Subgroups — C₂×#22559095 #30825560

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863475	C₂	1	2	yes	maximal, complemented
3	#22559095	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863125	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₃₆	C₄	1	1	yes	characteristic
8	C₂²×#11761611	—	2	2	no	—
9	C₂×C₄×D₂₆₈	C₂²	1	2	yes	—
10	C₄×D₅₃₆	C₄	1	4	yes	—
11	C₄×D₅₃₆	C₂²	1	4	yes	—
12	#16863475	—	2	4	no	—
13	#16863475	C₂²	1	8	yes	complemented
14	C₂×#14523977	C₄	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14523965	C₄	1	2	yes	—
17	C₂×(C₂×D₆₇)⋊C₈	—	2	2	no	—
18	#16863149	C₂²	1	4	yes	complemented
19	#16863125	C₂²	1	8	yes	complemented
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₈²×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×D₂₆₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₆₇	—	2	2	no	—
26	C₂×D₅₃₆	—	2	4	no	—
27	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
28	C₂×#11761611	—	2	8	no	—
29	C₂×#11761611	—	4	8	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₄×D₂₆₈	—	2	2	no	—
32	C₄×D₂₆₈	—	2	2	no	—
33	C₄×D₂₆₈	C₂×C₄	1	4	yes	—
34	C₄×D₂₆₈	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
36	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
37	#14523977	C₂×C₄	1	4	yes	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
40	#14523965	C₂×C₄	1	8	yes	—
41	(C₂×D₆₇)⋊C₈	—	2	8	no	—
42	C₂²×C₈×C₆₇	—	2	1	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₆₇	D₄	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	2	yes	—
52	C₄×C₈×C₆₇	C₂³	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂³×D₆₇	—	2	2	no	—
55	C₂×D₂₆₈	—	2	4	no	—
56	C₂×D₂₆₈	—	2	4	no	—
57	C₂×D₂₆₈	—	2	2	no	—
58	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
59	C₂²×Dic₆₇	—	2	2	no	—
60	C₂×C₄×D₆₇	—	2	4	no	—
61	C₂×C₄×D₆₇	—	2	4	no	—
62	C₂×C₄×D₆₇	—	4	4	no	—
63	C₂×C₄×D₆₇	—	4	4	no	—
64	D₅₃₆	—	2	8	no	—
65	#11761611	—	4	16	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	4	4	no	—
68	C₄⋊Dic₆₇	—	2	2	no	—
69	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
70	(C₂×D₆₇)⋊C₄	—	2	4	no	—
71	(C₂×D₆₇)⋊C₄	—	2	4	no	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	M₁₆	1	4	yes	complemented
80	C₂×C₈×C₆₇	D₈	1	4	yes	complemented
81	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₆₇	#41	1	1	yes	characteristic
88	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
90	C₂²×D₆₇	—	2	4	no	—
91	C₂²×D₆₇	—	4	4	no	—
92	C₂²×D₆₇	—	4	4	no	—
93	C₂²×D₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	#131	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₆₇	#135	1	2	yes	—
102	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₆₇	—	2	4	no	—
106	C₂×Dic₆₇	—	4	4	no	—
107	D₂₆₈	—	2	4	no	—
108	D₂₆₈	—	2	4	no	—
109	D₂₆₈	—	4	4	no	—
110	C₄×D₆₇	—	4	4	no	—
111	C₄×D₆₇	—	4	4	no	—
112	C₄×D₆₇	—	4	8	no	—
113	C₈×C₆₇	—	2	4	no	—
114	C₈×C₆₇	—	2	2	no	—
115	C₈×C₆₇	—	2	2	no	—
116	C₈×C₆₇	C₂×M₁₆	1	4	yes	complemented
117	C₈×C₆₇	C₂×D₈	1	4	yes	complemented
118	C₆₇⋊C₈	—	4	4	no	—
119	C₂²×C₆₇	#436	1	2	yes	—
