Subgroups — #30749091

Order 17152.

362 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569143	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558897	C₂	1	2	yes	maximal
4	#22558084	C₂	1	2	yes	maximal
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863526	—	2	1	no	—
8	#16863526	C₄	1	2	yes	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	—
11	C₂×#14523979	—	2	1	no	—
12	C₂×#14523979	—	2	1	no	—
13	#16863422	—	2	1	no	—
14	#16863420	C₂²	1	2	yes	—
15	#16863420	C₂²	1	1	yes	characteristic
16	#16863351	—	2	1	no	—
17	#16863144	—	2	2	no	—
18	#16863143	—	2	2	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#335×C₆₇	—	2	1	no	—
22	#335×C₆₇	—	2	1	no	—
23	C₂×#11761612	—	2	1	no	—
24	C₂×#11761612	—	2	2	no	—
25	#14524043	D₄	1	1	yes	characteristic
26	#14524043	D₄	1	1	yes	characteristic
27	#14524035	—	2	1	no	—
28	#14524022	—	2	1	no	—
29	#14524022	Q₈	1	2	yes	—
30	C₄×Dic₁₃₄	—	2	1	no	—
31	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
32	#14524002	—	2	1	no	—
33	#14523979	—	4	2	no	—
34	#14523979	—	4	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
39	Dic₆₇⋊C₈	—	2	2	no	—
40	Dic₆₇⋊C₈	—	2	2	no	—
41	#14523953	—	2	2	no	—
42	C₈×Dic₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	—	2	2	no	—
48	C₄⋊C₈×C₆₇	—	2	2	no	—
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂×Dic₁₃₄	—	2	1	no	—
54	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
55	#11761612	—	4	1	no	—
56	#11761612	—	4	1	no	—
57	#11761612	—	4	2	no	—
58	#11761612	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊Dic₆₇	—	2	1	no	—
63	C₄⋊Dic₆₇	—	2	1	no	—
64	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
65	Dic₆₇⋊C₄	—	2	2	no	—
66	Dic₆₇⋊C₄	—	2	2	no	—
67	C₂²⋊Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	8	2	no	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
81	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	#41	1	1	yes	characteristic
86	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	4	1	no	—
90	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	Dic₁₃₄	—	4	1	no	—
103	Dic₁₃₄	—	4	1	no	—
104	Dic₁₃₄	—	4	2	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	8	2	no	—
108	C₈×C₆₇	#103	1	2	yes	—
109	C₈×C₆₇	#103	1	2	yes	—
110	C₆₇⋊C₈	—	4	2	no	—
111	C₆₇⋊C₈	—	8	2	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	#380	1	1	yes	characteristic
115	C₂²×C₆₇	#475	1	1	yes	characteristic
116	C₂²×C₆₇	#477	1	1	yes	characteristic
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
121	C₄×C₆₇	#417	1	1	yes	characteristic
122	C₄×C₆₇	#460	1	1	yes	characteristic
123	C₄×C₆₇	#462	1	1	yes	characteristic
124	C₄×C₆₇	#472	1	1	yes	characteristic
125	C₄×C₆₇	#473	1	1	yes	characteristic
126	Dic₆₇	—	4	2	no	—
127	Dic₆₇	—	4	2	no	—
128	#8685	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	1	no	—
130	C₂×C₆₇	—	2	1	no	—
131	C₂×C₆₇	#1803	1	1	yes	characteristic
132	C₂×C₆₇	#1813	1	1	yes	characteristic
133	C₂×C₆₇	#1816	1	1	yes	characteristic
134	#2960	—	67	1	no	—
135	#1848	—	67	1	no	—
136	C₂×#335	D₆₇	1	1	yes	characteristic
137	#1550	—	67	2	no	—
138	#1292	—	67	2	no	—
139	C₆₇	#8685	1	1	yes	characteristic, complemented, Sylow
140	#533	—	67	1	no	—
141	#446	—	67	2	no	—
142	#446	—	134	1	no	—
143	C₈×Q₈	—	67	2	no	—
144	C₈×Q₈	—	134	1	no	—
145	C₂×#108	—	134	1	no	—
146	C₂×#108	—	134	1	no	—
147	#424	—	134	1	no	—
148	#423	—	67	1	no	—
149	#423	—	67	2	no	—
