Subgroups — C₂×Q₈×#11745664 #29453077

Order 15104.

2154 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11745664	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16837533	C₂	1	6	yes	maximal, complemented
3	C₂×Q₈×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×#16837426	C₂	1	3	yes	maximal
5	C₂×C₄×#11745664	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11745664	C₂²	1	3	yes	—
8	C₂²×#11745664	C₂²	1	1	yes	characteristic
9	#16837533	C₂²	1	48	yes	complemented
10	Q₈×C₅₉⋊C₈	C₂²	1	16	yes	complemented
11	#16837426	C₂²	1	24	yes	—
12	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	6	yes	—
13	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	6	yes	—
14	C₄×#11745664	C₂²	1	24	yes	—
15	C₂×#14273638	C₂²	1	6	yes	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	6	yes	—
17	C₂³×Q₈×C₅₉	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₉	C₄	1	3	yes	—
19	C₂×C₄×Q₈×C₅₉	C₄	1	8	yes	—
20	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₉	C₂²	1	3	yes	—
23	C₂×C₄×Q₈×C₅₉	C₂²	1	8	yes	—
24	C₂²×C₄⋊C₄×C₅₉	C₄	1	3	yes	—
25	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
26	C₂×#11745664	—	2	12	no	—
27	C₂×#11745664	Q₈	1	8	yes	direct product factor
28	C₂×#11745664	C₂³	1	12	yes	—
29	C₂×#11745664	C₂³	1	4	yes	—
30	C₂²×C₅₉⋊C₈	C₂³	1	6	yes	—
31	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
32	C₄⋊(C₅₉⋊C₈)	C₂³	1	24	yes	—
33	C₄⋊(C₅₉⋊C₈)	C₂³	1	24	yes	—
34	#14273638	C₂³	1	24	yes	—
35	C₄×C₅₉⋊C₈	C₂³	1	24	yes	—
36	C₂²×Q₈×C₅₉	—	2	8	no	—
37	C₂²×Q₈×C₅₉	C₂×C₄	1	8	yes	—
38	C₂²×Q₈×C₅₉	C₂×C₄	1	3	yes	—
39	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₉	C₂×C₄	1	3	yes	—
41	C₂³×C₄×C₅₉	C₂×C₄	1	3	yes	—
42	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₅₉	C₂×C₄	1	12	yes	—
44	C₄×Q₈×C₅₉	C₂×C₄	1	24	yes	—
45	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
46	C₄×Q₈×C₅₉	C₂×C₄	1	8	yes	—
47	C₄×Q₈×C₅₉	C₂³	1	4	yes	—
48	C₄×Q₈×C₅₉	C₂³	1	12	yes	—
49	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
50	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
51	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
52	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	24	yes	—
53	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
54	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
55	C₂×C₄²×C₅₉	C₂×C₄	1	12	yes	—
56	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
57	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
58	C₂×C₄²×C₅₉	C₂³	1	12	yes	—
59	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	—	2	32	no	—
61	C₂×Q₈×C₅₉	—	2	12	no	—
62	C₂×Q₈×C₅₉	M₁₆	1	8	yes	complemented
63	C₂×Q₈×C₅₉	C₂²×C₄	1	12	yes	—
64	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅₉	—	2	12	no	—
