Subgroups — C₁₁₃×#15202 #43283222

Order 28928.

1738 conjugacy classes of subgroups.

The subgroup lattice has 246 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×#389×C₁₁₃	C₂	1	3	yes	maximal
3	C₁₁₃×#1446	C₂	1	2	yes	maximal
4	C₁₁₃×#1444	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	3	yes	—
10	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
11	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	6	yes	—
12	C₂²×C₄²×C₁₁₃	C₂²	1	3	yes	—
13	#389×C₁₁₃	—	2	12	no	—
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	Q₈	1	4	yes	—
20	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
21	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
22	C₂³×C₄×C₁₁₃	C₂³	1	6	yes	—
23	C₂³×C₄×C₁₁₃	C₂³	1	3	yes	—
24	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₄×Q₈×C₁₁₃	—	2	32	no	—
26	C₂×C₄⋊C₄×C₁₁₃	—	2	12	no	—
27	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
28	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
29	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
30	C₂×C₄⋊C₄×C₁₁₃	—	2	12	no	—
31	C₂×C₄⋊C₄×C₁₁₃	—	2	6	no	—
32	C₂×C₄⋊C₄×C₁₁₃	—	4	12	no	—
33	C₂×C₄⋊C₄×C₁₁₃	Q₈	1	12	yes	complemented
34	C₂×C₄²×C₁₁₃	—	2	12	no	—
35	C₂×C₄²×C₁₁₃	D₄	1	12	yes	—
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	C₂²×C₄×C₁₁₃	—	2	6	no	—
44	C₂²×C₄×C₁₁₃	—	2	6	no	—
45	C₂²×C₄×C₁₁₃	—	2	6	no	—
46	C₂²×C₄×C₁₁₃	—	2	6	no	—
47	C₂²×C₄×C₁₁₃	—	2	12	no	—
48	C₂²×C₄×C₁₁₃	—	2	6	no	—
49	C₂²×C₄×C₁₁₃	—	2	3	no	—
50	C₂²×C₄×C₁₁₃	—	2	12	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	2	no	—
56	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
57	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
58	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
59	C₂²×C₄×C₁₁₃	C₂×Q₈	1	12	yes	—
60	C₂²×C₄×C₁₁₃	C₂×Q₈	1	6	yes	—
61	C₂²×C₄×C₁₁₃	C₂×Q₈	1	6	yes	—
62	C₂²×C₄×C₁₁₃	#41	1	6	yes	—
63	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
64	C₄⋊C₄×C₁₁₃	—	2	24	no	—
65	C₄⋊C₄×C₁₁₃	—	2	24	no	—
66	C₄⋊C₄×C₁₁₃	—	4	24	no	—
67	C₄⋊C₄×C₁₁₃	—	4	4	no	—
68	C₄²×C₁₁₃	—	2	24	no	—
69	C₂³×C₁₁₃	C₂²×D₄	1	3	yes	—
70	C₂³×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂³×C₁₁₃	C₂²×Q₈	1	3	yes	—
72	C₂³×C₁₁₃	C₂×#41	1	1	yes	characteristic
73	C₂³×C₁₁₃	C₂×#41	1	3	yes	—
74	C₂³×C₁₁₃	H(5,𝔽₂)	1	3	yes	—
75	C₂³×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
76	Q₈×C₁₁₃	—	2	32	no	—
77	C₂×C₄×C₁₁₃	—	2	6	no	—
78	C₂×C₄×C₁₁₃	—	2	12	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	6	no	—
85	C₂×C₄×C₁₁₃	—	2	24	no	—
86	C₂×C₄×C₁₁₃	—	4	6	no	—
87	C₂×C₄×C₁₁₃	—	4	6	no	—
88	C₂×C₄×C₁₁₃	—	4	6	no	—
89	C₂×C₄×C₁₁₃	—	4	6	no	—
90	C₂×C₄×C₁₁₃	—	4	12	no	—
91	C₂×C₄×C₁₁₃	—	4	2	no	—
92	C₂×C₄×C₁₁₃	—	4	6	no	—
93	C₂×C₄×C₁₁₃	—	4	6	no	—
94	C₂×C₄×C₁₁₃	#120	1	4	yes	—
95	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	12	yes	—
96	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	12	yes	—
97	C₂×C₄×C₁₁₃	C₄⋊Q₈	1	12	yes	—
98	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
99	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	3	yes	—
100	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	3	yes	—
101	C₂²×C₁₁₃	#525	1	3	yes	—
102	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	3	yes	—
103	C₂²×C₁₁₃	#535	1	3	yes	—
104	C₂²×C₁₁₃	#537	1	3	yes	—
105	C₂²×C₁₁₃	#554	1	6	yes	—
106	C₂²×C₁₁₃	#557	1	1	yes	characteristic
107	C₂²×C₁₁₃	#557	1	3	yes	—
108	C₂²×C₁₁₃	Q₈²	1	3	yes	—
109	C₂²×C₁₁₃	#559	1	3	yes	—
110	C₄×C₁₁₃	—	2	12	no	—
111	C₄×C₁₁₃	—	4	12	no	—
112	C₄×C₁₁₃	—	4	6	no	—
113	C₄×C₁₁₃	—	8	2	no	—
114	C₄×C₁₁₃	#395	1	8	yes	—
115	#15202	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₁₁₃	C₂×#395	1	1	yes	characteristic
117	C₂×C₁₁₃	#2567	1	3	yes	—
118	C₂×C₁₁₃	#2580	1	3	yes	—
119	C₂×C₁₁₃	#2582	1	3	yes	—
120	C₂×C₁₁₃	#2584	1	3	yes	—
121	C₂×C₁₁₃	#2615	1	1	yes	characteristic
122	C₂×C₁₁₃	#2848	1	1	yes	characteristic
123	C₂²×C₄×Q₈	C₂×C₁₁₃	1	1	yes	characteristic
124	C₂×#389	C₂×C₁₁₃	1	3	yes	—
125	#1446	C₂×C₁₁₃	1	2	yes	—
126	#1444	C₂×C₁₁₃	1	6	yes	—
127	C₄⋊#95	C₂×C₁₁₃	1	3	yes	—
