Subgroups — C₂×Q₈×#11745689 #29495764

Order 15104.

7522 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11745689	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16838131	C₂	1	3	yes	maximal, complemented
3	C₂×#16838113	C₂	1	9	yes	maximal, complemented
4	C₂×Q₈×D₂₃₆	C₂	1	3	yes	maximal, complemented
5	C₂×#16837957	C₂	1	9	yes	maximal, complemented
6	C₂×C₄×#11745689	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈×D₅₉	C₂	1	3	yes	maximal, complemented
8	C₂×Q₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11745689	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×D₅₉	C₂²	1	3	yes	—
11	#16838131	C₂²	1	48	yes	complemented
12	#16838113	C₂²	1	144	yes	complemented
13	Q₈×D₂₃₆	C₂²	1	48	yes	complemented
14	#16837957	C₂²	1	144	yes	complemented
15	C₂×#14273727	C₂²	1	9	yes	—
16	C₂×#14273718	C₂²	1	18	yes	—
17	C₂×#14273707	C₂²	1	18	yes	—
18	C₂×#14273703	C₂²	1	18	yes	—
19	C₄×#11745689	C₂²	1	48	yes	—
20	C₂×Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×Dic₅₉	C₂²	1	3	yes	—
22	C₂×Q₈×Dic₅₉	C₂²	1	6	yes	—
23	C₂×C₄×Dic₁₁₈	C₂²	1	9	yes	—
24	C₂×#14273692	C₂²	1	9	yes	—
25	C₄×Q₈×D₅₉	C₂²	1	48	yes	complemented
26	C₂×C₄×D₂₃₆	C₂²	1	9	yes	—
27	C₂×#14273689	C₂²	1	9	yes	—
28	C₂×#14273685	C₂²	1	9	yes	—
29	C₂×C₄⋊C₄×D₅₉	C₂²	1	9	yes	—
30	C₂×C₄²×D₅₉	C₂²	1	9	yes	—
31	Q₈²×C₅₉	C₂²	1	16	yes	complemented
32	C₂×C₄⋊Q₈×C₅₉	C₂²	1	9	yes	—
33	C₂×C₄×Q₈×C₅₉	C₂²	1	3	yes	—
34	C₂×C₄×Q₈×C₅₉	C₂²	1	3	yes	—
35	C₂×#11745689	Q₈	1	16	yes	direct product factor
36	C₂×#11745689	C₂³	1	8	yes	—
37	C₂²×Dic₁₁₈	C₂³	1	9	yes	—
38	C₂×Q₈×D₅₉	—	2	24	no	—
39	C₂×Q₈×D₅₉	C₂³	1	24	yes	—
40	C₂²×D₂₃₆	C₂³	1	3	yes	—
41	C₂²×C₄×D₅₉	C₂³	1	9	yes	—
42	C₂²×C₄×D₅₉	C₂³	1	9	yes	—
43	C₂²×C₄×D₅₉	C₂³	1	3	yes	—
44	#14273727	C₂³	1	72	yes	—
45	#14273718	C₂³	1	144	yes	—
46	#14273707	C₂³	1	144	yes	—
47	#14273703	C₂³	1	144	yes	—
48	Q₈×Dic₅₉	C₂³	1	48	yes	—
49	Q₈×Dic₅₉	C₂³	1	8	yes	—
50	Q₈×Dic₅₉	C₂³	1	24	yes	—
51	C₄×Dic₁₁₈	C₂³	1	72	yes	—
52	#14273692	C₂³	1	72	yes	—
53	C₄×D₂₃₆	C₂³	1	72	yes	—
54	#14273689	C₂³	1	72	yes	—
55	#14273685	C₂³	1	72	yes	—
56	C₂×C₄⋊Dic₅₉	C₂³	1	18	yes	—
57	C₂×C₄⋊Dic₅₉	C₂³	1	9	yes	—
58	C₂×C₄⋊Dic₅₉	C₂³	1	3	yes	—
59	C₂×Dic₅₉⋊C₄	C₂³	1	18	yes	—
60	C₄⋊C₄×D₅₉	C₂³	1	72	yes	—
