Subgroups — #30748977

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558637	C₂	1	2	yes	maximal
4	#22558104	C₂	1	2	yes	maximal
5	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	#16863526	—	2	1	no	—
8	#16863526	—	2	1	no	—
9	#16863526	C₄	1	2	yes	—
10	#16863526	C₄	1	2	yes	—
11	#16863449	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	—	2	1	no	—
14	C₂×#14523953	—	2	1	no	—
15	#16863149	—	2	1	no	—
16	#16863147	—	2	1	no	—
17	#16863126	—	2	4	no	—
18	C₂×#108×C₆₇	—	2	1	no	—
19	C₂×#108×C₆₇	—	2	1	no	—
20	#423×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₄×Dic₁₃₄	—	2	1	no	—
25	C₄×Dic₁₃₄	—	2	1	no	—
26	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₃₄	D₄	1	2	yes	—
28	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
29	C₂×Dic₆₇⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₆₇	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	Dic₆₇⋊C₈	—	2	2	no	—
36	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
37	#14523953	—	2	2	no	—
38	#14523953	—	2	2	no	—
39	C₄×C₆₇⋊C₈	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	#108×C₆₇	—	4	2	no	—
45	#108×C₆₇	—	4	2	no	—
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄⋊C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×Dic₁₃₄	—	2	1	no	—
53	C₂×Dic₁₃₄	—	4	2	no	—
54	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₆₇	—	2	1	no	—
56	C₂²×Dic₆₇	—	2	1	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₂×C₆₇⋊C₈	—	4	1	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
63	Dic₆₇⋊C₄	—	2	2	no	—
64	Dic₆₇⋊C₄	—	2	2	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	M₁₆×C₆₇	—	4	2	no	—
71	M₁₆×C₆₇	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
80	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
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	Dic₁₃₄	—	2	4	no	—
103	Dic₁₃₄	—	4	2	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	8	2	no	—
107	C₈×C₆₇	#103	1	2	yes	—
108	C₈×C₆₇	#103	1	2	yes	—
109	C₆₇⋊C₈	—	4	2	no	—
110	C₆₇⋊C₈	—	8	2	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	#380	1	1	yes	characteristic
114	C₂²×C₆₇	#475	1	1	yes	characteristic
115	C₂²×C₆₇	#477	1	1	yes	characteristic
116	C₄×C₆₇	—	2	1	no	—
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	#430	1	2	yes	—
122	C₄×C₆₇	#460	1	1	yes	characteristic
123	C₄×C₆₇	#462	1	1	yes	characteristic
124	Dic₆₇	—	4	2	no	—
125	Dic₆₇	—	4	2	no	—
126	#8663	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	1	no	—
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	#1803	1	1	yes	characteristic
130	C₂×C₆₇	#1811	1	1	yes	characteristic
131	C₂×C₆₇	#1814	1	1	yes	characteristic
132	C₂×#446	—	67	1	no	—
133	#1848	D₆₇	1	1	yes	characteristic
134	#1564	—	67	1	no	—
135	#1487	—	67	2	no	—
136	#1301	—	67	2	no	—
137	C₆₇	#8663	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄×Q₈	—	67	1	no	—
139	#446	—	67	2	no	—
140	#446	—	67	2	no	—
141	#446	—	134	1	no	—
142	#446	—	134	1	no	—
143	#431	—	67	2	no	—
144	C₂×#108	—	2	1	no	—
145	C₂×#108	—	2	1	no	—
146	#423	C₂×D₆₇	1	2	yes	—
147	C₂×C₄⋊C₈	—	67	1	no	—
148	C₂×C₄⋊C₈	—	134	1	no	—
149	C₂×#97	—	134	1	no	—
150	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
151	#335	—	134	1	no	—
152	#334	—	134	1	no	—
153	Q₈⋊C₈	—	134	4	no	—
154	C₂²×Q₈	—	67	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	—	134	1	no	—
157	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄×Q₈	—	67	2	no	—
160	C₄×Q₈	—	67	2	no	—
161	C₄×Q₈	—	134	1	no	—
162	C₄×Q₈	—	134	1	no	—
163	C₂×C₄⋊C₄	—	67	1	no	—
164	C₂×C₄⋊C₄	—	134	1	no	—
165	C₂×C₄²	—	134	1	no	—
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	#108	—	4	2	no	—
168	#108	—	4	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	67	2	no	—
172	C₄⋊C₈	—	134	2	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	C₄⋊C₈	—	134	1	no	—
176	C₂²⋊C₈	—	134	2	no	—
177	#97	—	134	2	no	—
178	#97	—	134	2	no	—
179	C₄×C₈	—	134	2	no	—
180	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
182	C₄×C₈	D₂₆₈	1	1	yes	characteristic
183	C₄×C₈	D₂₆₈	1	1	yes	characteristic
184	C₂×Q₈	—	268	2	no	—
185	C₂×Q₈	—	67	2	no	—
186	C₂×Q₈	—	134	1	no	—
187	C₂²×C₄	—	134	1	no	—
188	C₂²×C₄	—	134	1	no	—
189	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	#11761629	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	268	1	no	—
198	C₂×C₈	—	268	1	no	—
199	C₂×C₈	—	268	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
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	#11761616	1	2	yes	—
206	C₂×C₈	#11761618	1	2	yes	—
207	C₄⋊C₄	—	67	2	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	1	no	—
211	C₄²	—	134	2	no	—
212	C₄²	—	134	2	no	—
213	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₄²	D₄×D₆₇	1	1	yes	characteristic
215	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
216	C₄²	#11761627	1	1	yes	characteristic
217	C₂³	#14524019	1	1	yes	characteristic
218	Q₈	—	268	2	no	—
219	Q₈	—	134	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	134	2	no	—
225	C₂×C₄	—	134	2	no	—
226	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
228	C₂×C₄	#14524001	1	1	yes	characteristic
229	C₂×C₄	#14524014	1	1	yes	characteristic
230	C₂×C₄	#14524018	1	1	yes	characteristic
231	C₂×C₄	#14524023	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
234	C₂×C₄	#14524069	1	1	yes	characteristic
235	C₂×C₄	#14524074	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	268	2	no	—
240	C₈	—	536	2	no	—
241	C₈	#14523966	1	2	yes	—
242	C₈	#14523966	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#16863253	1	1	yes	characteristic
246	C₂²	#16863678	1	1	yes	characteristic
247	C₂²	#16863688	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	268	2	no	—
252	C₄	—	268	2	no	—
253	C₄	C₂×#14523966	1	1	yes	characteristic
254	C₄	#16863395	1	1	yes	characteristic
255	C₄	#16863443	1	2	yes	—
256	C₄	#16863593	1	1	yes	characteristic
257	C₄	#16863603	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#22560211	1	1	yes	characteristic
261	C₂	#22560248	1	1	yes	characteristic
262	C₂	#22560264	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#8663
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
137. C₆₇	#8663	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
#22558957	2
#22558637	2
#22558104	2
C₆₇×#1848	2
#8663	67