Subgroups — #30748902

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 296 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	#22560206	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558092	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₆₇	—	2	1	no	—
7	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
8	C₈×D₂₆₈	—	2	2	no	—
9	C₈×D₂₆₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14523963	—	2	2	no	—
13	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
14	#16863158	—	2	2	no	—
15	#16863149	—	2	2	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	C₂²	1	2	yes	—
18	#340×C₆₇	—	2	1	no	—
19	#340×C₆₇	—	2	1	no	—
20	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₆₇	—	2	1	no	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂×C₈×D₆₇	—	2	4	no	—
25	C₂×C₈×D₆₇	—	4	4	no	—
26	C₄×D₂₆₈	—	2	1	no	—
27	C₄×D₂₆₈	—	2	1	no	—
28	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
29	C₄×D₂₆₈	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
31	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
36	#14523963	—	4	4	no	—
37	(C₂×D₆₇)⋊C₈	—	2	4	no	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₂³×D₆₇	—	2	1	no	—
47	C₂×D₂₆₈	—	2	1	no	—
48	C₂×D₂₆₈	—	2	1	no	—
49	C₂×D₂₆₈	—	2	1	no	—
50	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
51	C₂×D₂₆₈	D₈	1	2	yes	—
52	C₂×D₂₆₈	Q₁₆	1	2	yes	—
53	C₂²×Dic₆₇	—	2	1	no	—
54	C₂×C₄×D₆₇	—	2	2	no	—
55	C₂×C₄×D₆₇	—	2	2	no	—
56	C₂×C₄×D₆₇	—	4	2	no	—
57	C₂×C₄×D₆₇	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₈×D₆₇	—	4	8	no	—
64	C₄⋊Dic₆₇	—	2	1	no	—
65	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
66	(C₂×D₆₇)⋊C₄	—	2	2	no	—
67	(C₂×D₆₇)⋊C₄	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	8	2	no	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₂³×C₆₇	#121	1	1	yes	characteristic
86	C₂²×D₆₇	—	2	2	no	—
87	C₂²×D₆₇	—	2	2	no	—
88	C₂²×D₆₇	—	2	2	no	—
89	C₂²×D₆₇	—	4	2	no	—
90	C₂²×D₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	8	2	no	—
93	C₂×C₄×C₆₇	—	8	2	no	—
94	C₂×C₄×C₆₇	—	8	2	no	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#135	1	2	yes	—
105	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
106	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	4	2	no	—
109	D₂₆₈	—	2	2	no	—
110	D₂₆₈	—	2	2	no	—
111	D₂₆₈	—	2	2	no	—
112	D₂₆₈	#107	1	4	yes	—
113	C₄×D₆₇	—	4	2	no	—
114	C₄×D₆₇	—	4	2	no	—
115	C₄×D₆₇	—	4	4	no	—
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	—	2	2	no	—
118	C₆₇⋊C₈	—	4	2	no	—
119	C₆₇⋊C₈	—	8	2	no	—
120	C₂²×C₆₇	#380	1	1	yes	characteristic
121	C₂²×C₆₇	#463	1	2	yes	—
122	C₂²×C₆₇	#466	1	1	yes	characteristic
123	C₂²×C₆₇	#467	1	1	yes	characteristic
124	C₂²×C₆₇	#471	1	2	yes	—
125	C₂×D₆₇	—	2	4	no	—
126	C₂×D₆₇	—	4	2	no	—
127	C₂×D₆₇	—	4	2	no	—
128	C₂×D₆₇	—	4	4	no	—
129	C₂×D₆₇	—	4	4	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	16	2	no	—
133	C₄×C₆₇	#416	1	1	yes	characteristic
134	C₄×C₆₇	#418	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
136	C₄×C₆₇	#427	1	1	yes	characteristic
137	C₄×C₆₇	#466	1	1	yes	characteristic
138	C₄×C₆₇	#467	1	1	yes	characteristic
139	C₄×C₆₇	#478	1	2	yes	—
140	Dic₆₇	—	4	2	no	—
141	#8648	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	#1681	1	2	yes	—
143	C₂×C₆₇	#1683	1	2	yes	—
144	C₂×C₆₇	#1804	1	1	yes	characteristic
145	C₂×C₆₇	#1808	1	1	yes	characteristic
146	C₂×C₆₇	#1815	1	1	yes	characteristic
147	D₆₇	—	4	4	no	—
148	C₂×C₈×D₄	—	67	1	no	—
149	#1834	D₆₇	1	1	yes	characteristic, complemented
150	#1802	—	67	2	no	—
151	C₂×#335	—	67	1	no	—
152	#1296	—	67	2	no	—
153	C₆₇	#8648	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₈	—	134	1	no	—
155	C₂×C₄×D₄	—	67	1	no	—
156	C₈×D₄	—	67	4	no	—
157	C₈×D₄	—	134	2	no	—
158	C₂×C₄⋊C₈	—	67	1	no	—
159	C₂×C₄⋊C₈	—	67	2	no	—
160	C₂×#102	—	134	2	no	—
161	C₂×C₂²⋊C₈	—	134	1	no	—
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	#384	C₂×D₆₇	1	2	yes	—
164	#340	—	2	1	no	—
165	#340	—	2	1	no	—
166	#340	—	134	2	no	—
167	#335	—	134	2	no	—