61	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	18	yes	—
62	C₂×C₄×Dic₅₉	C₂³	1	9	yes	—
63	C₂×C₄×Dic₅₉	C₂³	1	3	yes	—
64	C₂×C₄×Dic₅₉	C₂³	1	9	yes	—
65	C₄²×D₅₉	C₂³	1	72	yes	—
66	C₂²×Q₈×C₅₉	C₂³	1	6	yes	—
67	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₅₉	C₂³	1	72	yes	—
70	C₄×Q₈×C₅₉	C₂³	1	24	yes	—
71	C₄×Q₈×C₅₉	C₂³	1	24	yes	—
72	C₂×C₄⋊C₄×C₅₉	C₂³	1	9	yes	—
73	C₂×C₄⋊C₄×C₅₉	C₂³	1	9	yes	—
74	C₂×C₄²×C₅₉	C₂³	1	9	yes	—
75	C₂³×D₅₉	C₂⁴	1	3	yes	—
76	C₂×Q₈×C₅₉	—	4	12	no	—
77	C₂×Q₈×C₅₉	C₂×Q₈	1	8	yes	complemented
78	C₂×Q₈×C₅₉	#41	1	8	yes	complemented
79	C₂×Q₈×C₅₉	C₂⁴	1	24	yes	—
80	C₂×Q₈×C₅₉	C₂⁴	1	4	yes	—
81	C₂×Q₈×C₅₉	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₅₉	C₂⁴	1	9	yes	—
83	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
84	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
85	C₂²×Dic₅₉	C₂⁴	1	3	yes	—
86	C₂²×Dic₅₉	C₂⁴	1	9	yes	—
87	C₂²×Dic₅₉	C₂⁴	1	1	yes	characteristic
88	#11745689	C₂×Q₈	1	64	yes	direct product factor
89	Q₈×D₅₉	—	2	96	no	—
90	C₂×D₂₃₆	C₂×Q₈	1	24	yes	complemented
91	C₂×D₂₃₆	C₂⁴	1	12	yes	—
92	C₂×C₄×D₅₉	—	2	36	no	—
93	C₂×C₄×D₅₉	—	4	18	no	—
94	C₂×C₄×D₅₉	C₂×Q₈	1	24	yes	complemented
95	C₂×C₄×D₅₉	C₂⁴	1	36	yes	—
96	C₂×C₄×D₅₉	C₂⁴	1	36	yes	—
97	C₂×C₄×D₅₉	C₂⁴	1	12	yes	—
98	C₂×Dic₁₁₈	—	2	36	no	—
99	C₂×Dic₁₁₈	C₂⁴	1	36	yes	—
100	C₄⋊C₄×C₅₉	C₂⁴	1	36	yes	—
101	C₄⋊C₄×C₅₉	C₂⁴	1	36	yes	—
102	C₄²×C₅₉	C₂⁴	1	36	yes	—
103	(C₂×D₅₉)⋊C₄	C₂⁴	1	72	yes	—
104	C₄⋊Dic₅₉	C₂⁴	1	12	yes	—
105	C₄⋊Dic₅₉	C₂⁴	1	36	yes	—
106	C₄⋊Dic₅₉	C₂⁴	1	72	yes	—
107	Dic₅₉⋊C₄	C₂⁴	1	72	yes	—
108	C₄×Dic₅₉	C₂⁴	1	12	yes	—
109	C₄×Dic₅₉	C₂⁴	1	36	yes	—
110	C₄×Dic₅₉	C₂⁴	1	36	yes	—
111	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
112	C₂²×D₅₉	—	2	6	no	—
113	C₂²×D₅₉	—	2	6	no	—
114	C₂²×D₅₉	C₂²×Q₈	1	12	yes	—
115	C₂²×D₅₉	C₂⁵	1	6	yes	—
116	Q₈×C₅₉	—	4	24	no	—
117	Q₈×C₅₉	C₂²×Q₈	1	16	yes	complemented
118	Q₈×C₅₉	C₂×#41	1	16	yes	complemented
119	C₂×C₄×C₅₉	—	2	18	no	—
120	C₂×C₄×C₅₉	C₂²×Q₈	1	12	yes	—
121	C₂×C₄×C₅₉	C₂×#41	1	12	yes	—
122	C₂×C₄×C₅₉	C₂⁵	1	18	yes	—
123	C₂×C₄×C₅₉	C₂⁵	1	6	yes	—
124	C₂×C₄×C₅₉	C₂⁵	1	6	yes	—
125	C₂×Dic₅₉	—	2	18	no	—
126	C₂×Dic₅₉	—	2	6	no	—
127	C₂×Dic₅₉	C₂²×Q₈	1	4	yes	—
128	C₂×Dic₅₉	C₂⁵	1	6	yes	—
129	C₂×Dic₅₉	C₂⁵	1	18	yes	—
130	C₂×Dic₅₉	C₂⁵	1	2	yes	—
