Subgroups — C₂×#11756490 #14481204

Order 1920.

2008 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91835	C₂	1	2	yes	maximal
3	C₂×Q₈×Dic₁₅	C₂	1	2	yes	maximal, complemented
4	#11756490	C₂	1	16	yes	maximal, direct product factor
5	C₂×#91787	C₂	1	4	yes	maximal
6	C₂×#91778	C₂	1	4	yes	maximal, complemented
7	C₂×#91761	C₂	1	1	yes	maximal, characteristic
8	C₂×#91759	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66065	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	#91835	C₂²	1	16	yes	—
16	Q₈×Dic₁₅	C₂²	1	16	yes	complemented
17	C₂×#6714	C₂²	1	4	yes	—
18	C₂×#6714	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₅	C₂²	1	4	yes	—
21	C₂×#6712	C₂²	1	4	yes	—
22	C₂×#6712	C₂²	1	4	yes	—
23	C₂×#6712	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₅	C₂²	1	4	yes	—
26	#91787	C₂²	1	32	yes	—
27	#91778	C₂²	1	32	yes	complemented
28	#91761	C₂²	1	8	yes	—
29	#91759	C₂²	1	8	yes	complemented
30	C₂×#5731	—	5	1	no	maximal, Hall
31	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
32	C₂×#4092	—	3	2	no	—
33	C₂×Q₈×Dic₅	—	3	2	no	—
34	#66065	—	3	16	no	—
35	C₂×#4044	—	3	4	no	—
36	C₂×#4035	—	3	4	no	—
37	C₂×#4018	—	3	1	no	—
38	C₂×#4016	—	3	1	no	—
39	C₂³×D₁₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₅×Q₈	—	2	8	no	—
41	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
43	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
44	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₅	C₂³	1	2	yes	—
46	C₂²×Dic₁₅	C₂³	1	4	yes	—
47	C₂²×Dic₁₅	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₅	—	2	4	no	—
49	C₂×C₄×D₁₅	C₂³	1	4	yes	—
50	C₃×C₅×C₄⋊C₄	C₂³	1	16	yes	—
51	C₃×C₄²×C₅	C₂³	1	4	yes	—
52	#6714	C₂³	1	8	yes	—
53	#6714	C₂³	1	16	yes	—
54	C₄⋊Dic₁₅	C₂³	1	8	yes	—
55	C₄⋊Dic₁₅	C₂³	1	16	yes	—
56	#6712	C₂³	1	16	yes	—
57	#6712	C₂³	1	8	yes	—
58	#6712	C₂³	1	16	yes	—
59	C₄×Dic₁₅	C₂³	1	4	yes	—
60	C₄×Dic₁₅	C₂³	1	16	yes	—
61	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
62	C₂×#946	—	5	2	no	—
63	C₂×Q₈×Dic₃	—	5	2	no	—
64	#5731	—	5	16	no	—
65	C₂×#898	—	5	4	no	—
66	C₂×#889	—	5	4	no	—
67	C₂×#872	—	5	1	no	—
68	C₂×#870	—	5	1	no	—
69	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
70	C₂²×C₄×D₅	—	3	1	no	—
71	C₅×C₄⋊Q₈	C₂×D₃	1	8	yes	complemented
72	C₂×C₅×C₄⋊C₄	C₂×D₃	1	4	yes	—
73	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	#4092	—	3	16	no	—
75	Q₈×Dic₅	—	3	16	no	—
76	C₂×#685	—	3	2	no	—
77	C₂×#685	—	3	4	no	—
78	C₂×C₄⋊Dic₅	—	3	2	no	—
79	C₂×C₄⋊Dic₅	—	3	4	no	—
80	C₂×#683	—	3	4	no	—
81	C₂×#683	—	3	2	no	—
82	C₂×#683	—	3	4	no	—
83	C₂×C₄×Dic₅	—	3	1	no	—
84	C₂×C₄×Dic₅	—	3	4	no	—
85	#4044	—	3	32	no	—
86	#4035	—	3	32	no	—
87	#4018	—	3	8	no	—
88	#4016	—	3	8	no	—
89	C₂×#541	—	15	1	no	Sylow
90	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
91	C₂²×D₁₅	—	2	2	no	—
