Subgroups — C₂×#22560234 #30855388

Order 17152.

1790 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523975	C₂	1	2	yes	maximal, complemented
2	C₂×#16863256	C₂	1	2	yes	maximal
3	#22560234	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863159	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
7	C₂²×C₂²⋊Dic₆₇	—	2	2	no	—
8	C₂×#14523975	—	2	8	no	—
9	C₂×#14523975	C₂²	1	8	yes	complemented
10	#16863256	—	2	4	no	—
11	#16863256	C₂²	1	8	yes	—
12	C₂×C₆₇⋊#95	—	2	2	no	—
13	#16863159	C₂²	1	8	yes	complemented
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	—	2	8	no	—
16	C₂³×C₈×C₆₇	C₄	1	8	yes	complemented
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	C₂³×Dic₆₇	—	2	4	no	—
21	C₂×C₄⋊Dic₆₇	—	2	8	no	—
22	C₂×C₄⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	8	no	—
24	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
25	C₂×C₂²⋊Dic₆₇	—	2	8	no	—
26	C₂×C₂²⋊Dic₆₇	—	4	8	no	—
27	#14523975	—	2	16	no	—
28	C₆₇⋊#95	—	2	8	no	—
29	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	—	2	4	no	—
31	C₂³×C₄×C₆₇	—	2	4	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	D₄	1	2	yes	—
35	C₂³×C₄×C₆₇	Q₈	1	4	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	32	no	—
40	C₂²×C₈×C₆₇	—	2	16	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₆₇	D₄	1	8	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×Dic₆₇	—	2	8	no	—
48	C₂²×Dic₆₇	—	4	8	no	—
49	C₂²×Dic₆₇	—	4	8	no	—
50	C₂²×Dic₆₇	—	4	8	no	—
51	C₄⋊Dic₆₇	—	2	8	no	—
52	C₄⋊Dic₆₇	—	2	8	no	—
53	C₄⋊Dic₆₇	—	4	8	no	—
54	C₂²⋊Dic₆₇	—	4	16	no	—
55	C₂⁴×C₆₇	—	2	4	no	—
56	C₂⁴×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
58	C₂⁴×C₆₇	SD₁₆	1	8	yes	—
59	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
60	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	8	no	—
63	C₂²×C₄×C₆₇	—	2	8	no	—
64	C₂²×C₄×C₆₇	—	2	8	no	—
65	C₂²×C₄×C₆₇	—	2	16	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	4	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₆₇	—	2	8	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	32	no	—
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	16	yes	complemented
83	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
86	C₂³×C₆₇	—	2	4	no	—
87	C₂³×C₆₇	—	2	4	no	—
88	C₂³×C₆₇	—	2	8	no	—
89	C₂³×C₆₇	—	2	8	no	—
90	C₂³×C₆₇	—	2	32	no	—
91	C₂³×C₆₇	#106	1	16	yes	complemented
92	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₆₇	#121	1	4	yes	—
95	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	—
96	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
97	C₂³×C₆₇	#135	1	4	yes	—
98	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	16	no	—
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₆₇	#117	1	4	yes	—
109	C₂×C₄×C₆₇	#121	1	4	yes	—
110	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
115	C₂×Dic₆₇	—	4	8	no	—
116	C₂×Dic₆₇	—	4	8	no	—
117	C₂×Dic₆₇	—	4	8	no	—
118	C₂×Dic₆₇	—	8	16	no	—
119	C₈×C₆₇	—	2	4	no	—
120	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	complemented
121	C₂²×C₆₇	—	2	32	no	—
122	C₂²×C₆₇	—	2	4	no	—
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	2	16	no	—
125	C₂²×C₆₇	#380	1	4	yes	—
126	C₂²×C₆₇	C₂×#106	1	4	yes	—
127	C₂²×C₆₇	C₂×#106	1	8	yes	complemented
128	C₂²×C₆₇	#427	1	4	yes	—
129	C₂²×C₆₇	#465	1	16	yes	—
130	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×#121	1	2	yes	—
132	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
133	C₂²×C₆₇	C₂×#135	1	2	yes	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	#380	1	4	yes	—
136	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
138	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
139	Dic₆₇	—	8	8	no	—
140	C₂×#1807	—	67	1	no	maximal, Sylow
141	C₂×C₆₇	—	2	8	no	—
142	C₂×C₆₇	#1807	1	8	yes	complemented
143	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
144	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#465	1	4	yes	—
147	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
148	C₂²×#106	—	67	2	no	—
149	C₂×#380	—	67	2	no	—
150	#1807	—	67	8	no	—
151	C₂×#340	—	67	2	no	—
152	C₆₇	C₂×#1807	1	1	yes	characteristic, complemented, Sylow
153	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
154	C₂³×C₈	—	2	8	no	—
155	C₂³×C₈	Dic₆₇	1	8	yes	complemented
156	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₆₇	1	4	yes	—
158	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂²×C₄⋊C₄	—	67	4	no	—
160	C₂²×C₂²⋊C₄	—	134	2	no	—
161	C₂×#106	—	67	8	no	—
162	C₂×#106	—	134	8	no	—
163	#380	—	67	8	no	—
164	#380	—	134	4	no	—
165	C₂×#95	—	134	2	no	—
166	#340	—	67	8	no	—
167	C₂⁵	D₂₆₈	1	1	yes	characteristic
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	134	4	no	—
171	C₂³×C₄	Dic₁₃₄	1	4	yes	—
172	C₂³×C₄	D₂₆₈	1	2	yes	—
