Subgroups — C₂×#22560247 #30855445

Order 17152.

1566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	2	yes	maximal, complemented
2	C₂×#16863360	C₂	1	2	yes	maximal
3	#22560247	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863161	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	4	yes	—
7	C₂×#14523979	—	2	8	no	—
8	C₂×#14523979	C₂²	1	8	yes	complemented
9	#16863360	—	2	4	no	—
10	#16863360	C₂²	1	8	yes	—
11	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
12	#16863161	C₂²	1	8	yes	complemented
13	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	—	2	8	no	—
15	C₂³×C₈×C₆₇	C₄	1	4	yes	—
16	C₂³×C₈×C₆₇	C₄	1	4	yes	—
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	4	no	—
21	C₂×#11761612	—	2	8	no	—
22	C₂×#11761612	—	2	8	no	—
23	C₂×#11761612	C₂³	1	8	yes	—
24	C₂²×C₆₇⋊C₈	—	2	4	no	—
25	#14523979	—	2	16	no	—
26	C₂²⋊(C₆₇⋊C₈)	—	2	16	no	—
27	C₂⁵×C₆₇	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	—	2	4	no	—
29	C₂³×C₄×C₆₇	—	2	4	no	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	D₄	1	2	yes	—
33	C₂³×C₄×C₆₇	D₄	1	2	yes	—
34	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	16	no	—
40	C₂²×C₈×C₆₇	—	2	16	no	—
41	C₂²×C₈×C₆₇	—	2	16	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	D₄	1	8	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	#11761612	—	2	8	no	—
51	#11761612	—	2	8	no	—
52	#11761612	—	4	8	no	—
53	C₂×C₆₇⋊C₈	—	2	8	no	—
54	C₂×C₆₇⋊C₈	—	4	8	no	—
55	C₂⁴×C₆₇	—	2	8	no	—
56	C₂⁴×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	8	no	—
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	—	2	8	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	32	no	—
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
91	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
95	C₂³×C₆₇	—	2	32	no	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	—	2	4	no	—
98	C₂³×C₆₇	—	2	4	no	—
99	C₂³×C₆₇	—	2	4	no	—
100	C₂³×C₆₇	—	2	4	no	—
101	C₂³×C₆₇	—	2	8	no	—
102	C₂³×C₆₇	—	2	8	no	—
103	C₂³×C₆₇	#108	1	8	yes	complemented
104	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₆₇	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	8	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	2	8	no	—
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#117	1	2	yes	—
123	C₂×C₄×C₆₇	#117	1	2	yes	—
124	C₂×C₄×C₆₇	#121	1	4	yes	—
125	C₂×C₄×C₆₇	#121	1	4	yes	—
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
131	C₈×C₆₇	—	2	4	no	—
132	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
133	C₆₇⋊C₈	—	4	8	no	—
134	C₂²×C₆₇	—	2	16	no	—
135	C₂²×C₆₇	—	2	16	no	—
136	C₂²×C₆₇	—	2	16	no	—
137	C₂²×C₆₇	—	2	8	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	4	no	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	4	no	—
142	C₂²×C₆₇	#380	1	2	yes	—
143	C₂²×C₆₇	#380	1	2	yes	—
144	C₂²×C₆₇	C₂×#108	1	4	yes	—
145	C₂²×C₆₇	C₂×#108	1	8	yes	complemented
146	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	#380	1	2	yes	—
150	C₄×C₆₇	#380	1	2	yes	—
151	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#121	1	2	yes	—
154	C₂×#1810	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	2	8	no	—
158	C₂×C₆₇	#1810	1	4	yes	complemented
159	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
160	C₂×C₆₇	C₂²×#108	1	2	yes	—
161	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
162	C₂²×#108	—	67	2	no	—
163	C₂×#407	—	67	2	no	—
164	#1810	—	67	8	no	—
165	C₂×#341	—	67	2	no	—
166	C₆₇	C₂×#1810	1	1	yes	characteristic, complemented, Sylow
167	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
168	C₂²×M₁₆	—	67	4	no	—
169	C₂³×C₈	—	2	8	no	—
170	C₂³×C₈	Dic₆₇	1	4	yes	—
171	C₂³×C₈	Dic₆₇	1	4	yes	—
172	C₂³×C₈	C₂×D₆₇	1	4	yes	—
173	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
174	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
175	C₂×#108	—	67	8	no	—
176	C₂×#108	—	134	8	no	—
177	#407	—	67	8	no	—
178	#407	—	134	4	no	—
179	C₂×C₂²⋊C₈	—	134	4	no	—
180	#341	—	67	8	no	—
181	C₂⁵	Dic₁₃₄	1	1	yes	characteristic
182	C₂³×C₄	—	2	4	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
185	C₂³×C₄	Dic₁₃₄	1	2	yes	—
186	C₂³×C₄	D₂₆₈	1	2	yes	—
187	C₂³×C₄	D₂₆₈	1	2	yes	—
188	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
189	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