92	C₂²×D₁₅	—	2	4	no	—
93	C₂²×D₁₅	C₂⁴	1	2	yes	—
94	C₃×C₅×Q₈	—	2	16	no	—
95	C₂×C₃×C₄×C₅	—	2	8	no	—
96	C₂×C₃×C₄×C₅	—	2	4	no	—
97	C₂×C₃×C₄×C₅	#41	1	4	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
100	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
101	C₂×Dic₁₅	—	2	2	no	—
102	C₂×Dic₁₅	—	2	4	no	—
103	C₂×Dic₁₅	—	2	8	no	—
104	C₂×Dic₁₅	C₂⁴	1	4	yes	—
105	C₂×Dic₁₅	C₂⁴	1	8	yes	—
106	C₂×Dic₁₅	C₂⁴	1	2	yes	—
107	C₄×D₁₅	—	2	8	no	—
108	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
109	C₂²×C₄×D₃	—	5	1	no	—
110	C₃×C₄⋊Q₈	C₂×D₅	1	8	yes	complemented
111	C₂×C₃×C₄⋊C₄	C₂×D₅	1	4	yes	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	#946	—	5	16	no	—
114	Q₈×Dic₃	—	5	16	no	—
115	C₂×(C₂×D₃)⋊C₄	—	5	2	no	—
116	C₂×(C₂×D₃)⋊C₄	—	5	4	no	—
117	C₂×C₄⋊Dic₃	—	5	2	no	—
118	C₂×C₄⋊Dic₃	—	5	4	no	—
119	C₂×Dic₃⋊C₄	—	5	4	no	—
120	C₂×Dic₃⋊C₄	—	5	2	no	—
121	C₂×Dic₃⋊C₄	—	5	4	no	—
122	C₂×C₄×Dic₃	—	5	1	no	—
123	C₂×C₄×Dic₃	—	5	4	no	—
124	#898	—	5	32	no	—
125	#889	—	5	32	no	—
126	#872	—	5	8	no	—
127	#870	—	5	8	no	—
128	C₂³×D₅	—	3	1	no	—
129	C₂×C₅×Q₈	—	2	8	no	—
130	C₂×C₅×Q₈	C₂²×D₃	1	8	yes	—
131	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
132	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂²×Dic₅	—	3	1	no	—
135	C₂²×Dic₅	—	3	2	no	—
136	C₂²×Dic₅	—	3	4	no	—
137	C₂×C₄×D₅	—	3	4	no	—
138	C₂×C₄×D₅	—	6	4	no	—
139	C₅×C₄⋊C₄	C₂²×D₃	1	16	yes	—
140	C₄²×C₅	C₂²×D₃	1	4	yes	—
141	#685	—	3	8	no	—
142	#685	—	3	16	no	—
143	C₄⋊Dic₅	—	3	8	no	—
144	C₄⋊Dic₅	—	3	16	no	—
145	#683	—	3	16	no	—
146	#683	—	3	8	no	—
147	#683	—	3	16	no	—
148	C₄×Dic₅	—	3	4	no	—
149	C₄×Dic₅	—	3	16	no	—
150	#541	—	15	16	no	—
151	C₂×C₄⋊Q₈	D₁₅	1	1	yes	characteristic, complemented
152	C₂×#126	—	15	4	no	—
153	C₂×#125	—	15	4	no	—
154	C₂×#125	—	15	1	no	—
155	C₂×C₂²⋊Q₈	—	15	2	no	—
156	C₂×C₄×Q₈	—	15	2	no	—
157	C₂×#117	—	15	1	no	—
158	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
159	C₂²×C₃×C₅	2₋¹⁺⁴	1	4	yes	—
160	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
161	C₂×D₁₅	—	2	2	no	—
162	C₂×D₁₅	—	2	4	no	—
163	C₂×D₁₅	—	4	4	no	—
164	C₃×C₄×C₅	—	2	8	no	—
165	C₃×C₄×C₅	—	2	4	no	—
166	C₃×C₄×C₅	C₂×#41	1	4	yes	—
167	Dic₁₅	—	2	2	no	—
168	Dic₁₅	—	2	4	no	—
169	Dic₁₅	—	2	8	no	—
170	C₂³×D₃	—	5	1	no	—
171	C₂×C₃×Q₈	—	2	8	no	—
172	C₂×C₃×Q₈	C₂²×D₅	1	8	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
174	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
175	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
176	C₂²×Dic₃	—	5	2	no	—
177	C₂²×Dic₃	—	5	4	no	—
178	C₂²×Dic₃	—	5	1	no	—
179	C₂×C₄×D₃	—	5	4	no	—
180	C₂×C₄×D₃	—	10	4	no	—
181	C₃×C₄⋊C₄	C₂²×D₅	1	16	yes	—
182	C₃×C₄²	C₂²×D₅	1	4	yes	—
183	(C₂×D₃)⋊C₄	—	5	8	no	—
184	(C₂×D₃)⋊C₄	—	5	16	no	—
185	C₄⋊Dic₃	—	5	8	no	—
186	C₄⋊Dic₃	—	5	16	no	—
187	Dic₃⋊C₄	—	5	16	no	—