131	D₂₃₆	C₂²×Q₈	1	48	yes	complemented
132	C₄×D₅₉	—	2	72	no	—
133	C₄×D₅₉	—	4	36	no	—
134	C₄×D₅₉	C₂²×Q₈	1	48	yes	complemented
135	Dic₁₁₈	—	2	72	no	—
136	C₂×Q₈×#41	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	C₂³×Q₈	1	2	yes	—
138	C₂²×C₅₉	C₂²×#41	1	2	yes	—
139	C₂²×C₅₉	#585	1	2	yes	—
140	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
141	C₂×D₅₉	—	2	6	no	—
142	C₂×D₅₉	—	2	6	no	—
143	C₂×D₅₉	—	2	24	no	—
144	C₂×D₅₉	C₂³×Q₈	1	12	yes	—
145	C₄×C₅₉	—	2	18	no	—
146	C₄×C₅₉	C₂³×Q₈	1	12	yes	—
147	C₄×C₅₉	C₂²×#41	1	12	yes	—
148	Dic₅₉	—	2	18	no	—
149	Dic₅₉	—	2	6	no	—
150	Dic₅₉	C₂³×Q₈	1	4	yes	—
151	Q₈×#41	—	59	32	no	—
152	C₂×Q₈²	D₅₉	1	1	yes	characteristic, complemented
153	C₂×#557	—	59	3	no	—
154	C₂×#554	—	59	9	no	—
155	C₂×D₄×Q₈	—	59	3	no	—
156	C₂×#533	—	59	9	no	—
157	C₂×C₄×#41	—	59	3	no	—
158	C₂²×C₄×Q₈	—	59	3	no	—
159	C₂×C₅₉	Q₈×#41	1	4	yes	complemented
160	C₂×C₅₉	C₂⁴×Q₈	1	1	yes	characteristic
161	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
162	C₂×C₅₉	C₂×#585	1	1	yes	characteristic
163	D₅₉	—	2	12	no	—
164	C₂²×#41	—	59	1	no	—
165	C₂³×Q₈	—	59	3	no	—
166	Q₈²	C₂×D₅₉	1	16	yes	complemented
167	#557	—	59	48	no	—
168	#554	—	59	144	no	—
169	D₄×Q₈	—	59	48	no	—
170	#533	—	59	144	no	—
171	C₂×C₄⋊Q₈	—	59	9	no	—
172	C₂×C₄⋊Q₈	C₂×D₅₉	1	9	yes	—
173	C₂×#125	—	59	18	no	—
174	C₂×C₂²⋊Q₈	—	59	18	no	—
175	C₂×C₂²⋊Q₈	—	59	18	no	—
176	C₄×#41	—	59	48	no	—
177	C₂×C₄×Q₈	—	59	1	no	—
178	C₂×C₄×Q₈	—	59	3	no	—
179	C₂×C₄×Q₈	—	59	6	no	—
180	C₂×C₄×Q₈	—	59	48	no	—
181	C₂×C₄×Q₈	—	59	9	no	—
182	C₂×C₄×Q₈	—	59	9	no	—
183	C₂×C₄×Q₈	C₂×D₅₉	1	3	yes	—
184	C₂×C₄×Q₈	C₂×D₅₉	1	3	yes	—
185	C₂×C₄×D₄	—	59	9	no	—
186	C₂×C₄×D₄	—	59	9	no	—
187	C₂×#117	—	59	9	no	—
188	C₂²×C₄⋊C₄	—	59	9	no	—
189	C₂²×C₄²	—	59	9	no	—
190	C₅₉	C₂×Q₈×#41	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	59	16	no	—
192	C₂×#41	—	59	8	no	—
193	C₂²×Q₈	—	59	24	no	—
194	C₂²×Q₈	—	59	9	no	—
195	C₂²×Q₈	—	118	24	no	—
196	C₂²×Q₈	C₂²×D₅₉	1	6	yes	—
197	C₂²×Q₈	C₂²×D₅₉	1	1	yes	characteristic
198	C₂²×Q₈	C₂²×D₅₉	1	1	yes	characteristic
199	C₂²×D₄	—	59	3	no	—
200	C₂³×C₄	—	59	3	no	—
201	C₂³×C₄	—	59	9	no	—
202	C₂³×C₄	—	59	9	no	—
203	C₄⋊Q₈	—	59	72	no	—
204	C₄⋊Q₈	C₂²×D₅₉	1	72	yes	—
205	#125	—	59	144	no	—
206	C₂²⋊Q₈	—	59	144	no	—
207	C₂²⋊Q₈	—	59	144	no	—