128	C₁₁₃	#15202	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	—	2	12	no	—
131	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	3	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	3	yes	—
134	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	6	yes	—
135	C₂²×C₄²	C₂²×C₁₁₃	1	3	yes	—
136	#389	—	2	12	no	—
137	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
138	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
139	C₂×#95	C₂²×C₁₁₃	1	6	yes	—
140	C₂²×Q₈	—	2	6	no	—
141	C₂²×Q₈	—	2	6	no	—
142	C₂²×Q₈	Q₈×C₁₁₃	1	4	yes	—
143	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
144	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
145	C₂³×C₄	C₂³×C₁₁₃	1	6	yes	—
146	C₂³×C₄	C₂³×C₁₁₃	1	3	yes	—
147	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
148	C₄×Q₈	—	2	32	no	—
149	C₂×C₄⋊C₄	—	2	12	no	—
150	C₂×C₄⋊C₄	—	2	6	no	—
151	C₂×C₄⋊C₄	—	2	6	no	—
152	C₂×C₄⋊C₄	—	2	6	no	—
153	C₂×C₄⋊C₄	—	2	12	no	—
154	C₂×C₄⋊C₄	—	2	6	no	—
155	C₂×C₄⋊C₄	—	4	12	no	—
156	C₂×C₄⋊C₄	Q₈×C₁₁₃	1	12	yes	complemented
157	C₂×C₄²	—	2	12	no	—
158	C₂×C₄²	D₄×C₁₁₃	1	12	yes	—
159	#95	—	2	12	no	—
160	#95	—	2	12	no	—
161	#95	—	4	6	no	—
162	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	2	24	no	—
165	C₂×Q₈	—	2	24	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	2	6	no	—
169	C₂²×C₄	—	2	6	no	—
170	C₂²×C₄	—	2	12	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	3	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	3	no	—
176	C₂²×C₄	—	2	6	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
180	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
181	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
182	C₂²×C₄	C₂×Q₈×C₁₁₃	1	12	yes	—
183	C₂²×C₄	C₂×Q₈×C₁₁₃	1	6	yes	—
184	C₂²×C₄	C₂×Q₈×C₁₁₃	1	6	yes	—
185	C₂²×C₄	#41×C₁₁₃	1	6	yes	—
186	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
187	C₄⋊C₄	—	2	24	no	—
188	C₄⋊C₄	—	2	24	no	—
189	C₄⋊C₄	—	4	24	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄²	—	2	24	no	—
192	C₂³	C₂²×D₄×C₁₁₃	1	3	yes	—
193	C₂³	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
194	C₂³	C₂²×Q₈×C₁₁₃	1	3	yes	—
195	C₂³	C₂×#41×C₁₁₃	1	1	yes	characteristic
196	C₂³	C₂×#41×C₁₁₃	1	3	yes	—
197	C₂³	H(5,𝔽₂)×C₁₁₃	1	3	yes	—
198	C₂³	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
199	Q₈	—	2	32	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	12	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	—	4	6	no	—
211	C₂×C₄	—	4	6	no	—
212	C₂×C₄	—	4	6	no	—
213	C₂×C₄	—	4	12	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	6	no	—
216	C₂×C₄	—	4	6	no	—
217	C₂×C₄	#120×C₁₁₃	1	4	yes	—
218	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	12	yes	—
219	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	12	yes	—
220	C₂×C₄	C₄⋊Q₈×C₁₁₃	1	12	yes	—
221	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
222	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	3	yes	—
223	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	3	yes	—
224	C₂²	#525×C₁₁₃	1	3	yes	—
225	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	3	yes	—
226	C₂²	#535×C₁₁₃	1	3	yes	—
227	C₂²	#537×C₁₁₃	1	3	yes	—
228	C₂²	#554×C₁₁₃	1	6	yes	—
229	C₂²	#557×C₁₁₃	1	1	yes	characteristic
230	C₂²	#557×C₁₁₃	1	3	yes	—
231	C₂²	Q₈²×C₁₁₃	1	3	yes	—
232	C₂²	#559×C₁₁₃	1	3	yes	—
233	C₄	—	2	12	no	—
234	C₄	—	4	12	no	—
235	C₄	—	4	6	no	—
236	C₄	—	8	2	no	—
237	C₄	#395×C₁₁₃	1	8	yes	—
238	C₂	C₂×#395×C₁₁₃	1	1	yes	characteristic
239	C₂	C₁₁₃×#2567	1	3	yes	—
240	C₂	C₁₁₃×#2580	1	3	yes	—
241	C₂	C₁₁₃×#2582	1	3	yes	—
242	C₂	C₁₁₃×#2584	1	3	yes	—
243	C₂	C₁₁₃×#2615	1	1	yes	characteristic
244	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
33. C₂×C₄⋊C₄×C₁₁₃	Q₈	semidirect
115. #15202	C₁₁₃	direct (N × H)
128. C₁₁₃	#15202	direct (N × H)
156. C₂×C₄⋊C₄	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×#389×C₁₁₃	2
C₁₁₃×#1446	2
C₁₁₃×#1444	2
C₁₁₃×C₄⋊#95	2
#15202	113

Subgroups — C113×#15202 #43283222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#15202 #43283222