Subgroups — #30748921

Order 17152.

434 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863526	C₂	1	1	yes	maximal, characteristic
2	#22560366	C₂	1	1	yes	maximal, characteristic
3	#22560215	C₂	1	2	yes	maximal
4	#22558093	C₂	1	2	yes	maximal
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	#16863526	—	2	2	no	—
8	#16863526	C₄	1	4	yes	complemented
9	C₂×#14523975	—	2	1	no	—
10	C₂×#14523975	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×#14523968	—	2	2	no	—
14	C₂×#14523953	—	2	1	no	—
15	#16863266	C₂²	1	2	yes	—
16	#16863159	—	2	2	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#335×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₃₄	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
30	C₂×Dic₆₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	#14523975	—	4	2	no	—
33	#14523975	—	4	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₆₇⋊C₈	—	2	4	no	—
37	#14523968	—	4	4	no	—
38	#14523953	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂×Dic₁₃₄	—	2	2	no	—
48	C₂×Dic₁₃₄	—	2	2	no	—
49	C₂×Dic₁₃₄	—	2	1	no	—
50	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₆₇	—	2	1	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	4	1	no	—
56	C₂×C₆₇⋊C₈	—	4	1	no	—
57	C₄⋊Dic₆₇	—	2	1	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	8	2	no	—
63	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
64	Dic₆₇⋊C₄	—	2	2	no	—
65	Dic₆₇⋊C₄	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
79	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₂³×C₆₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#137	1	2	yes	—
97	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
98	C₂×Dic₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	2	2	no	—
100	C₂×Dic₆₇	—	4	2	no	—
101	C₂×Dic₆₇	—	4	2	no	—
102	C₂×Dic₆₇	—	8	2	no	—
103	C₂×Dic₆₇	—	8	2	no	—
104	C₂×Dic₆₇	—	8	2	no	—
105	Dic₁₃₄	—	2	4	no	—
106	Dic₁₃₄	—	4	1	no	—
107	Dic₁₃₄	—	4	1	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	—	8	2	no	—
110	C₈×C₆₇	#103	1	4	yes	complemented
111	C₆₇⋊C₈	—	4	2	no	—
112	C₂²×C₆₇	#380	1	1	yes	characteristic
113	C₂²×C₆₇	#460	1	1	yes	characteristic
114	C₂²×C₆₇	#462	1	1	yes	characteristic
115	C₂²×C₆₇	#468	1	2	yes	—
116	C₂²×C₆₇	#471	1	2	yes	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
120	C₄×C₆₇	#417	1	1	yes	characteristic
121	C₄×C₆₇	#428	1	2	yes	—
122	C₄×C₆₇	#469	1	1	yes	characteristic
123	C₄×C₆₇	#470	1	1	yes	characteristic
124	C₄×C₆₇	#475	1	1	yes	characteristic
125	C₄×C₆₇	#477	1	1	yes	characteristic
126	Dic₆₇	—	4	2	no	—
127	Dic₆₇	—	4	2	no	—
128	Dic₆₇	—	16	2	no	—
129	#8651	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	#1686	1	2	yes	—
131	C₂×C₆₇	#1692	1	2	yes	—
132	C₂×C₆₇	#1803	1	1	yes	characteristic
133	C₂×C₆₇	#1809	1	1	yes	characteristic
134	C₂×C₆₇	#1815	1	1	yes	characteristic
135	C₂×#446	—	67	1	no	—
136	#1844	—	67	1	no	—
137	#1803	—	67	2	no	—
138	C₂×#335	D₆₇	1	1	yes	characteristic
139	#1296	—	67	2	no	—
140	C₆₇	#8651	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	—	67	1	no	—
142	#446	—	67	4	no	—
143	#446	—	134	2	no	—
144	C₂×#106	—	134	1	no	—
145	C₂×#106	—	134	1	no	—
146	C₂×C₄⋊C₈	—	67	1	no	—
147	C₂×C₄⋊C₈	—	134	1	no	—
148	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
149	C₂×#103	—	134	2	no	—
150	C₂×#97	—	134	1	no	—
151	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
152	#384	—	67	2	no	—
153	#340	—	134	2	no	—
154	#335	—	2	2	no	—
155	C₂²×Q₈	—	67	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	134	1	no	—
