Subgroups — #30765741

Order 17152.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#22562008	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558163	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₆₇	—	2	1	no	—
6	C₂×C₄×D₂₆₈	C₄	1	1	yes	characteristic
7	C₈×D₂₆₈	—	2	1	no	—
8	C₈×D₂₆₈	—	2	1	no	—
9	C₈×D₂₆₈	C₄	1	2	yes	—
10	C₈×D₂₆₈	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
13	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
14	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
15	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
18	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
19	C₄×C₁₆×C₆₇	—	2	1	no	—
20	C₂²×D₂₆₈	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₆₇	—	2	1	no	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	C₂×C₈×D₆₇	—	2	2	no	—
24	C₂×C₈×D₆₇	—	2	2	no	—
25	C₂×C₈×D₆₇	—	4	4	no	—
26	C₄×D₂₆₈	—	2	1	no	—
27	C₄×D₂₆₈	—	2	1	no	—
28	C₄×D₂₆₈	C₈	1	2	yes	—
29	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
31	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
35	(C₂×D₆₇)⋊C₈	—	2	2	no	—
36	(C₂×D₆₇)⋊C₈	—	2	2	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×D₆₇	—	2	1	no	—
46	C₂×D₂₆₈	—	2	1	no	—
47	C₂×D₂₆₈	—	2	2	no	—
48	C₂×D₂₆₈	C₁₆	1	4	yes	complemented
49	C₂×D₂₆₈	C₂×C₈	1	2	yes	—
50	C₂²×Dic₆₇	—	2	1	no	—
51	C₂×C₄×D₆₇	—	2	2	no	—
52	C₂×C₄×D₆₇	—	2	2	no	—
53	C₂×C₄×D₆₇	—	4	2	no	—
54	C₂×C₄×D₆₇	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₈×D₆₇	—	4	4	no	—
58	C₈×D₆₇	—	4	4	no	—
59	C₄⋊Dic₆₇	—	2	1	no	—
60	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
61	(C₂×D₆₇)⋊C₄	—	2	2	no	—
62	(C₂×D₆₇)⋊C₄	—	2	2	no	—
63	C₆₇⋊C₁₆	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	4	1	no	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
76	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₁₆×C₆₇	—	4	2	no	—
78	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂²×D₆₇	—	2	4	no	—
80	C₂²×D₆₇	—	2	2	no	—
81	C₂²×D₆₇	—	4	2	no	—
82	C₂²×D₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
90	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×Dic₆₇	—	2	2	no	—
95	C₂×Dic₆₇	—	4	2	no	—
96	D₂₆₈	—	2	4	no	—
97	D₂₆₈	—	2	2	no	—
98	D₂₆₈	C₂×C₁₆	1	4	yes	—
99	C₄×D₆₇	—	4	4	no	—
100	C₄×D₆₇	—	4	2	no	—
101	C₄×D₆₇	—	4	2	no	—
102	C₈×C₆₇	—	2	1	no	—
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₆₇⋊C₈	—	4	2	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	C₂²⋊C₁₆	1	2	yes	—
110	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂×D₆₇	—	2	4	no	—
112	C₂×D₆₇	—	4	8	no	—
113	C₂×D₆₇	—	4	2	no	—
114	C₂×D₆₇	—	4	2	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	C₂²⋊C₁₆	1	2	yes	—
119	C₄×C₆₇	#418	1	1	yes	characteristic
120	C₄×C₆₇	#421	1	1	yes	characteristic
121	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
122	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
123	Dic₆₇	—	4	2	no	—
124	#11629	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	#1478	1	1	yes	characteristic
127	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
128	C₂×C₆₇	#2117	1	1	yes	characteristic
129	D₆₇	—	4	4	no	—
130	C₂×C₈×D₄	—	67	1	no	—
131	#2162	—	67	1	no	—
132	#2162	D₆₇	1	1	yes	characteristic, complemented
133	D₄⋊C₁₆	—	67	4	no	—
134	C₆₇	#11629	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₈	—	134	1	no	—
136	C₂×C₄×D₄	—	67	1	no	—
137	C₈×D₄	—	67	2	no	—
138	C₈×D₄	—	67	2	no	—
139	C₈×D₄	—	134	1	no	—
140	C₈×D₄	—	134	1	no	—
141	C₂×C₄⋊C₈	—	67	1	no	—
142	C₂×C₂²⋊C₈	—	134	1	no	—
143	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
144	C₄⋊C₁₆	—	134	1	no	—
145	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	complemented