188	Dic₃⋊C₄	—	5	8	no	—
189	Dic₃⋊C₄	—	5	16	no	—
190	C₄×Dic₃	—	5	4	no	—
191	C₄×Dic₃	—	5	16	no	—
192	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₂²×D₅	—	3	2	no	—
194	C₂²×D₅	—	6	2	no	—
195	C₂²×D₅	—	6	4	no	—
196	C₅×Q₈	—	2	16	no	—
197	C₂×C₄×C₅	—	2	8	no	—
198	C₂×C₄×C₅	—	2	4	no	—
199	C₂×C₄×C₅	#237	1	4	yes	—
200	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
201	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
202	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
203	C₂×Dic₅	—	3	4	no	—
204	C₂×Dic₅	—	3	8	no	—
205	C₂×Dic₅	—	3	2	no	—
206	C₂×Dic₅	—	6	2	no	—
207	C₂×Dic₅	—	6	4	no	—
208	C₂×Dic₅	—	6	8	no	—
209	C₄×D₅	—	6	8	no	—
210	C₂²×Q₈	C₂×D₁₅	1	2	yes	—
211	C₂³×C₄	—	15	1	no	—
212	C₄⋊Q₈	C₂×D₁₅	1	8	yes	complemented
213	#126	—	15	32	no	—
214	#125	—	15	32	no	—
215	#125	—	15	8	no	—
216	C₂²⋊Q₈	—	15	16	no	—
217	C₄×Q₈	—	15	16	no	—
218	#117	—	15	8	no	—
219	C₂×C₄⋊C₄	—	15	2	no	—
220	C₂×C₄⋊C₄	—	15	2	no	—
221	C₂×C₄⋊C₄	—	15	4	no	—
222	C₂×C₄⋊C₄	—	15	4	no	—
223	C₂×C₄⋊C₄	—	15	4	no	—
224	C₂×C₄⋊C₄	C₂×D₁₅	1	4	yes	—
225	C₂×C₂²⋊C₄	—	15	2	no	—
226	C₂×C₂²⋊C₄	—	15	4	no	—
227	C₂×C₄²	—	15	1	no	—
228	C₂×C₄²	—	15	4	no	—
229	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
230	C₂×C₃×C₅	#541	1	4	yes	complemented
231	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
232	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	2	yes	—
233	D₁₅	—	4	2	no	—
234	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
235	C₂²×D₃	—	5	2	no	—
236	C₂²×D₃	—	10	2	no	—
237	C₂²×D₃	—	10	4	no	—
238	C₃×Q₈	—	2	16	no	—
239	C₂×C₃×C₄	—	2	8	no	—
240	C₂×C₃×C₄	—	2	4	no	—
241	C₂×C₃×C₄	#711	1	4	yes	—
242	C₂×C₃×C₄	C₂³×D₅	1	8	yes	—
243	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
244	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
245	C₂×Dic₃	—	5	4	no	—
246	C₂×Dic₃	—	5	8	no	—
247	C₂×Dic₃	—	5	2	no	—
248	C₂×Dic₃	—	10	2	no	—
249	C₂×Dic₃	—	10	4	no	—
250	C₂×Dic₃	—	10	8	no	—
251	C₄×D₃	—	10	8	no	—
252	C₂²×C₅	C₂×#237	1	2	yes	—
253	C₂²×C₅	#1007	1	4	yes	—
254	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
255	C₂×D₅	—	6	2	no	—
256	C₂×D₅	—	6	4	no	—
257	C₂×D₅	—	12	4	no	—
258	C₄×C₅	—	2	8	no	—
259	C₄×C₅	—	2	4	no	—
260	C₄×C₅	C₂×#237	1	4	yes	—
261	Dic₅	—	6	2	no	—
262	Dic₅	—	6	4	no	—
263	Dic₅	—	6	8	no	—
264	C₂⁴	—	15	1	no	—
265	C₂×Q₈	—	2	8	no	—
266	C₂×Q₈	C₂²×D₁₅	1	8	yes	—
267	C₂²×C₄	—	15	2	no	—
268	C₂²×C₄	—	15	4	no	—
269	C₂²×C₄	—	15	1	no	—
270	C₂²×C₄	—	15	4	no	—
271	C₂²×C₄	—	30	4	no	—
272	C₂²×C₄	C₂²×D₁₅	1	4	yes	—
273	C₂²×C₄	C₂²×D₁₅	1	2	yes	—
274	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
275	C₄⋊C₄	—	15	8	no	—
276	C₄⋊C₄	—	15	8	no	—
277	C₄⋊C₄	—	15	16	no	—
278	C₄⋊C₄	—	15	16	no	—
279	C₄⋊C₄	—	15	16	no	—
280	C₄⋊C₄	C₂²×D₁₅	1	16	yes	—
281	C₂²⋊C₄	—	15	8	no	—
282	C₂²⋊C₄	—	15	16	no	—
283	C₄²	—	15	4	no	—
