Subgroups — C₂×#22560364 #30856227

Order 17152.

1358 conjugacy classes of subgroups.

The subgroup lattice has 280 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	1	yes	maximal, characteristic
2	C₂²×#14523974	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863265	C₂	1	2	yes	maximal
5	#22560364	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂²×Dic₆₇⋊C₄	—	2	4	no	—
10	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×#14523975	—	2	4	no	—
12	C₂×#14523975	C₂²	1	4	yes	—
13	C₂×#14523974	—	2	8	no	—
14	C₂×#14523974	C₂²	1	8	yes	—
15	C₂×C₈×Dic₆₇	C₄	1	8	yes	complemented
16	C₂×C₈×Dic₆₇	C₂²	1	4	yes	—
17	#16863265	—	2	4	no	—
18	#16863265	C₂²	1	8	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#106×C₆₇	—	2	4	no	—
22	C₂×#106×C₆₇	C₂²	1	4	yes	—
23	C₂³×Dic₆₇	—	2	2	no	—
24	C₂³×Dic₆₇	D₄	1	2	yes	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	4	no	—
30	C₂×C₄⋊Dic₆₇	—	2	4	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
33	C₂×Dic₆₇⋊C₄	—	2	16	no	—
34	C₂×Dic₆₇⋊C₄	—	4	16	no	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₆₇	Q₈	1	4	yes	—
38	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
39	#14523975	—	2	8	no	—
40	#14523974	—	2	16	no	—
41	C₈×Dic₆₇	C₂×C₄	1	16	yes	complemented
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
52	#106×C₆₇	—	2	8	no	—
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	4	4	no	—
55	C₂²×Dic₆₇	—	4	8	no	—
56	C₂²×Dic₆₇	C₄⋊C₄	1	8	yes	—
57	C₂²×Dic₆₇	SD₁₆	1	16	yes	—
58	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
59	C₂×C₆₇⋊C₈	C₄⋊C₄	1	16	yes	complemented
60	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
62	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	—	4	4	no	—
66	Dic₆₇⋊C₄	—	4	32	no	—
67	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	—
68	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
69	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
70	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	—	4	4	no	—
73	C₂²×C₄×C₆₇	—	4	8	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
83	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	4	4	no	—
89	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₆₇	#127	1	2	yes	—
91	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
92	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	—
93	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	—	4	8	no	—
96	C₂×C₄×C₆₇	—	8	8	no	—
97	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₆₇	#127	1	2	yes	—
101	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₆₇	—	4	4	no	—
111	C₂×Dic₆₇	—	4	8	no	—
112	C₂×Dic₆₇	—	8	8	no	—
113	C₂×Dic₆₇	#106	1	32	yes	complemented
114	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
115	C₂×Dic₆₇	C₂×SD₁₆	1	16	yes	—
116	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	complemented
117	C₆₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	complemented
118	C₂²×C₆₇	#384	1	4	yes	—
119	C₂²×C₆₇	C₂×#106	1	8	yes	—
120	C₂²×C₆₇	#492	1	8	yes	—
121	C₂²×C₆₇	#498	1	8	yes	—
122	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
124	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₆₇	C₂²×SD₁₆	1	4	yes	—
126	C₄×C₆₇	—	8	4	no	—
127	C₄×C₆₇	#384	1	4	yes	—
128	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	Dic₆₇	—	8	4	no	—
133	Dic₆₇	C₂×#106	1	16	yes	complemented
134	C₂×#1844	—	67	1	no	maximal, Sylow
135	C₂×C₆₇	#1844	1	8	yes	complemented
136	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
137	C₂×C₆₇	C₂²×#106	1	2	yes	—
138	C₂×C₆₇	C₂×#492	1	2	yes	—
139	C₂×C₆₇	C₂×#498	1	2	yes	—
140	C₂²×#106	—	67	1	no	—
141	C₂²×#106	—	67	2	no	—
142	C₂²×#106	D₆₇	1	1	yes	characteristic
143	C₂²×C₄×C₈	—	67	1	no	—
144	C₂×#384	—	67	2	no	—
145	#1844	—	67	8	no	—
146	C₆₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₈	—	67	1	no	—
148	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	—	67	2	no	—
150	C₂²×C₄⋊C₄	—	134	4	no	—
151	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
152	C₂²×C₄²	—	67	1	no	—
153	C₂×#106	—	2	4	no	—
154	C₂×#106	—	67	4	no	—
155	C₂×#106	—	67	8	no	—
156	C₂×#106	—	134	8	no	—
157	C₂×#106	—	134	4	no	—
158	C₂×#106	C₂×D₆₇	1	4	yes	—
159	C₂×C₄×C₈	—	67	8	no	—
160	C₂×C₄×C₈	—	67	4	no	—
161	#384	—	67	8	no	—