66	C₂²×C₄×C₅₉	—	4	6	no	—
67	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
71	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₅₉	#41	1	4	yes	—
77	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
78	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂²×C₄	1	24	yes	—
80	C₄⋊C₄×C₅₉	C₂²×C₄	1	12	yes	—
81	C₄⋊C₄×C₅₉	C₂²×C₄	1	12	yes	—
82	C₄²×C₅₉	C₂²×C₄	1	6	yes	—
83	C₄²×C₅₉	C₂²×C₄	1	12	yes	—
84	C₄²×C₅₉	C₂²×C₄	1	6	yes	—
85	C₄²×C₅₉	C₂²×C₄	1	12	yes	—
86	C₄²×C₅₉	C₂⁴	1	6	yes	—
87	C₄²×C₅₉	C₂⁴	1	6	yes	—
88	#11745664	—	2	24	no	—
89	#11745664	C₂×Q₈	1	16	yes	direct product factor
90	C₂×C₅₉⋊C₈	—	2	12	no	—
91	C₂×C₅₉⋊C₈	C₂×Q₈	1	8	yes	complemented
92	C₂×C₅₉⋊C₈	C₂⁴	1	12	yes	—
93	C₂×C₅₉⋊C₈	C₂⁴	1	4	yes	—
94	C₂³×C₅₉	—	2	2	no	—
95	C₂³×C₅₉	—	2	2	no	—
96	C₂³×C₅₉	C₄×Q₈	1	4	yes	—
97	C₂³×C₅₉	C₂³×C₄	1	2	yes	—
98	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
99	Q₈×C₅₉	—	2	24	no	—
100	Q₈×C₅₉	C₂×M₁₆	1	16	yes	complemented
101	C₂×C₄×C₅₉	—	2	6	no	—
102	C₂×C₄×C₅₉	—	2	6	no	—
103	C₂×C₄×C₅₉	—	2	6	no	—
104	C₂×C₄×C₅₉	—	2	24	no	—
105	C₂×C₄×C₅₉	—	4	12	no	—
106	C₂×C₄×C₅₉	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₅₉	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₅₉	C₂×M₁₆	1	12	yes	—
109	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
110	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
112	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
115	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
116	C₂×C₄×C₅₉	C₂²×Q₈	1	4	yes	—
117	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
118	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
119	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
120	C₅₉⋊C₈	—	2	12	no	—
121	C₅₉⋊C₈	C₂²×Q₈	1	8	yes	complemented
122	C₂×Q₈×M₁₆	—	59	1	no	maximal, Sylow
123	C₂²×C₅₉	—	2	8	no	—
124	C₂²×C₅₉	—	2	2	no	—
125	C₂²×C₅₉	—	2	2	no	—
126	C₂²×C₅₉	C₂×C₄×Q₈	1	4	yes	—
127	C₂²×C₅₉	C₂×C₄×Q₈	1	2	yes	—
128	C₂²×C₅₉	C₂²×M₁₆	1	2	yes	—
129	C₂²×C₅₉	#568	1	2	yes	—
130	C₂²×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
131	C₄×C₅₉	—	2	6	no	—
132	C₄×C₅₉	—	2	6	no	—
133	C₄×C₅₉	—	2	6	no	—
134	C₄×C₅₉	C₂×C₄×Q₈	1	2	yes	—
135	C₄×C₅₉	C₂×C₄×Q₈	1	4	yes	—
136	C₄×C₅₉	C₂²×M₁₆	1	12	yes	—
137	C₄×C₅₉	C₂³×Q₈	1	1	yes	characteristic
138	C₄×C₅₉	C₂²×#41	1	1	yes	characteristic
139	C₂²×C₄×Q₈	D₅₉	1	1	yes	characteristic
140	Q₈×M₁₆	—	59	16	no	—
141	C₂×#446	—	59	6	no	—
142	C₂×C₈×Q₈	—	59	2	no	—
143	C₂×#423	—	59	3	no	—
144	C₂×C₄×M₁₆	—	59	3	no	—
145	C₂×C₅₉	—	2	4	no	—
146	C₂×C₅₉	Q₈×M₁₆	1	4	yes	complemented
147	C₂×C₅₉	C₂²×C₄×Q₈	1	1	yes	characteristic
148	C₂×C₅₉	C₂³×M₁₆	1	1	yes	characteristic
149	C₂×C₅₉	C₂×#568	1	1	yes	characteristic
