Subgroups — #30826398

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22568922	C₂	1	2	yes	maximal, complemented
3	#22559124	C₂	1	4	yes	maximal, complemented
4	#22559034	C₂	1	4	yes	maximal, complemented
5	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558661	C₂	1	2	yes	maximal, complemented
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524049	—	2	2	no	—
9	#16863946	C₂²	1	2	yes	—
10	#16863801	C₄	1	1	yes	characteristic
11	#16863773	C₄	1	1	yes	characteristic
12	#16863743	C₄	1	2	yes	—
13	#16863632	C₄	1	4	yes	—
14	C₄×D₅₃₆	C₂²	1	4	yes	—
15	#16863475	—	2	2	no	—
16	#16863475	C₂²	1	4	yes	—
17	C₈×D₂₆₈	—	2	2	no	—
18	C₈×D₂₆₈	C₂²	1	4	yes	—
19	#16863420	C₂²	1	2	yes	—
20	#16863362	—	2	2	no	—
21	#16863147	C₂²	1	4	yes	—
22	#16863125	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#322×C₆₇	C₂²	1	2	yes	—
27	#322×C₆₇	C₂²	1	2	yes	—
28	C₂×#11761628	—	2	2	no	—
29	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
30	#14524049	—	4	8	no	—
31	C₂×#11761612	—	2	2	no	—
32	C₂×#11761611	—	2	4	no	—
33	C₂×C₈×D₆₇	—	2	4	no	—
34	#14524043	C₂×C₄	1	2	yes	—
35	#14524040	C₂×C₄	1	2	yes	—
36	#14524031	—	2	2	no	—
37	#14524017	—	2	2	no	—
38	#14524017	C₂×C₄	1	4	yes	—
39	C₄×D₂₆₈	—	2	2	no	—
40	C₄×D₂₆₈	C₂³	1	4	yes	—
41	#14523977	C₂×C₄	1	4	yes	—
42	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
44	#14523965	C₂×C₄	1	8	yes	—
45	(C₂×D₆₇)⋊C₈	—	2	4	no	—
46	(C₂×D₆₇)⋊C₈	—	2	4	no	—
47	C₂²×C₈×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	D₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₇	—	2	1	no	—
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄×C₈×C₆₇	C₂³	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	#11761628	—	4	4	no	—
60	#11761628	—	4	4	no	—
61	C₂×Dic₁₃₄	—	2	2	no	—
62	C₂×D₂₆₈	—	2	2	no	—
63	C₂×D₂₆₈	—	2	2	no	—
64	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
65	C₂×C₂²⋊D₆₇	—	2	4	no	—
66	C₂×C₄×D₆₇	—	2	4	no	—
67	D₅₃₆	—	2	8	no	—
68	#11761612	—	4	4	no	—
69	#11761611	—	4	8	no	—
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₈×D₆₇	—	4	8	no	—
72	C₄⋊Dic₆₇	—	2	2	no	—
73	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
74	(C₂×D₆₇)⋊C₄	—	2	4	no	—
75	(C₂×D₆₇)⋊C₄	—	2	4	no	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂²×D₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	2	2	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	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₂²⋊D₆₇	—	4	8	no	—
103	C₂×Dic₆₇	—	2	4	no	—
104	D₂₆₈	—	2	8	no	—
105	D₂₆₈	—	4	4	no	—
106	D₂₆₈	—	4	4	no	—
107	C₄×D₆₇	—	4	4	no	—
108	C₄×D₆₇	—	4	4	no	—
109	Dic₁₃₄	—	4	4	no	—
110	C₈×C₆₇	—	2	2	no	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	—	4	2	no	—
114	C₈×C₆₇	#131	1	4	yes	complemented
115	C₆₇⋊C₈	—	4	4	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	#442	1	2	yes	—
119	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×D₆₇	—	4	4	no	—
121	C₂×D₆₇	—	4	4	no	—
122	C₂×D₆₇	—	4	4	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	#442	1	2	yes	—
126	C₄×C₆₇	#444	1	2	yes	—
127	C₄×C₆₇	C₂×#131	1	2	yes	—
128	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
129	Dic₆₇	—	4	4	no	—
130	#17962	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	2	2	no	—
132	C₂×C₆₇	#2931	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
135	D₆₇	—	8	4	no	—
136	#3169	—	67	1	no	—
137	#2931	—	67	2	no	—
138	#1594	—	67	4	no	—
139	#1580	—	67	4	no	—
140	#1566	—	67	1	no	—
141	#1492	—	67	2	no	—
142	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
143	C₆₇	#17962	1	1	yes	characteristic, complemented, Sylow
144	C₂×#131	—	134	2	no	—
145	#532	—	67	2	no	—
146	#500	—	67	1	no	—
