Subgroups — #30749490

Order 17152.

422 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	#22560360	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558044	C₂	1	1	yes	maximal, characteristic
4	#22558043	C₂	1	2	yes	maximal
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₆₇⋊C₈	—	2	2	no	—
8	C₂×Dic₆₇⋊C₈	—	2	2	no	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
14	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	Dic₆₇⋊C₈	—	4	4	no	—
29	Dic₆₇⋊C₈	—	4	4	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₈×Dic₆₇	—	2	2	no	—
32	C₈×Dic₆₇	—	2	2	no	—
33	C₈×Dic₆₇	—	2	2	no	—
34	C₈×Dic₆₇	C₈	1	4	yes	complemented
35	C₂²×C₈×C₆₇	—	2	2	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	—	2	2	no	—
42	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	—	2	1	no	—
47	C₂×C₆₇⋊C₈	—	2	1	no	—
48	C₂×C₆₇⋊C₈	—	2	1	no	—
49	C₂×C₆₇⋊C₈	—	2	1	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
54	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
55	C₄×Dic₆₇	—	2	1	no	—
56	C₄×Dic₆₇	—	2	1	no	—
57	C₄×Dic₆₇	—	2	2	no	—
58	C₄×Dic₆₇	—	2	2	no	—
59	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
60	C₄×Dic₆₇	M₁₆	1	2	yes	—
61	C₄×Dic₆₇	D₈	1	2	yes	—
62	C₄×Dic₆₇	SD₁₆	1	4	yes	—
63	C₄×Dic₆₇	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	no	—
69	C₂×C₈×C₆₇	—	2	1	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
76	C₄²×C₆₇	#41	1	2	yes	—
77	C₄²×C₆₇	#41	1	2	yes	—
78	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#124	1	2	yes	—
84	C₂×C₄×C₆₇	#125	1	2	yes	—
85	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×Dic₆₇	—	2	1	no	—
93	C₂×Dic₆₇	—	2	1	no	—
94	C₂×Dic₆₇	—	2	1	no	—
95	C₂×Dic₆₇	—	2	1	no	—
96	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
97	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
98	C₂×Dic₆₇	#106	1	2	yes	—
99	C₂×Dic₆₇	#106	1	2	yes	—
100	C₂×Dic₆₇	#107	1	2	yes	—
101	C₂×Dic₆₇	#107	1	2	yes	—
102	C₂×Dic₆₇	#108	1	2	yes	—
103	C₂×Dic₆₇	#108	1	2	yes	—
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	—	8	2	no	—
107	C₆₇⋊C₈	—	2	2	no	—
108	C₆₇⋊C₈	—	2	2	no	—
109	C₆₇⋊C₈	—	8	2	no	—
110	C₂²×C₆₇	#382	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₆₇	#423	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
116	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	#431	1	1	yes	characteristic
119	C₄×C₆₇	#431	1	1	yes	characteristic
120	C₄×C₆₇	#432	1	1	yes	characteristic
121	C₄×C₆₇	#432	1	1	yes	characteristic
122	C₄×C₆₇	#486	1	2	yes	—
123	C₄×C₆₇	#487	1	2	yes	—
124	Dic₆₇	—	2	2	no	—
125	Dic₆₇	—	2	2	no	—
126	Dic₆₇	#334	1	4	yes	—
127	Dic₆₇	#335	1	4	yes	—
128	#8784	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	C₂×#334	1	1	yes	characteristic
130	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
131	C₂×C₆₇	#1566	1	1	yes	characteristic
132	C₂×C₆₇	#1567	1	1	yes	characteristic
133	C₂×C₆₇	#1831	1	1	yes	characteristic
134	C₂×C₆₇	#1834	1	1	yes	characteristic
135	C₂×C₆₇	#1837	1	1	yes	characteristic
136	#1842	—	67	2	no	—
137	C₄²×C₈	—	67	1	no	—
138	#1277	—	67	2	no	—
139	#1277	—	67	1	no	—
140	#1277	D₆₇	1	1	yes	characteristic
141	C₆₇	#8784	1	1	yes	characteristic, complemented, Sylow
142	C₂×C₄⋊C₈	—	67	2	no	—
143	C₂×C₄⋊C₈	—	134	2	no	—
144	C₂×C₄⋊C₈	—	134	2	no	—
145	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
146	C₂×C₄×C₈	—	67	1	no	—
147	C₂×C₄×C₈	—	67	1	no	—
148	C₂×C₄×C₈	—	67	1	no	—
149	C₂×C₄×C₈	—	67	1	no	—
150	C₂×C₄×C₈	—	67	1	no	—