120	C₂²×C₆₇	C₄×D₈	1	2	yes	—
121	C₂²×C₆₇	#443	1	2	yes	—
122	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×D₆₇	—	4	4	no	—
124	C₂×D₆₇	—	4	4	no	—
125	C₂×D₆₇	—	4	4	no	—
126	C₂×D₆₇	—	8	8	no	—
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	#436	1	2	yes	—
130	C₄×C₆₇	C₄×D₈	1	2	yes	—
131	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
132	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
133	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
135	Dic₆₇	—	4	4	no	—
136	C₂×#1589	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	#1589	1	4	yes	complemented
138	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
139	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
140	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
141	D₆₇	—	8	4	no	—
142	C₂×C₄×D₈	—	67	1	no	—
143	C₂×#436	—	67	2	no	—
144	#1589	—	67	8	no	—
145	C₂×#335	—	67	1	no	—
146	C₂×D₄⋊C₈	—	67	2	no	—
147	C₂×C₈²	D₆₇	1	1	yes	characteristic, complemented
148	C₆₇	C₂×#1589	1	1	yes	characteristic, complemented, Sylow
149	C₂²×D₈	—	67	1	no	—
150	C₂²×M₁₆	—	134	2	no	—
151	C₂×C₄×D₄	—	67	2	no	—
152	C₄×D₈	—	67	4	no	—
153	C₄×D₈	—	67	4	no	—
154	#436	—	67	8	no	—
155	#436	—	134	4	no	—
156	C₂×#107	—	67	1	no	—
157	C₂×C₄⋊C₈	—	67	2	no	—
158	C₂×#102	—	67	2	no	—
159	C₂×C₂²⋊C₈	—	134	2	no	—
160	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	#335	—	67	4	no	—
164	D₄⋊C₈	—	67	8	no	—
165	C₈²	C₂×D₆₇	1	4	yes	complemented
166	C₂²×D₄	—	67	2	no	—
167	C₂³×C₄	—	134	2	no	—
168	C₂×D₈	—	67	4	no	—
169	C₂×D₈	—	134	4	no	—
170	C₂×M₁₆	—	268	8	no	—
171	C₂×M₁₆	—	134	8	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	134	2	no	—
174	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
175	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
177	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
178	C₄×D₄	—	67	4	no	—
179	C₄×D₄	—	67	4	no	—
180	C₄×D₄	—	134	2	no	—
181	C₄×D₄	—	134	2	no	—
182	C₂×C₄⋊C₄	—	67	2	no	—
183	C₂×C₂²⋊C₄	—	134	2	no	—
184	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
185	#107	—	67	4	no	—
186	C₄⋊C₈	—	67	4	no	—
187	C₄⋊C₈	—	134	2	no	—
188	#102	—	67	8	no	—
189	C₂²⋊C₈	—	134	8	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	C₄×D₆₇	1	2	yes	—
192	C₄×C₈	D₂₆₈	1	2	yes	—
193	C₄×C₈	C₂²×D₆₇	1	2	yes	—
194	C₄×C₈	C₂²×D₆₇	1	2	yes	—
195	C₄×C₈	C₂²×D₆₇	1	2	yes	—
196	C₂⁴	—	134	2	no	—
197	C₂×D₄	—	67	4	no	—
198	C₂×D₄	—	134	2	no	—
199	C₂×D₄	—	134	4	no	—
200	C₂×D₄	—	134	4	no	—
201	C₂²×C₄	—	268	4	no	—
202	C₂²×C₄	—	268	4	no	—
203	C₂²×C₄	—	134	2	no	—
204	C₂²×C₄	—	134	4	no	—
205	C₂²×C₄	—	134	4	no	—
206	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
208	C₂²×C₄	#11761628	1	1	yes	characteristic
209	D₈	—	134	8	no	—
210	M₁₆	—	268	16	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	268	4	no	—
216	C₂×C₈	—	134	4	no	—
217	C₂×C₈	#11761611	1	4	yes	complemented
218	C₂×C₈	D₅₃₆	1	4	yes	complemented
219	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
221	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