146	C₂²⋊C₁₆	—	2	1	no	—
147	C₂²⋊C₁₆	—	134	1	no	—
148	C₄×C₁₆	—	2	1	no	—
149	C₄×C₁₆	—	67	2	no	—
150	C₂²×D₄	—	67	1	no	—
151	C₂³×C₄	—	134	1	no	—
152	C₂²×C₈	—	268	4	no	—
153	C₂²×C₈	—	134	1	no	—
154	C₂²×C₈	—	134	2	no	—
155	C₂²×C₈	—	134	2	no	—
156	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄×D₄	—	67	2	no	—
159	C₄×D₄	—	67	2	no	—
160	C₄×D₄	—	134	1	no	—
161	C₄×D₄	—	134	1	no	—
162	C₂×C₄⋊C₄	—	67	1	no	—
163	C₂×C₂²⋊C₄	—	134	1	no	—
164	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	134	2	no	—
167	C₄⋊C₈	—	67	2	no	—
168	C₄⋊C₈	—	134	1	no	—
169	C₂²⋊C₈	—	134	2	no	—
170	C₂²⋊C₈	—	134	2	no	—
171	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
172	C₄×C₈	D₂₆₈	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
175	C₂⁴	—	134	1	no	—
176	C₂×D₄	—	67	4	no	—
177	C₂×D₄	—	67	2	no	—
178	C₂×D₄	—	134	1	no	—
179	C₂×D₄	—	134	2	no	—
180	C₂²×C₄	—	268	2	no	—
181	C₂²×C₄	—	268	2	no	—
182	C₂²×C₄	—	134	1	no	—
183	C₂²×C₄	—	134	2	no	—
184	C₂²×C₄	—	134	2	no	—
185	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
187	C₂²×C₄	#11761611	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	268	4	no	—
191	C₂×C₈	—	268	2	no	—
192	C₂×C₈	—	268	4	no	—
193	C₂×C₈	—	134	2	no	—
194	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
195	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
198	C₄⋊C₄	—	67	2	no	—
199	C₄⋊C₄	—	134	1	no	—
200	C₂²⋊C₄	—	134	2	no	—
201	C₂²⋊C₄	—	134	2	no	—
202	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₄²	C₈×D₆₇	1	1	yes	characteristic
204	C₄²	#11761611	1	1	yes	characteristic
205	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
206	C₁₆	—	4	2	no	—
207	C₁₆	—	268	2	no	—
208	C₂³	—	268	2	no	—
209	C₂³	—	268	2	no	—
210	C₂³	—	134	2	no	—
211	C₂³	—	134	4	no	—
212	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
213	D₄	—	67	4	no	—
214	D₄	—	134	2	no	—
215	D₄	—	134	4	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	268	4	no	—
220	C₂×C₄	—	268	2	no	—
221	C₂×C₄	—	268	2	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	134	2	no	—
224	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
225	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
226	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
228	C₂×C₄	#14523982	1	2	yes	—
229	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
231	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	268	2	no	—
236	C₈	#14524068	1	2	yes	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	268	8	no	—
240	C₂²	—	268	2	no	—
241	C₂²	—	268	2	no	—
242	C₂²	—	134	4	no	—
243	C₂²	(C₂×D₆₇)⋊C₁₆	1	2	yes	—
244	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	268	2	no	—
249	C₄	(C₂×D₆₇)⋊C₁₆	1	2	yes	—
250	C₄	#16863400	1	1	yes	characteristic
251	C₄	#16863412	1	1	yes	characteristic
252	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
253	C₄	C₂×#14523982	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	268	4	no	—
256	C₂	#22558599	1	1	yes	characteristic
257	C₂	C₂×(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
258	C₂	#22561838	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2162	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₂₆₈	C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2162	D₆₇
hypocenter	C₆₇	#11629
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #22562008	C₂	semidirect
3. #22558163	C₂	semidirect
4. C₆₇×#2162	C₂	semidirect
15. C₄×C₆₇⋊C₁₆	C₂²	semidirect
17. C₄⋊C₁₆×C₆₇	C₂²	semidirect
48. C₂×D₂₆₈	C₁₆	semidirect
132. #2162	D₆₇	semidirect
134. C₆₇	#11629	semidirect
145. C₄⋊C₁₆	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₆₈	2
#22562008	2
#22558163	2
C₆₇×#2162	2
#11629	67