91	C₂×C₄×C₁₁₃	—	4	4	no	—
92	C₂×C₄×C₁₁₃	—	4	4	no	—
93	C₂×C₄×C₁₁₃	—	4	8	no	—
94	C₂×C₄×C₁₁₃	—	4	8	no	—
95	C₂×C₄×C₁₁₃	—	4	4	no	—
96	C₂×C₄×C₁₁₃	—	4	8	no	—
97	C₂×C₄×C₁₁₃	—	4	16	no	—
98	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
99	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₁₃	C₄×D₄	1	8	yes	—
101	C₂×C₄×C₁₁₃	#121	1	8	yes	—
102	C₂×C₄×C₁₁₃	#124	1	4	yes	—
103	C₂×C₄×C₁₁₃	#127	1	4	yes	—
104	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₁₃	C₂×C₄×D₄	1	2	yes	—
106	C₂²×C₁₁₃	#519	1	4	yes	—
107	C₂²×C₁₁₃	C₂×#121	1	2	yes	—
108	C₂²×C₁₁₃	C₂×#124	1	1	yes	characteristic
109	C₂²×C₁₁₃	C₂×#127	1	1	yes	characteristic
110	C₂²×C₁₁₃	#536	1	4	yes	—
111	C₂²×C₁₁₃	#536	1	4	yes	—
112	C₂²×C₁₁₃	#536	1	8	yes	—
113	C₂²×C₁₁₃	#541	1	8	yes	—
114	C₄×C₁₁₃	—	2	8	no	—
115	C₄×C₁₁₃	—	4	8	no	—
116	C₄×C₁₁₃	—	4	8	no	—
117	C₄×C₁₁₃	—	8	4	no	—
118	C₄×C₁₁₃	#390	1	8	yes	—
119	#14274	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₁₁₃	C₂×#390	1	1	yes	characteristic
121	C₂×C₁₁₃	#2310	1	2	yes	—
122	C₂×C₁₁₃	#2327	1	4	yes	—
123	C₂×C₁₁₃	#2621	1	4	yes	—
124	C₂×C₁₁₃	#2642	1	2	yes	—
125	C₂×C₁₁₃	#2658	1	2	yes	—
126	C₂²×C₄⋊Q₈	C₂×C₁₁₃	1	1	yes	characteristic
127	C₂×#391	C₂×C₁₁₃	1	4	yes	—
128	C₂×#382	C₂×C₁₁₃	1	1	yes	characteristic
129	C₂×#376	C₂×C₁₁₃	1	1	yes	characteristic
130	#1445	C₂×C₁₁₃	1	8	yes	—
131	C₁₁₃	#14274	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×Q₈	C₂²×C₁₁₃	1	2	yes	—
133	C₂×C₄⋊Q₈	—	2	4	no	—
134	C₂×C₄⋊Q₈	—	2	4	no	—
135	C₂×C₄⋊Q₈	C₄×C₁₁₃	1	8	yes	complemented
136	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
137	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	4	yes	—
138	C₂²×C₄²	C₂²×C₁₁₃	1	2	yes	—
139	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
140	#391	—	2	16	no	—
141	#382	—	2	4	no	—
142	#376	—	2	4	no	—
143	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
144	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
145	C₂×#95	C₂²×C₁₁₃	1	16	yes	—
146	C₂²×Q₈	—	2	4	no	—
147	C₂²×Q₈	—	2	4	no	—
148	C₂²×Q₈	—	2	8	no	—
149	C₂²×Q₈	—	2	8	no	—
150	C₂²×Q₈	C₂×C₄×C₁₁₃	1	8	yes	—
151	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
152	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
153	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
154	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
155	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	32	no	—
157	C₂×C₄⋊C₄	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	8	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	8	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	16	yes	—
162	C₂×C₄²	—	2	4	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
165	C₂×C₄²	D₄×C₁₁₃	1	8	yes	—
166	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
167	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
168	#95	—	2	8	no	—
169	#95	—	2	8	no	—
170	#95	—	2	32	no	—
171	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
172	C₂×Q₈	—	2	16	no	—
173	C₂×Q₈	—	2	32	no	—
174	C₂×Q₈	—	4	16	no	—
175	C₂×Q₈	—	4	8	no	—
176	C₂×Q₈	—	4	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	C₂²×C₄×C₁₁₃	1	8	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
193	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
194	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
197	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
198	C₄⋊C₄	—	2	32	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₄²	—	2	16	no	—
201	C₄²	C₂²⋊C₄×C₁₁₃	1	16	yes	—
202	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
203	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
204	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
205	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
206	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
207	C₂³	2₋¹⁺⁴×C₁₁₃	1	8	yes	—
208	Q₈	—	4	32	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	16	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
226	C₂×C₄	C₄×D₄×C₁₁₃	1	8	yes	—
227	C₂×C₄	#121×C₁₁₃	1	8	yes	—
228	C₂×C₄	#124×C₁₁₃	1	4	yes	—
229	C₂×C₄	#127×C₁₁₃	1	4	yes	—
230	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
231	C₂²	C₂×C₄×D₄×C₁₁₃	1	2	yes	—
232	C₂²	#519×C₁₁₃	1	4	yes	—
233	C₂²	C₂×#121×C₁₁₃	1	2	yes	—
234	C₂²	C₂×#124×C₁₁₃	1	1	yes	characteristic
235	C₂²	C₂×#127×C₁₁₃	1	1	yes	characteristic
236	C₂²	#536×C₁₁₃	1	4	yes	—
237	C₂²	#536×C₁₁₃	1	4	yes	—
238	C₂²	#536×C₁₁₃	1	8	yes	—
239	C₂²	#541×C₁₁₃	1	8	yes	—
240	C₄	—	2	8	no	—
241	C₄	—	4	8	no	—
242	C₄	—	4	8	no	—
243	C₄	—	8	4	no	—
244	C₄	#390×C₁₁₃	1	8	yes	—
245	C₂	C₂×#390×C₁₁₃	1	1	yes	characteristic
246	C₂	C₁₁₃×#2310	1	2	yes	—
247	C₂	C₁₁₃×#2327	1	4	yes	—
248	C₂	C₁₁₃×#2621	1	4	yes	—
249	C₂	C₁₁₃×#2642	1	2	yes	—
250	C₂	C₁₁₃×#2658	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₁₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Q₈×C₁₁₃	C₄	semidirect
119. #14274	C₁₁₃	direct (N × H)
131. C₁₁₃	#14274	direct (N × H)
135. C₂×C₄⋊Q₈	C₄×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
C₂×#382×C₁₁₃	2
C₂×#376×C₁₁₃	2
C₁₁₃×#1445	2
#14274	113

Subgroups — C113×#14274 #43282294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#14274 #43282294