284	C₄²	—	15	16	no	—
285	C₄²	C₂²×D₁₅	1	4	yes	—
286	C₃×C₅	C₂×#541	1	1	yes	characteristic, complemented, Hall
287	C₂²×C₃	C₂×#711	1	2	yes	—
288	C₂²×C₃	#4147	1	4	yes	—
289	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
290	C₂×D₃	—	10	2	no	—
291	C₂×D₃	—	10	4	no	—
292	C₂×D₃	—	20	4	no	—
293	C₃×C₄	—	2	8	no	—
294	C₃×C₄	—	2	4	no	—
295	C₃×C₄	C₂×#711	1	4	yes	—
296	Dic₃	—	10	2	no	—
297	Dic₃	—	10	4	no	—
298	Dic₃	—	10	8	no	—
299	C₂×C₅	#5731	1	4	yes	complemented
300	C₂×C₅	C₂²×#237	1	1	yes	characteristic
301	C₂×C₅	C₂×#1007	1	2	yes	—
302	D₅	—	12	2	no	—
303	C₂³	—	15	2	no	—
304	C₂³	—	30	2	no	—
305	C₂³	—	30	4	no	—
306	C₂³	C₂³×D₁₅	1	1	yes	characteristic
307	Q₈	—	2	16	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	15	4	no	—
311	C₂×C₄	—	15	8	no	—
312	C₂×C₄	—	15	2	no	—
313	C₂×C₄	—	30	2	no	—
314	C₂×C₄	—	30	4	no	—
315	C₂×C₄	—	30	8	no	—
316	C₂×C₄	—	30	8	no	—
317	C₂×C₄	#6819	1	4	yes	—
318	C₂×C₄	C₂³×D₁₅	1	8	yes	—
319	C₂×C₄	C₂³×D₁₅	1	4	yes	—
320	C₂×C₄	C₂³×D₁₅	1	2	yes	—
321	C₂×C₃	#66065	1	4	yes	complemented
322	C₂×C₃	C₂²×#711	1	1	yes	characteristic
323	C₂×C₃	C₂×#4147	1	2	yes	—
324	D₃	—	20	2	no	—
325	C₅	C₂×#5731	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	30	2	no	—
327	C₂²	—	30	4	no	—
328	C₂²	—	60	4	no	—
329	C₂²	C₂×#6819	1	2	yes	—
330	C₂²	#92098	1	4	yes	—
331	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
332	C₄	—	2	8	no	—
333	C₄	—	2	4	no	—
334	C₄	—	30	2	no	—
335	C₄	—	30	4	no	—
336	C₄	—	30	8	no	—
337	C₄	C₂×#6819	1	4	yes	—
338	C₃	C₂×#66065	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	60	2	no	—
340	C₂	#11756490	1	4	yes	direct product factor
341	C₂	C₂²×#6819	1	1	yes	characteristic
342	C₂	C₂×#92098	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×D₁₅	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	C₂×#541
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₂×Q₈×Dic₁₅	C₂	semidirect
4. #11756490	C₂	direct (N × H)
6. C₂×#91778	C₂	semidirect
8. C₂×#91759	C₂	semidirect
12. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
16. Q₈×Dic₁₅	C₂²	semidirect
27. #91778	C₂²	semidirect
29. #91759	C₂²	semidirect
31. C₂×C₅×C₄⋊Q₈	D₃	semidirect
61. C₂×C₃×C₄⋊Q₈	D₅	semidirect
71. C₅×C₄⋊Q₈	C₂×D₃	semidirect
110. C₃×C₄⋊Q₈	C₂×D₅	semidirect
151. C₂×C₄⋊Q₈	D₁₅	semidirect
212. C₄⋊Q₈	C₂×D₁₅	semidirect
230. C₂×C₃×C₅	#541	semidirect
286. C₃×C₅	C₂×#541	semidirect
299. C₂×C₅	#5731	semidirect
321. C₂×C₃	#66065	semidirect
325. C₅	C₂×#5731	semidirect
338. C₃	C₂×#66065	semidirect
340. C₂	#11756490	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×#91835	2
C₂×Q₈×Dic₁₅	2
#11756490	2
C₂×#91787	2
C₂×#91778	2
C₂×#91761	2
C₂×#91759	2
C₂×#66065	3
C₂×#5731	5

Subgroups — C2×#11756490 #14481204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11756490 #14481204