151	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
152	C₄³	—	67	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	67	1	no	—
155	C₂²×C₈	—	67	1	no	—
156	C₂²×C₈	—	134	2	no	—
157	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
159	C₂×C₄²	—	67	1	no	—
160	C₂×C₄²	—	67	1	no	—
161	C₂×C₄²	—	67	1	no	—
162	C₂×C₄²	—	67	1	no	—
163	C₂×C₄²	—	67	1	no	—
164	C₂×C₄²	—	67	1	no	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	268	4	no	—
169	C₄⋊C₈	—	268	4	no	—
170	C₄⋊C₈	—	134	2	no	—
171	C₄⋊C₈	—	134	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	67	4	no	—
174	C₄×C₈	—	134	2	no	—
175	C₄×C₈	—	134	2	no	—
176	C₄×C₈	—	134	2	no	—
177	C₄×C₈	—	134	2	no	—
178	C₂²×C₄	—	67	1	no	—
179	C₂²×C₄	—	67	1	no	—
180	C₂²×C₄	—	67	1	no	—
181	C₂²×C₄	—	67	1	no	—
182	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	268	2	no	—
193	C₂×C₈	—	268	2	no	—
194	C₂×C₈	—	268	2	no	—
195	C₂×C₈	—	67	2	no	—
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	134	1	no	—
198	C₂×C₈	—	134	1	no	—
199	C₂×C₈	—	134	1	no	—
200	C₂×C₈	—	134	1	no	—
201	C₂×C₈	C₈×D₆₇	1	2	yes	—
202	C₂×C₈	#11761611	1	2	yes	—
203	C₄²	—	67	2	no	—
204	C₄²	—	67	2	no	—
205	C₄²	—	67	2	no	—
206	C₄²	—	67	4	no	—
207	C₄²	—	67	2	no	—
208	C₄²	—	134	2	no	—
209	C₄²	—	134	2	no	—
210	C₄²	—	134	1	no	—
211	C₄²	—	134	1	no	—
212	C₄²	#11761627	1	2	yes	—
213	C₄²	#11761628	1	2	yes	—
214	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	67	2	no	—
218	C₂×C₄	—	67	2	no	—
219	C₂×C₄	—	67	2	no	—
220	C₂×C₄	—	67	2	no	—
221	C₂×C₄	—	67	2	no	—
222	C₂×C₄	—	67	2	no	—
223	C₂×C₄	—	67	2	no	—
224	C₂×C₄	—	67	2	no	—
225	C₂×C₄	—	134	1	no	—
226	C₂×C₄	—	134	1	no	—
227	C₂×C₄	—	134	1	no	—
228	C₂×C₄	—	134	1	no	—
229	C₂×C₄	#14523999	1	1	yes	characteristic
230	C₂×C₄	#14524000	1	1	yes	characteristic
231	C₂×C₄	#14524029	1	2	yes	—
232	C₂×C₄	#14524033	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
234	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
237	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
238	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
239	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	536	2	no	—
244	C₈	—	134	2	no	—
245	C₈	—	134	2	no	—
246	C₂²	#16863260	1	1	yes	characteristic
247	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
248	C₂²	#16863418	1	1	yes	characteristic
249	C₂²	#106×D₆₇	1	1	yes	characteristic
250	C₂²	#107×D₆₇	1	1	yes	characteristic
251	C₂²	#108×D₆₇	1	1	yes	characteristic
252	C₂²	#108×D₆₇	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	67	4	no	—
255	C₄	—	67	4	no	—
256	C₄	—	134	2	no	—
257	C₄	—	134	2	no	—
258	C₄	#16863446	1	1	yes	characteristic
259	C₄	#16863447	1	1	yes	characteristic
260	C₄	#16863450	1	1	yes	characteristic
261	C₄	#16863451	1	1	yes	characteristic
262	C₄	#16863741	1	2	yes	—
263	C₄	#16863745	1	2	yes	—
264	C₂	#334×D₆₇	1	1	yes	characteristic
265	C₂	#335×D₆₇	1	1	yes	characteristic
266	C₂	#22558963	1	1	yes	characteristic
267	C₂	#22558966	1	1	yes	characteristic
268	C₂	#22560324	1	1	yes	characteristic
269	C₂	#22560335	1	1	yes	characteristic
270	C₂	#22560345	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1277	D₆₇
hypocenter	C₆₇	#8784
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₆₇	C₈	semidirect
141. C₆₇	#8784	semidirect

Maximal subgroups

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

subgroup	index
#22560360	2
C₄×C₈×Dic₆₇	2
#22558044	2
#22558043	2
C₆₇×#1277	2
#8784	67