150	C₂³×Q₈	Dic₅₉	1	1	yes	characteristic
151	C₂²×M₁₆	—	59	1	no	—
152	C₂²×M₁₆	—	59	3	no	—
153	C₂×C₄×Q₈	Dic₅₉	1	3	yes	—
154	C₂×C₄×Q₈	Dic₅₉	1	8	yes	—
155	C₂×C₄×Q₈	Dic₅₉	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂×D₅₉	1	3	yes	—
158	C₂×C₄×Q₈	C₂×D₅₉	1	8	yes	—
159	C₂²×C₄⋊C₄	Dic₅₉	1	3	yes	—
160	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
161	#446	—	59	48	no	—
162	C₈×Q₈	—	59	16	no	—
163	#423	—	59	24	no	—
164	C₂×C₄⋊C₈	—	59	6	no	—
165	C₂×C₄⋊C₈	—	59	6	no	—
166	C₄×M₁₆	—	59	24	no	—
167	C₂×#97	—	59	6	no	—
168	C₂×C₄×C₈	—	59	6	no	—
169	C₅₉	C₂×Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	2	8	no	—
171	C₂²×Q₈	C₂×Dic₅₉	1	8	yes	—
172	C₂²×Q₈	C₂×Dic₅₉	1	3	yes	—
173	C₂²×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₅₉	1	3	yes	—
175	C₂³×C₄	C₂×Dic₅₉	1	3	yes	—
176	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
177	C₂×M₁₆	—	59	8	no	—
178	C₂×M₁₆	—	59	4	no	—
179	C₂×M₁₆	—	59	12	no	—
180	C₂×M₁₆	—	118	12	no	—
181	C₂²×C₈	—	59	2	no	—
182	C₂²×C₈	—	59	6	no	—
183	C₄×Q₈	C₂×Dic₅₉	1	12	yes	—
184	C₄×Q₈	C₂×Dic₅₉	1	24	yes	—
185	C₄×Q₈	C₂×Dic₅₉	1	4	yes	—
186	C₄×Q₈	C₂×Dic₅₉	1	8	yes	—
187	C₄×Q₈	C₂²×D₅₉	1	4	yes	—
188	C₄×Q₈	C₂²×D₅₉	1	12	yes	—
189	C₂×C₄⋊C₄	C₂×Dic₅₉	1	6	yes	—
190	C₂×C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₅₉	1	24	yes	—
193	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
194	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
195	C₂×C₄²	C₂×Dic₅₉	1	12	yes	—
196	C₂×C₄²	C₂²×D₅₉	1	12	yes	—
197	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
198	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
199	C₄⋊C₈	—	59	24	no	—
200	C₄⋊C₈	—	59	24	no	—
201	#97	—	59	24	no	—
202	C₄×C₈	—	59	24	no	—
203	C₂⁴	C₂²×Dic₅₉	1	1	yes	characteristic
204	C₂×Q₈	—	2	32	no	—
205	C₂×Q₈	—	2	12	no	—
206	C₂×Q₈	#11745664	1	8	yes	direct product factor
207	C₂×Q₈	C₂²×Dic₅₉	1	12	yes	—
208	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	6	no	—
211	C₂²×C₄	Q₈×D₅₉	1	4	yes	—
212	C₂²×C₄	#11745689	1	4	yes	—
213	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
214	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
215	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
216	C₂²×C₄	C₂²×Dic₅₉	1	12	yes	—
217	C₂²×C₄	C₂²×Dic₅₉	1	12	yes	—
218	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
221	C₂²×C₄	C₂³×D₅₉	1	2	yes	—
222	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
223	M₁₆	—	59	16	no	—
224	M₁₆	—	118	24	no	—
225	C₂×C₈	—	59	8	no	—
226	C₂×C₈	—	59	4	no	—
227	C₂×C₈	—	59	12	no	—
228	C₂×C₈	—	118	12	no	—
229	C₄⋊C₄	C₂²×Dic₅₉	1	24	yes	—
230	C₄⋊C₄	C₂²×Dic₅₉	1	12	yes	—
231	C₄⋊C₄	C₂²×Dic₅₉	1	12	yes	—
232	C₄²	C₂²×Dic₅₉	1	12	yes	—
233	C₄²	C₂²×Dic₅₉	1	6	yes	—
234	C₄²	C₂²×Dic₅₉	1	12	yes	—