158	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
159	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
160	C₄×Q₈	—	67	2	no	—
161	C₄×Q₈	—	67	2	no	—
162	C₄×Q₈	—	134	1	no	—
163	C₄×Q₈	—	134	1	no	—
164	C₂×C₄⋊C₄	—	67	1	no	—
165	C₂×C₄⋊C₄	—	134	2	no	—
166	C₂×C₄⋊C₄	—	134	2	no	—
167	C₂×C₄⋊C₄	—	134	2	no	—
168	C₂×C₄⋊C₄	—	134	1	no	—
169	C₂×C₄²	—	134	1	no	—
170	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
171	#106	—	268	2	no	—
172	#106	—	268	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	67	2	no	—
176	C₄⋊C₈	—	134	4	no	—
177	C₄⋊C₈	—	134	1	no	—
178	#103	—	268	4	no	—
179	#97	—	134	4	no	—
180	C₄×C₈	C₄×D₆₇	1	2	yes	—
181	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
182	C₂×Q₈	—	67	2	no	—
183	C₂×Q₈	—	134	2	no	—
184	C₂×Q₈	—	134	2	no	—
185	C₂×Q₈	—	134	1	no	—
186	C₂²×C₄	—	268	2	no	—
187	C₂²×C₄	—	134	1	no	—
188	C₂²×C₄	—	134	1	no	—
189	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
191	C₂²×C₄	#11761629	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	268	1	no	—
198	C₂×C₈	—	268	1	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
202	C₂×C₈	#11761617	1	2	yes	—
203	C₂×C₈	Dic₂₆₈	1	2	yes	—
204	C₄⋊C₄	—	268	2	no	—
205	C₄⋊C₄	—	268	2	no	—
206	C₄⋊C₄	—	268	2	no	—
207	C₄⋊C₄	—	268	2	no	—
208	C₄⋊C₄	—	536	2	no	—
209	C₄⋊C₄	—	67	2	no	—
210	C₄⋊C₄	—	134	2	no	—
211	C₄⋊C₄	—	134	1	no	—
212	C₄⋊C₄	—	134	2	no	—
213	C₄²	—	134	2	no	—
214	C₄²	—	134	2	no	—
215	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
217	C₄²	Q₈×D₆₇	1	1	yes	characteristic
218	C₄²	#11761627	1	1	yes	characteristic
219	C₂³	#14524018	1	1	yes	characteristic
220	Q₈	—	268	1	no	—
221	Q₈	—	268	1	no	—
222	Q₈	—	134	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	268	2	no	—
225	C₂×C₄	—	268	2	no	—
226	C₂×C₄	—	536	2	no	—
227	C₂×C₄	—	536	2	no	—
228	C₂×C₄	—	536	2	no	—
229	C₂×C₄	—	134	2	no	—
230	C₂×C₄	—	134	2	no	—
231	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
233	C₂×C₄	#14524001	1	1	yes	characteristic
234	C₂×C₄	#14524014	1	1	yes	characteristic
235	C₂×C₄	#14524019	1	1	yes	characteristic
236	C₂×C₄	#14524023	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
238	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
239	C₂×C₄	#14524067	1	2	yes	—
240	C₂×C₄	#14524071	1	1	yes	characteristic
241	C₂×C₄	#14524075	1	2	yes	—
242	C₂×C₄	#14524076	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	268	2	no	—
246	C₈	#14523968	1	4	yes	complemented
247	C₂²	#16863253	1	1	yes	characteristic
248	C₂²	#16863597	1	1	yes	characteristic
249	C₂²	#16863607	1	1	yes	characteristic
250	C₂²	#16863643	1	2	yes	—
251	C₂²	#16863660	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	268	2	no	—
255	C₄	—	268	2	no	—
256	C₄	—	1072	2	no	—
257	C₄	C₂×#14523968	1	1	yes	characteristic
258	C₄	#16863398	1	1	yes	characteristic
259	C₄	#16863437	1	2	yes	—
260	C₄	#16863646	1	1	yes	characteristic
261	C₄	#16863651	1	1	yes	characteristic
262	C₄	#16863682	1	1	yes	characteristic
263	C₄	#16863692	1	1	yes	characteristic
264	C₂	#22559682	1	2	yes	—
265	C₂	#22559716	1	2	yes	—
266	C₂	#22560215	1	1	yes	characteristic
267	C₂	#22560243	1	1	yes	characteristic
268	C₂	#22560265	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#8651
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. #16863526	C₄	semidirect
110. C₈×C₆₇	#103	semidirect
140. C₆₇	#8651	semidirect
246. C₈	#14523968	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863526	2
#22560366	2
#22560215	2
#22558093	2
C₂×#335×C₆₇	2
#8651	67