208	C₄×Q₈	—	59	8	no	—
209	C₄×Q₈	—	59	24	no	—
210	C₄×Q₈	—	59	48	no	—
211	C₄×Q₈	—	59	72	no	—
212	C₄×Q₈	—	59	72	no	—
213	C₄×Q₈	C₂²×D₅₉	1	24	yes	—
214	C₄×Q₈	C₂²×D₅₉	1	24	yes	—
215	C₄×D₄	—	59	72	no	—
216	C₄×D₄	—	59	72	no	—
217	#117	—	59	72	no	—
218	C₂×C₄⋊C₄	—	59	3	no	—
219	C₂×C₄⋊C₄	—	59	18	no	—
220	C₂×C₄⋊C₄	—	59	9	no	—
221	C₂×C₄⋊C₄	—	59	18	no	—
222	C₂×C₄⋊C₄	—	59	72	no	—
223	C₂×C₄⋊C₄	C₂²×D₅₉	1	9	yes	—
224	C₂×C₄⋊C₄	C₂²×D₅₉	1	9	yes	—
225	C₂×C₂²⋊C₄	—	59	18	no	—
226	C₂×C₄²	—	59	9	no	—
227	C₂×C₄²	—	59	9	no	—
228	C₂×C₄²	—	59	3	no	—
229	C₂×C₄²	—	59	72	no	—
230	C₂×C₄²	C₂²×D₅₉	1	9	yes	—
231	C₂⁴	—	59	3	no	—
232	#41	—	59	64	no	—
233	C₂×Q₈	—	4	12	no	—
234	C₂×Q₈	—	59	36	no	—
235	C₂×Q₈	—	118	96	no	—
236	C₂×Q₈	—	118	36	no	—
237	C₂×Q₈	Q₈×D₅₉	1	8	yes	complemented
238	C₂×Q₈	#11745689	1	8	yes	direct product factor
239	C₂×Q₈	C₂³×D₅₉	1	24	yes	—
240	C₂×Q₈	C₂³×D₅₉	1	4	yes	—
241	C₂×Q₈	C₂³×D₅₉	1	4	yes	—
242	C₂×D₄	—	59	24	no	—
243	C₂×D₄	—	59	12	no	—
244	C₂²×C₄	—	59	1	no	—
245	C₂²×C₄	—	59	24	no	—
246	C₂²×C₄	—	59	12	no	—
247	C₂²×C₄	—	59	3	no	—
248	C₂²×C₄	—	59	9	no	—
249	C₂²×C₄	—	59	36	no	—
250	C₂²×C₄	—	59	36	no	—
251	C₂²×C₄	—	118	36	no	—
252	C₂²×C₄	—	236	18	no	—
253	C₂²×C₄	C₂³×D₅₉	1	9	yes	—
254	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
255	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
256	C₄⋊C₄	—	59	12	no	—
257	C₄⋊C₄	—	59	72	no	—
258	C₄⋊C₄	—	59	36	no	—
259	C₄⋊C₄	—	59	72	no	—
260	C₄⋊C₄	C₂³×D₅₉	1	36	yes	—
261	C₄⋊C₄	C₂³×D₅₉	1	36	yes	—
262	C₂²⋊C₄	—	59	72	no	—
263	C₄²	—	59	12	no	—
264	C₄²	—	59	36	no	—
265	C₄²	—	59	36	no	—
266	C₄²	C₂³×D₅₉	1	36	yes	—
267	C₂³	—	59	12	no	—
268	C₂³	—	59	6	no	—
269	C₂³	—	118	6	no	—
270	C₂³	—	118	6	no	—
271	C₂³	C₂⁴×D₅₉	1	1	yes	characteristic
272	Q₈	—	4	24	no	—
273	Q₈	—	118	72	no	—
274	Q₈	C₂×Q₈×D₅₉	1	16	yes	complemented
275	Q₈	C₂×#11745689	1	16	yes	direct product factor
276	D₄	—	59	48	no	—
277	C₂×C₄	—	2	18	no	—
278	C₂×C₄	—	59	4	no	—
279	C₂×C₄	—	59	2	no	—
280	C₂×C₄	—	59	48	no	—
281	C₂×C₄	—	59	6	no	—
282	C₂×C₄	—	59	18	no	—
283	C₂×C₄	—	118	18	no	—
284	C₂×C₄	—	118	6	no	—
285	C₂×C₄	—	118	72	no	—
286	C₂×C₄	—	236	36	no	—
287	C₂×C₄	C₂×Q₈×D₅₉	1	12	yes	—
288	C₂×C₄	C₂×#11745689	1	12	yes	—
289	C₂×C₄	C₂⁴×D₅₉	1	18	yes	—