162	#384	—	134	4	no	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	—	67	2	no	—
165	C₂³×C₄	—	134	2	no	—
166	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
167	C₂²×C₈	—	67	4	no	—
168	C₂²×C₈	—	67	4	no	—
169	C₂²×C₈	—	67	4	no	—
170	C₂²×C₈	—	67	2	no	—
171	C₂²×C₈	Dic₁₃₄	1	4	yes	—
172	C₂²×C₈	C₄×D₆₇	1	4	yes	—
173	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
174	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	268	16	no	—
179	C₂×C₄⋊C₄	—	67	4	no	—
180	C₂×C₄⋊C₄	—	134	16	no	—
181	C₂×C₄⋊C₄	—	134	2	no	—
182	C₂×C₄⋊C₄	—	134	4	no	—
183	C₂×C₄⋊C₄	—	134	4	no	—
184	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
185	C₂×C₄²	—	67	4	no	—
186	C₂×C₄²	—	67	2	no	—
187	C₂×C₄²	—	67	4	no	—
188	C₂×C₄²	—	67	2	no	—
189	#106	—	2	8	no	—
190	#106	—	134	16	no	—
191	#106	—	134	8	no	—
192	C₄×C₈	—	67	16	no	—
193	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	268	4	no	—
198	C₂²×C₄	—	268	8	no	—
199	C₂²×C₄	—	67	8	no	—
200	C₂²×C₄	—	67	16	no	—
201	C₂²×C₄	—	67	4	no	—
202	C₂²×C₄	—	134	4	no	—
203	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
204	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₆₇	1	2	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
207	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
208	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
209	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
210	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
211	C₂×C₈	—	67	16	no	—
212	C₂×C₈	—	67	4	no	—
213	C₂×C₈	—	67	4	no	—
214	C₂×C₈	—	67	4	no	—
215	C₂×C₈	Dic₆₇⋊C₄	1	16	yes	complemented
216	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
217	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
218	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	268	32	no	—
223	C₄⋊C₄	—	268	4	no	—
224	C₄⋊C₄	—	134	4	no	—
225	C₄⋊C₄	—	134	4	no	—
226	C₄²	—	67	8	no	—
227	C₄²	—	67	4	no	—
228	C₄²	—	67	4	no	—
229	C₂³	C₄⋊C₄×D₆₇	1	2	yes	—
230	C₂³	#14524039	1	2	yes	—
231	C₂³	#14524042	1	2	yes	—
232	C₂³	SD₁₆×D₆₇	1	8	yes	—
233	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	8	no	—
237	C₂×C₄	—	268	8	no	—
238	C₂×C₄	—	268	4	no	—
239	C₂×C₄	—	536	8	no	—
240	C₂×C₄	—	67	8	no	—
241	C₂×C₄	—	67	16	no	—
242	C₂×C₄	—	67	32	no	—
243	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
244	C₂×C₄	C₄⋊C₄×D₆₇	1	4	yes	—
245	C₂×C₄	C₂×Dic₆₇⋊C₄	1	4	yes	—
246	C₂×C₄	#14524039	1	2	yes	—
247	C₂×C₄	#14524041	1	4	yes	—
248	C₂×C₄	#14524042	1	2	yes	—
249	C₂×C₄	#14524042	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
256	C₈	—	67	8	no	—
257	C₈	C₂×Dic₆₇⋊C₄	1	8	yes	complemented
258	C₂²	#16863265	1	4	yes	—
259	C₂²	#106×D₆₇	1	8	yes	—
260	C₂²	#16863767	1	8	yes	—
261	C₂²	#16863793	1	8	yes	—
262	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
263	C₂²	C₂×#14524039	1	1	yes	characteristic
264	C₂²	C₂×#14524042	1	1	yes	characteristic
265	C₂²	C₂×SD₁₆×D₆₇	1	4	yes	—
266	C₄	—	8	4	no	—
267	C₄	—	536	4	no	—
268	C₄	—	67	16	no	—
269	C₄	#16863265	1	4	yes	—
270	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
271	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
272	C₄	C₂×#14524041	1	1	yes	characteristic
273	C₄	C₂×#14524042	1	1	yes	characteristic
274	C₂	#22560364	1	8	yes	direct product factor
275	C₂	C₂×#16863265	1	1	yes	characteristic
276	C₂	C₂×#106×D₆₇	1	2	yes	—
277	C₂	C₂×#16863767	1	2	yes	—
278	C₂	C₂×#16863793	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#106×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#106	D₆₇
hypocenter	C₆₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
5. #22560364	C₂	direct (N × H)
15. C₂×C₈×Dic₆₇	C₄	semidirect
41. C₈×Dic₆₇	C₂×C₄	semidirect
59. C₂×C₆₇⋊C₈	C₄⋊C₄	semidirect
82. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
113. C₂×Dic₆₇	#106	semidirect
116. C₈×C₆₇	C₂×C₄⋊C₄	semidirect
117. C₆₇⋊C₈	C₂×C₄⋊C₄	semidirect
133. Dic₆₇	C₂×#106	semidirect
135. C₂×C₆₇	#1844	semidirect
146. C₆₇	C₂×#1844	semidirect
215. C₂×C₈	Dic₆₇⋊C₄	semidirect
257. C₈	C₂×Dic₆₇⋊C₄	semidirect
274. C₂	#22560364	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
C₂²×#14523974	2
C₂²×C₈×Dic₆₇	2
C₂×#16863265	2
#22560364	2
C₂²×#106×C₆₇	2
C₂×#1844	67

Subgroups — C2×#22560364 #30856227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560364 #30856227