Subgroups — #30749095

Order 17152.

414 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	#22560347	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558886	C₂	1	2	yes	maximal
4	#22558083	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863161	—	2	1	no	—
9	#16863161	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863142	—	2	2	no	—
13	#16863140	—	2	2	no	—
14	C₂×#131×C₆₇	—	2	1	no	—
15	#532×C₆₇	C₂²	1	1	yes	characteristic
16	#436×C₆₇	—	2	1	no	—
17	#436×C₆₇	C₄	1	2	yes	—
18	C₈×D₄×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	C₄	1	2	yes	—
20	#423×C₆₇	C₂²	1	1	yes	characteristic
21	#408×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	1	no	—
30	#131×C₆₇	—	4	4	no	—
31	C₂×M₁₆×C₆₇	—	2	1	no	—
32	C₂×M₁₆×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
37	#127×C₆₇	D₄	1	1	yes	characteristic
38	#124×C₆₇	—	2	1	no	—
39	#121×C₆₇	—	2	1	no	—
40	#121×C₆₇	Q₈	1	2	yes	—
41	C₄×D₄×C₆₇	—	2	1	no	—
42	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	—	2	1	no	—
45	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₆₇	—	2	2	no	—
47	C₂²⋊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	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	8	2	no	—
57	#41×C₆₇	—	4	2	no	—
58	#41×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	2	1	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	2	1	no	—
62	C₂×D₄×C₆₇	—	2	1	no	—
63	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
68	M₁₆×C₆₇	—	4	4	no	—
69	M₁₆×C₆₇	—	4	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	4	4	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₂²⋊C₄×C₆₇	—	2	2	no	—
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	—	2	2	no	—
87	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
88	Q₈×C₆₇	—	4	2	no	—
89	D₄×C₆₇	—	4	2	no	—
90	D₄×C₆₇	—	4	4	no	—
91	D₄×C₆₇	—	4	2	no	—
92	D₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	1	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#135	1	2	yes	—
105	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	2	1	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	4	2	no	—
111	C₆₇⋊C₈	—	8	2	no	—
112	C₆₇⋊C₈	—	8	2	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	4	2	no	—
116	C₂²×C₆₇	—	4	2	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	#380	1	1	yes	characteristic
119	C₂²×C₆₇	#478	1	2	yes	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	4	2	no	—
124	C₄×C₆₇	#416	1	1	yes	characteristic
125	C₄×C₆₇	#418	1	1	yes	characteristic
126	C₄×C₆₇	#463	1	2	yes	—
127	C₄×C₆₇	#471	1	2	yes	—
128	#8686	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	8	2	no	—
131	C₂×C₆₇	#1804	1	1	yes	characteristic
132	C₂×C₆₇	#1813	1	1	yes	characteristic
133	C₂×C₆₇	#1815	1	1	yes	characteristic
134	#2931	D₆₇	1	1	yes	characteristic
135	#1837	—	67	1	no	—
136	C₂×#334	—	67	1	no	—
137	#1547	—	67	2	no	—
138	#1292	—	67	2	no	—
139	C₆₇	#8686	1	1	yes	characteristic, complemented, Sylow
140	C₂×#131	—	2	1	no	—
141	#532	C₂×D₆₇	1	1	yes	characteristic
142	#436	—	2	1	no	—
143	#436	Dic₆₇	1	2	yes	—
144	C₈×D₄	—	2	1	no	—
145	C₈×D₄	Dic₆₇	1	2	yes	—
146	#423	—	67	2	no	—
147	#423	C₂×D₆₇	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	67	2	no	—
149	#408	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
151	#341	—	134	1	no	—
152	#341	—	134	1	no	—
153	#334	—	134	1	no	—
154	#334	—	134	1	no	—
155	#332	—	134	2	no	—
156	#331	—	134	2	no	—
157	C₂×#41	—	2	1	no	—
158	#131	—	4	4	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	134	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	134	2	no	—
164	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
167	#127	D₂₆₈	1	1	yes	characteristic
168	#124	—	2	1	no	—
169	#121	—	2	1	no	—
170	#121	Dic₁₃₄	1	2	yes	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	134	2	no	—
176	C₄⋊C₈	—	134	2	no	—
177	C₄⋊C₈	—	134	2	no	—
178	C₄⋊C₈	—	134	2	no	—
179	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
183	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
196	C₂²×C₄	#11761628	1	1	yes	characteristic
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	2	no	—
199	M₁₆	—	268	2	no	—
200	M₁₆	—	268	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	—	268	2	no	—
206	C₂×C₈	—	268	2	no	—
207	C₂×C₈	—	536	2	no	—
208	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
219	C₄²	#11761628	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	#14524022	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
234	C₂×C₄	#14524002	1	1	yes	characteristic
235	C₂×C₄	#14524017	1	1	yes	characteristic
236	C₂×C₄	#14524017	1	1	yes	characteristic
237	C₂×C₄	#14524022	1	1	yes	characteristic
238	C₂×C₄	#14524063	1	2	yes	—
239	C₂×C₄	#14524069	1	2	yes	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	536	2	no	—
246	C₈	—	536	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	#16863256	1	1	yes	characteristic
253	C₂²	#16863696	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	#16863392	1	1	yes	characteristic
259	C₄	#16863401	1	1	yes	characteristic
260	C₄	#16863611	1	2	yes	—
261	C₄	#16863659	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#22560219	1	1	yes	characteristic
265	C₂	#22560259	1	1	yes	characteristic
266	C₂	#22560268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2931	C₂
socle	C₂²×C₆₇	#380
prime radical	#532×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2931	D₆₇
hypocenter	C₆₇	#8686
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #22560347	C₂	semidirect
2. C₂×#16863147	C₂	semidirect
4. #22558083	C₂	semidirect
139. C₆₇	#8686	semidirect

Maximal subgroups

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

subgroup	index
#22560347	2
C₂×#16863147	2
#22558886	2
#22558083	2
C₆₇×#2931	2
#8686	67