290	C₂×C₄	C₂⁴×D₅₉	1	6	yes	—
291	C₂×C₄	C₂⁴×D₅₉	1	6	yes	—
292	C₂²	—	59	12	no	—
293	C₂²	—	118	6	no	—
294	C₂²	—	118	6	no	—
295	C₂²	—	118	24	no	—
296	C₂²	C₂²×Q₈×D₅₉	1	2	yes	—
297	C₂²	C₂²×#11745689	1	2	yes	—
298	C₂²	#16838237	1	2	yes	—
299	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
300	C₄	—	2	18	no	—
301	C₄	—	59	4	no	—
302	C₄	—	118	18	no	—
303	C₄	—	118	6	no	—
304	C₄	C₂²×Q₈×D₅₉	1	12	yes	—
305	C₄	C₂²×#11745689	1	12	yes	—
306	C₂	—	118	12	no	—
307	C₂	Q₈×#11745689	1	4	yes	direct product factor
308	C₂	C₂³×Q₈×D₅₉	1	1	yes	characteristic
309	C₂	C₂³×#11745689	1	1	yes	characteristic
310	C₂	C₂×#16838237	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₉
commutator, cosocle	C₂²×C₅₉	C₂⁶
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂×Q₈²×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁵
prime radical	C₂²×#11745689	C₂²
hypercenter	C₂×Q₈²	D₅₉
hypocenter	C₅₉	C₂×Q₈×#41
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈×#11745689	C₂	direct (N × H)
2. C₂×#16838131	C₂	semidirect
3. C₂×#16838113	C₂	semidirect
4. C₂×Q₈×D₂₃₆	C₂	semidirect
5. C₂×#16837957	C₂	semidirect
7. C₂×C₄×Q₈×D₅₉	C₂	semidirect
8. C₂×Q₈²×C₅₉	C₂	semidirect
11. #16838131	C₂²	semidirect
12. #16838113	C₂²	semidirect
13. Q₈×D₂₃₆	C₂²	semidirect
14. #16837957	C₂²	semidirect
25. C₄×Q₈×D₅₉	C₂²	semidirect
31. Q₈²×C₅₉	C₂²	semidirect
35. C₂×#11745689	Q₈	direct (N × H)
77. C₂×Q₈×C₅₉	C₂×Q₈	semidirect
78. C₂×Q₈×C₅₉	#41	semidirect
88. #11745689	C₂×Q₈	direct (N × H)
90. C₂×D₂₃₆	C₂×Q₈	semidirect
94. C₂×C₄×D₅₉	C₂×Q₈	semidirect
117. Q₈×C₅₉	C₂²×Q₈	semidirect
118. Q₈×C₅₉	C₂×#41	semidirect
131. D₂₃₆	C₂²×Q₈	semidirect
134. C₄×D₅₉	C₂²×Q₈	semidirect
152. C₂×Q₈²	D₅₉	semidirect
159. C₂×C₅₉	Q₈×#41	semidirect
166. Q₈²	C₂×D₅₉	semidirect
190. C₅₉	C₂×Q₈×#41	semidirect
237. C₂×Q₈	Q₈×D₅₉	semidirect
238. C₂×Q₈	#11745689	direct (N × H)
274. Q₈	C₂×Q₈×D₅₉	semidirect
275. Q₈	C₂×#11745689	direct (N × H)
307. C₂	Q₈×#11745689	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×#11745689	2
C₂×#16838131	2
C₂×#16838113	2
C₂×Q₈×D₂₃₆	2
C₂×#16837957	2
C₂×C₄×#11745689	2
C₂×C₄×Q₈×D₅₉	2
C₂×Q₈²×C₅₉	2
C₂×Q₈×#41	59

Subgroups — C2×Q8×#11745689 #29495764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#11745689 #29495764