235	C₄²	C₂²×Dic₅₉	1	6	yes	—
236	C₄²	C₂³×D₅₉	1	6	yes	—
237	C₄²	C₂³×D₅₉	1	6	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	Q₈×Dic₅₉	1	4	yes	—
241	C₂³	C₂³×Dic₅₉	1	2	yes	—
242	C₂³	C₂³×Dic₅₉	1	1	yes	characteristic
243	Q₈	—	2	24	no	—
244	Q₈	C₂×#11745664	1	16	yes	direct product factor
245	C₂×C₄	—	2	6	no	—
246	C₂×C₄	—	2	6	no	—
247	C₂×C₄	—	2	6	no	—
248	C₂×C₄	—	2	24	no	—
249	C₂×C₄	—	4	12	no	—
250	C₂×C₄	Q₈×Dic₅₉	1	8	yes	—
251	C₂×C₄	Q₈×Dic₅₉	1	4	yes	—
252	C₂×C₄	C₂×#11745664	1	12	yes	—
253	C₂×C₄	C₂³×Dic₅₉	1	6	yes	—
254	C₂×C₄	C₂³×Dic₅₉	1	6	yes	—
255	C₂×C₄	C₂³×Dic₅₉	1	6	yes	—
256	C₂×C₄	C₂³×Dic₅₉	1	6	yes	—
257	C₂×C₄	C₂³×Dic₅₉	1	2	yes	—
258	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
259	C₂×C₄	C₂×Q₈×D₅₉	1	2	yes	—
260	C₂×C₄	C₂×Q₈×D₅₉	1	4	yes	—
261	C₂×C₄	C₂×#11745689	1	2	yes	—
262	C₂×C₄	C₂×#11745689	1	4	yes	—
263	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
264	C₈	—	59	8	no	—
265	C₈	—	118	12	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	8	no	—
269	C₂²	C₂×Q₈×Dic₅₉	1	4	yes	—
270	C₂²	C₂×Q₈×Dic₅₉	1	2	yes	—
271	C₂²	C₂²×#11745664	1	2	yes	—
272	C₂²	#16838169	1	2	yes	—
273	C₂²	C₂⁴×Dic₅₉	1	1	yes	characteristic
274	C₄	—	2	6	no	—
275	C₄	—	2	6	no	—
276	C₄	—	2	6	no	—
277	C₄	C₂×Q₈×Dic₅₉	1	4	yes	—
278	C₄	C₂×Q₈×Dic₅₉	1	2	yes	—
279	C₄	C₂²×#11745664	1	12	yes	—
280	C₄	C₂²×Q₈×D₅₉	1	1	yes	characteristic
281	C₄	C₂²×#11745689	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	Q₈×#11745664	1	4	yes	direct product factor
284	C₂	C₂²×Q₈×Dic₅₉	1	1	yes	characteristic
285	C₂	C₂³×#11745664	1	1	yes	characteristic
286	C₂	C₂×#16838169	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₂²×C₄×Q₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂³×C₄
prime radical	C₂⁴×C₅₉	C₂²×C₄
cosocle	C₂×C₄×C₅₉	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₅₉
hypocenter	C₅₉	C₂×Q₈×M₁₆
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11745664	C₂	direct (N × H)
2. C₂×#16837533	C₂	semidirect
3. C₂×Q₈×C₅₉⋊C₈	C₂	semidirect
9. #16837533	C₂²	semidirect
10. Q₈×C₅₉⋊C₈	C₂²	semidirect
27. C₂×#11745664	Q₈	direct (N × H)
62. C₂×Q₈×C₅₉	M₁₆	semidirect
89. #11745664	C₂×Q₈	direct (N × H)
91. C₂×C₅₉⋊C₈	C₂×Q₈	semidirect
100. Q₈×C₅₉	C₂×M₁₆	semidirect
121. C₅₉⋊C₈	C₂²×Q₈	semidirect
146. C₂×C₅₉	Q₈×M₁₆	semidirect
169. C₅₉	C₂×Q₈×M₁₆	semidirect
206. C₂×Q₈	#11745664	direct (N × H)
244. Q₈	C₂×#11745664	direct (N × H)
283. C₂	Q₈×#11745664	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#11745664	2
C₂×#16837533	2
C₂×Q₈×C₅₉⋊C₈	2
C₂×#16837426	2
C₂×C₄×#11745664	2
C₂²×C₄×Q₈×C₅₉	2
C₂×Q₈×M₁₆	59

Subgroups — C2×Q8×#11745664 #29453077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#11745664 #29453077