190	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
191	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
192	C₂×M₁₆	—	67	8	no	—
193	C₂×M₁₆	—	134	8	no	—
194	C₂×M₁₆	—	134	8	no	—
195	C₂×M₁₆	—	134	4	no	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	—	2	16	no	—
198	C₂²×C₈	—	2	16	no	—
199	C₂²×C₈	—	2	16	no	—
200	C₂²×C₈	—	134	4	no	—
201	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
202	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
203	C₂²×C₈	C₂×Dic₆₇	1	16	yes	—
204	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
205	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
206	C₂²×C₈	C₂²⋊D₆₇	1	8	yes	—
207	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
208	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
209	#108	—	134	16	no	—
210	C₂²⋊C₈	—	134	16	no	—
211	C₂⁴	—	2	8	no	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	C₄⋊Dic₆₇	1	2	yes	—
214	C₂⁴	C₄⋊Dic₆₇	1	2	yes	—
215	C₂⁴	C₂×Dic₁₃₄	1	2	yes	—
216	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
230	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
231	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
232	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
235	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
236	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
238	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
239	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
240	C₂²×C₄	#11761628	1	4	yes	—
241	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
242	M₁₆	—	268	8	no	—
243	M₁₆	—	134	8	no	—
244	M₁₆	—	134	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	32	no	—
249	C₂×C₈	—	268	8	no	—
250	C₂×C₈	—	134	8	no	—
251	C₂×C₈	C₂²⋊Dic₆₇	1	8	yes	—
252	C₂×C₈	C₂²⋊Dic₆₇	1	8	yes	—
253	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
254	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
255	C₂×C₈	C₂×C₂²⋊D₆₇	1	8	yes	—
256	C₂³	—	2	32	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	2	8	no	—
264	C₂³	#14523979	1	8	yes	complemented
265	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
266	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
267	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
268	C₂³	#14524022	1	4	yes	—
269	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	2	yes	—
279	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
283	C₂×C₄	#14524002	1	2	yes	—
284	C₂×C₄	#14524002	1	2	yes	—
285	C₂×C₄	#14524017	1	4	yes	—
286	C₂×C₄	#14524017	1	4	yes	—
287	C₂×C₄	#14524022	1	4	yes	—
288	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
290	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
291	C₂×C₄	C₂×#11761628	1	2	yes	—
292	C₈	—	2	4	no	—
293	C₈	—	268	8	no	—
294	C₈	C₂×C₂²⋊Dic₆₇	1	8	yes	—
295	C₂²	—	2	16	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	16	no	—
302	C₂²	—	2	8	no	—
303	C₂²	#16863256	1	2	yes	—
304	C₂²	#16863256	1	2	yes	—
305	C₂²	C₂×#14523979	1	4	yes	—
306	C₂²	C₂×#14523979	1	8	yes	complemented
307	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
308	C₂²	C₂×#14524022	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	#16863256	1	2	yes	—
311	C₄	#16863256	1	2	yes	—
312	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
313	C₄	C₂×#14524002	1	1	yes	characteristic
314	C₄	C₂×#14524017	1	2	yes	—
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	8	no	—
318	C₂	#22560247	1	4	yes	direct product factor
319	C₂	C₂×#16863256	1	1	yes	characteristic
320	C₂	C₂²×#14523979	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁵×C₆₇	Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂×#1810
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523979	C₂	semidirect
3. #22560247	C₂	direct (N × H)
4. C₂×#16863161	C₂	semidirect
8. C₂×#14523979	C₂²	semidirect
12. #16863161	C₂²	semidirect
103. C₂³×C₆₇	#108	semidirect
145. C₂²×C₆₇	C₂×#108	semidirect
158. C₂×C₆₇	#1810	semidirect
166. C₆₇	C₂×#1810	semidirect
264. C₂³	#14523979	semidirect
306. C₂²	C₂×#14523979	semidirect
318. C₂	#22560247	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523979	2
C₂×#16863360	2
#22560247	2
C₂×#16863161	2
C₂⁴×C₈×C₆₇	2
C₂×#1810	67

Subgroups — C2×#22560247 #30855445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560247 #30855445