168	C₂²×D₄	—	67	1	no	—
169	C₂³×C₄	—	134	1	no	—
170	C₂²×C₈	—	268	4	no	—
171	C₂²×C₈	—	134	2	no	—
172	C₂²×C₈	—	134	1	no	—
173	C₂²×C₈	—	134	4	no	—
174	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
176	C₄×D₄	—	67	2	no	—
177	C₄×D₄	—	67	2	no	—
178	C₄×D₄	—	134	1	no	—
179	C₄×D₄	—	134	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	67	1	no	—
184	C₂×C₂²⋊C₄	—	134	1	no	—
185	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
186	C₄⋊C₈	—	67	2	no	—
187	C₄⋊C₈	—	134	2	no	—
188	C₄⋊C₈	—	134	1	no	—
189	C₄⋊C₈	—	134	2	no	—
190	#102	—	268	4	no	—
191	C₂²⋊C₈	—	134	4	no	—
192	C₄×C₈	C₄×D₆₇	1	2	yes	—
193	C₄×C₈	D₂₆₈	1	2	yes	—
194	C₂⁴	—	134	1	no	—
195	C₂×D₄	—	67	2	no	—
196	C₂×D₄	—	67	2	no	—
197	C₂×D₄	—	67	2	no	—
198	C₂×D₄	—	134	1	no	—
199	C₂×D₄	—	134	1	no	—
200	C₂×D₄	—	134	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	268	2	no	—
203	C₂²×C₄	—	268	2	no	—
204	C₂²×C₄	—	134	1	no	—
205	C₂²×C₄	—	134	2	no	—
206	C₂²×C₄	—	134	2	no	—
207	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂²×C₄	#11761627	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	268	8	no	—
213	C₂×C₈	—	268	2	no	—
214	C₂×C₈	—	268	2	no	—
215	C₂×C₈	—	268	2	no	—
216	C₂×C₈	—	268	2	no	—
217	C₂×C₈	—	134	2	no	—
218	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
219	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	8	2	no	—
225	C₄⋊C₄	—	67	2	no	—
226	C₄⋊C₄	—	134	1	no	—
227	C₂²⋊C₄	—	134	2	no	—
228	C₂²⋊C₄	—	134	2	no	—
229	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
230	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
231	C₄²	Q₈×D₆₇	1	1	yes	characteristic
232	C₄²	#11761629	1	1	yes	characteristic
233	C₂³	—	268	2	no	—
234	C₂³	—	268	2	no	—
235	C₂³	—	134	2	no	—
236	C₂³	—	134	2	no	—
237	C₂³	—	134	2	no	—
238	C₂³	#14524014	1	1	yes	characteristic
239	D₄	—	67	4	no	—
240	D₄	—	134	2	no	—
241	D₄	—	134	2	no	—
242	D₄	—	134	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	268	2	no	—
248	C₂×C₄	—	268	2	no	—
249	C₂×C₄	—	268	2	no	—
250	C₂×C₄	—	268	4	no	—
251	C₂×C₄	—	134	2	no	—
252	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
254	C₂×C₄	#14524001	1	1	yes	characteristic
255	C₂×C₄	#14524018	1	1	yes	characteristic
256	C₂×C₄	#14524019	1	1	yes	characteristic
257	C₂×C₄	#14524023	1	1	yes	characteristic
258	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
259	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
260	C₂×C₄	#14524062	1	1	yes	characteristic
261	C₂×C₄	#14524063	1	2	yes	—
262	C₂×C₄	#14524066	1	1	yes	characteristic
263	C₂×C₄	#14524068	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	268	2	no	—
267	C₈	—	536	2	no	—
268	C₂²	—	268	2	no	—
269	C₂²	—	268	4	no	—
270	C₂²	—	268	4	no	—
271	C₂²	—	268	2	no	—
272	C₂²	—	134	4	no	—
273	C₂²	#16863253	1	1	yes	characteristic
274	C₂²	#16863608	1	2	yes	—
275	C₂²	#16863629	1	1	yes	characteristic
276	C₂²	#16863634	1	1	yes	characteristic
277	C₂²	#16863656	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	268	2	no	—
281	C₄	—	16	2	no	—
282	C₄	#16863390	1	1	yes	characteristic
283	C₄	#16863400	1	1	yes	characteristic
284	C₄	#107×D₆₇	1	1	yes	characteristic
285	C₄	#16863433	1	1	yes	characteristic
286	C₄	#16863633	1	1	yes	characteristic
287	C₄	#16863638	1	1	yes	characteristic
288	C₄	#16863693	1	2	yes	—
289	C₂	—	268	4	no	—
290	C₂	#22559643	1	2	yes	—
291	C₂	#22559657	1	2	yes	—
292	C₂	#22560216	1	1	yes	characteristic
293	C₂	#22560235	1	1	yes	characteristic
294	C₂	#22560269	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1834	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂²×D₂₆₈	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1834	D₆₇
hypocenter	C₆₇	#8648
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
3. C₂×#16863149	C₂	semidirect
4. #22558092	C₂	semidirect
5. C₆₇×#1834	C₂	semidirect
9. C₈×D₂₆₈	C₄	semidirect
149. #1834	D₆₇	semidirect
153. C₆₇	#8648	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
#22560206	2
C₂×#16863149	2
#22558092	2
C₆₇×#1834	2
#8648	67