222	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
223	C₄⋊C₄	—	67	4	no	—
224	C₄⋊C₄	—	134	2	no	—
225	C₂²⋊C₄	—	134	4	no	—
226	C₂²⋊C₄	—	134	4	no	—
227	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
228	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
229	C₄²	#11761628	1	1	yes	characteristic
230	C₄²	C₂³×D₆₇	1	1	yes	characteristic
231	C₂³	—	268	4	no	—
232	C₂³	—	268	4	no	—
233	C₂³	—	268	4	no	—
234	C₂³	—	134	4	no	—
235	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
236	D₄	—	268	4	no	—
237	D₄	—	134	4	no	—
238	D₄	—	134	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	268	4	no	—
241	C₂×C₄	—	268	4	no	—
242	C₂×C₄	—	268	4	no	—
243	C₂×C₄	—	268	8	no	—
244	C₂×C₄	—	134	4	no	—
245	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
246	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
247	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
248	C₂×C₄	C₂×#11761611	1	2	yes	—
249	C₂×C₄	#14524049	1	2	yes	—
250	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
251	C₂×C₄	#14524065	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
254	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	268	4	no	—
259	C₈	C₂×#11761611	1	4	yes	complemented
260	C₈	C₂×D₅₃₆	1	4	yes	complemented
261	C₂²	—	268	4	no	—
262	C₂²	—	268	4	no	—
263	C₂²	—	268	4	no	—
264	C₂²	—	536	8	no	—
265	C₂²	#16863475	1	2	yes	—
266	C₂²	C₄×D₅₃₆	1	2	yes	—
267	C₂²	#16863514	1	2	yes	—
268	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	268	4	no	—
272	C₄	#16863475	1	2	yes	—
273	C₄	C₄×D₅₃₆	1	2	yes	—
274	C₄	C₂²×#11761611	1	1	yes	characteristic
275	C₄	C₂×#14524049	1	1	yes	characteristic
276	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
277	C₄	C₂×#14524065	1	1	yes	characteristic
278	C₂	—	536	4	no	—
279	C₂	#22559095	1	4	yes	direct product factor
280	C₂	C₂×#16863475	1	1	yes	characteristic
281	C₂	C₂×C₄×D₅₃₆	1	1	yes	characteristic
282	C₂	C₂×#16863514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₅₃₆	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	C₂×#1589
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. C₂×#16863475	C₂	semidirect
3. #22559095	C₂	direct (N × H)
4. C₂×#16863149	C₂	semidirect
5. C₂×#16863125	C₂	semidirect
6. C₂×C₈²×C₆₇	C₂	semidirect
13. #16863475	C₂²	semidirect
18. #16863149	C₂²	semidirect
19. #16863125	C₂²	semidirect
23. C₈²×C₆₇	C₂²	semidirect
79. C₂×C₈×C₆₇	M₁₆	semidirect
80. C₂×C₈×C₆₇	D₈	semidirect
116. C₈×C₆₇	C₂×M₁₆	semidirect
117. C₈×C₆₇	C₂×D₈	semidirect
137. C₂×C₆₇	#1589	semidirect
147. C₂×C₈²	D₆₇	semidirect
148. C₆₇	C₂×#1589	semidirect
165. C₈²	C₂×D₆₇	semidirect
217. C₂×C₈	#11761611	semidirect
218. C₂×C₈	D₅₃₆	semidirect
259. C₈	C₂×#11761611	semidirect
260. C₈	C₂×D₅₃₆	semidirect
279. C₂	#22559095	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×C₄×D₅₃₆	2
C₂×#16863475	2
#22559095	2
C₂×#16863149	2
C₂×#16863125	2
C₂×C₈²×C₆₇	2
C₂×#1589	67

Subgroups — C2×#22559095 #30825560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559095 #30825560