173	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
175	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
176	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	32	no	—
179	C₂²×C₈	—	2	16	no	—
180	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
181	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
182	C₂²×C₈	C₂×Dic₆₇	1	16	yes	complemented
183	C₂²×C₈	C₂²⋊D₆₇	1	8	yes	—
184	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
185	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
186	C₂×C₄⋊C₄	—	67	8	no	—
187	C₂×C₄⋊C₄	—	134	4	no	—
188	C₂×C₄⋊C₄	—	134	8	no	—
189	C₂×C₄⋊C₄	—	134	8	no	—
190	C₂×C₂²⋊C₄	—	268	8	no	—
191	C₂×C₂²⋊C₄	—	134	8	no	—
192	#106	—	134	16	no	—
193	#95	—	134	8	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	C₄⋊Dic₆₇	1	4	yes	—
197	C₂⁴	#11761617	1	8	yes	—
198	C₂⁴	C₂×D₂₆₈	1	2	yes	—
199	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	268	8	no	—
208	C₂²×C₄	—	268	8	no	—
209	C₂²×C₄	—	268	8	no	—
210	C₂²×C₄	—	134	8	no	—
211	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
212	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
213	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
214	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
216	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
217	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
218	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
219	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
220	C₂²×C₄	#11761628	1	4	yes	—
221	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	32	no	—
225	C₂×C₈	C₂²⋊Dic₆₇	1	16	yes	complemented
226	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
227	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
228	C₂×C₈	C₂×C₂²⋊D₆₇	1	8	yes	—
229	C₄⋊C₄	—	268	8	no	—
230	C₄⋊C₄	—	134	8	no	—
231	C₄⋊C₄	—	134	8	no	—
232	C₂²⋊C₄	—	268	16	no	—
233	C₂³	—	2	8	no	—
234	C₂³	—	2	32	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	8	no	—
238	C₂³	#14523975	1	16	yes	complemented
239	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
240	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
241	C₂³	#14524017	1	4	yes	—
242	C₂³	C₂×#11761617	1	8	yes	—
243	C₂³	C₂×#11761617	1	4	yes	—
244	C₂³	#14524065	1	4	yes	—
245	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	268	8	no	—
253	C₂×C₄	—	268	8	no	—
254	C₂×C₄	—	268	8	no	—
255	C₂×C₄	—	536	16	no	—
256	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
259	C₂×C₄	#14524002	1	4	yes	—
260	C₂×C₄	#14524017	1	4	yes	—
261	C₂×C₄	#14524022	1	8	yes	—
262	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
264	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761628	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	C₂×C₂²⋊Dic₆₇	1	8	yes	complemented
268	C₂²	—	2	4	no	—
269	C₂²	—	2	16	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	32	no	—
272	C₂²	#16863256	1	4	yes	—
273	C₂²	C₂×#14523975	1	4	yes	—
274	C₂²	C₂×#14523975	1	8	yes	complemented
275	C₂²	#16863434	1	4	yes	—
276	C₂²	#16863625	1	16	yes	—
277	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
278	C₂²	C₂×#14524017	1	2	yes	—
279	C₂²	C₂²×#11761617	1	2	yes	—
280	C₂²	C₂×#14524065	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	536	8	no	—
283	C₄	#16863256	1	4	yes	—
284	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
285	C₄	C₂×#14524002	1	1	yes	characteristic
286	C₄	C₂×#14524022	1	2	yes	—
287	C₂	—	2	8	no	—
288	C₂	#22560234	1	8	yes	direct product factor
289	C₂	C₂×#16863256	1	1	yes	characteristic
290	C₂	C₂²×#14523975	1	1	yes	characteristic
291	C₂	C₂×#16863434	1	1	yes	characteristic
292	C₂	C₂×#16863625	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂×D₄
prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂×#1807
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523975	C₂	semidirect
3. #22560234	C₂	direct (N × H)
4. C₂×#16863159	C₂	semidirect
9. C₂×#14523975	C₂²	semidirect
13. #16863159	C₂²	semidirect
16. C₂³×C₈×C₆₇	C₄	semidirect
43. C₂²×C₈×C₆₇	C₂×C₄	semidirect
82. C₂×C₈×C₆₇	C₂²⋊C₄	semidirect
91. C₂³×C₆₇	#106	semidirect
120. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
127. C₂²×C₆₇	C₂×#106	semidirect
142. C₂×C₆₇	#1807	semidirect
152. C₆₇	C₂×#1807	semidirect
155. C₂³×C₈	Dic₆₇	semidirect
182. C₂²×C₈	C₂×Dic₆₇	semidirect
225. C₂×C₈	C₂²⋊Dic₆₇	semidirect
238. C₂³	#14523975	semidirect
267. C₈	C₂×C₂²⋊Dic₆₇	semidirect
274. C₂²	C₂×#14523975	semidirect
288. C₂	#22560234	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×#14523975	2
C₂×#16863256	2
#22560234	2
C₂×#16863159	2
C₂⁴×C₈×C₆₇	2
C₂×#1807	67

Subgroups — C2×#22560234 #30855388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560234 #30855388