150	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
151	#405	—	134	1	no	—
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	#335	—	2	1	no	—
154	#335	—	2	1	no	—
155	#333	—	134	2	no	—
156	#332	—	134	2	no	—
157	C₂×M₁₆	—	134	2	no	—
158	C₂×M₁₆	—	134	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
161	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
162	C₄⋊Q₈	—	67	1	no	—
163	C₄⋊Q₈	—	67	1	no	—
164	#125	—	134	1	no	—
165	C₂²⋊Q₈	—	67	2	no	—
166	C₂²⋊Q₈	—	134	1	no	—
167	C₄×Q₈	—	67	2	no	—
168	C₄×Q₈	—	134	1	no	—
169	#117	—	134	1	no	—
170	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
171	#108	—	268	2	no	—
172	#108	—	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	2	no	—
177	C₄⋊C₈	—	134	2	no	—
178	C₄⋊C₈	—	134	2	no	—
179	C₄⋊C₈	—	134	1	no	—
180	C₄⋊C₈	—	134	2	no	—
181	#97	—	134	2	no	—
182	C₄×C₈	—	134	2	no	—
183	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
184	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂×Q₈	—	67	2	no	—
188	C₂×Q₈	—	134	1	no	—
189	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	#11761627	1	1	yes	characteristic
192	M₁₆	—	268	1	no	—
193	M₁₆	—	268	1	no	—
194	M₁₆	—	268	2	no	—
195	M₁₆	—	268	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	8	2	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	—	134	2	no	—
202	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	#11761617	1	2	yes	—
207	C₂×C₈	Dic₂₆₈	1	2	yes	—
208	C₄⋊C₄	—	67	2	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	1	no	—
211	C₄⋊C₄	—	134	1	no	—
212	C₄⋊C₄	—	134	1	no	—
213	C₄⋊C₄	—	134	2	no	—
214	C₂²⋊C₄	—	134	2	no	—
215	C₄²	—	134	2	no	—
216	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₄²	D₄×D₆₇	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₄²	#11761629	1	1	yes	characteristic
220	C₂³	#14524023	1	1	yes	characteristic
221	Q₈	—	268	2	no	—
222	Q₈	—	268	1	no	—
223	Q₈	—	268	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	134	2	no	—
228	C₂×C₄	—	134	2	no	—
229	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
231	C₂×C₄	#14524001	1	1	yes	characteristic
232	C₂×C₄	#14524014	1	1	yes	characteristic
233	C₂×C₄	#14524018	1	1	yes	characteristic
234	C₂×C₄	#14524019	1	1	yes	characteristic
235	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
236	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
237	C₂×C₄	#14524071	1	1	yes	characteristic
238	C₂×C₄	#14524076	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	8	2	no	—
242	C₈	—	268	2	no	—
243	C₈	—	536	2	no	—
244	C₈	#14523968	1	2	yes	—
245	C₈	#14523968	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#16863253	1	1	yes	characteristic
249	C₂²	#16863682	1	1	yes	characteristic
250	C₂²	#16863692	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	268	2	no	—
255	C₄	—	268	2	no	—
256	C₄	C₂×#14523968	1	1	yes	characteristic
257	C₄	#16863398	1	1	yes	characteristic
258	C₄	#16863597	1	1	yes	characteristic
259	C₄	#16863607	1	1	yes	characteristic
260	C₄	#16863663	1	1	yes	characteristic
261	C₄	#16863668	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	#22560215	1	1	yes	characteristic
265	C₂	#22560256	1	1	yes	characteristic
266	C₂	#22560274	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#8685
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
139. C₆₇	#8685	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
#22560381	2
#22558897	2
#22558084	2
C₂×#335×C₆₇	2
#8685	67