147	#493	—	67	1	no	—
148	#486	—	67	2	no	—
149	#466	—	67	4	no	—
150	C₄×D₈	—	67	4	no	—
151	#436	—	67	4	no	—
152	#436	—	134	2	no	—
153	C₈×D₄	—	67	4	no	—
154	C₈×D₄	—	134	2	no	—
155	#423	—	67	2	no	—
156	#408	—	134	2	no	—
157	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	#334	—	67	4	no	—
161	D₄⋊C₈	—	67	8	no	—
162	#322	C₂×D₆₇	1	2	yes	—
163	#322	C₂×D₆₇	1	2	yes	—
164	C₂×#41	—	134	2	no	—
165	C₂×D₈	—	67	4	no	—
166	#131	—	268	8	no	—
167	C₂×M₁₆	—	134	2	no	—
168	C₂×M₁₆	—	134	4	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	134	4	no	—
171	C₂²×C₈	C₄×D₆₇	1	2	yes	—
172	C₂²×C₈	D₂₆₈	1	2	yes	—
173	C₄⋊Q₈	—	67	2	no	—
174	#127	—	67	2	no	—
175	#124	—	134	2	no	—
176	#121	—	67	4	no	—
177	#121	—	134	2	no	—
178	C₄×D₄	—	67	4	no	—
179	C₄×D₄	—	134	2	no	—
180	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
181	#107	—	67	4	no	—
182	C₄⋊C₈	—	67	4	no	—
183	C₄⋊C₈	—	134	2	no	—
184	#102	—	67	8	no	—
185	C₂²⋊C₈	—	134	4	no	—
186	C₂²⋊C₈	—	134	4	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₄×C₈	D₂₆₈	1	2	yes	—
191	C₄×C₈	C₂²×D₆₇	1	2	yes	—
192	C₄×C₈	C₂²×D₆₇	1	2	yes	—
193	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	#41	—	268	4	no	—
196	#41	—	268	4	no	—
197	C₂×Q₈	—	134	2	no	—
198	C₂×D₄	—	67	4	no	—
199	C₂×D₄	—	134	4	no	—
200	C₂×D₄	—	134	2	no	—
201	C₂×D₄	—	134	2	no	—
202	C₂²×C₄	—	134	4	no	—
203	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
205	C₂²×C₄	#11761628	1	1	yes	characteristic
206	D₈	—	134	8	no	—
207	M₁₆	—	268	8	no	—
208	M₁₆	—	268	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	268	8	no	—
214	C₂×C₈	—	134	4	no	—
215	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
217	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
218	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
219	C₄⋊C₄	—	67	4	no	—
220	C₄⋊C₄	—	134	2	no	—
221	C₂²⋊C₄	—	134	4	no	—
222	C₂²⋊C₄	—	134	4	no	—
223	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
225	C₄²	#11761628	1	1	yes	characteristic
226	C₄²	C₂³×D₆₇	1	1	yes	characteristic
227	C₂³	—	134	4	no	—
228	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
229	Q₈	—	268	4	no	—
230	D₄	—	268	8	no	—
231	D₄	—	268	4	no	—
232	D₄	—	268	4	no	—
233	D₄	—	134	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	268	4	no	—
238	C₂×C₄	—	134	4	no	—
239	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
240	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
241	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
242	C₂×C₄	#14524071	1	4	yes	—
243	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	268	4	no	—
251	C₈	#14524049	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	268	4	no	—
255	C₂²	—	268	4	no	—
256	C₂²	—	268	4	no	—
257	C₂²	#16863509	1	2	yes	—
258	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	268	4	no	—
262	C₄	#16863509	1	2	yes	—
263	C₄	#16863519	1	2	yes	—
264	C₄	C₂×#14524049	1	2	yes	—
265	C₄	C₂×#14524071	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	536	4	no	—
268	C₂	#22568922	1	1	yes	characteristic
269	C₂	C₂×#16863509	1	1	yes	characteristic
270	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570788	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#17962
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #22568922	C₂	semidirect
3. #22559124	C₂	semidirect
4. #22559034	C₂	semidirect
5. #22558965	C₂	semidirect
6. #22558661	C₂	semidirect
7. C₂×#322×C₆₇	C₂	semidirect
114. C₈×C₆₇	#131	semidirect
142. C₂×#322	D₆₇	semidirect
143. C₆₇	#17962	semidirect
251. C₈	#14524049	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22568922	2
#22559124	2
#22559034	2
#22558965	2
#22558661	2
C₂×#322×C₆₇	2
#17962	67