Subgroups — #30748769

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#22560202	C₂	1	2	yes	maximal
3	#22558968	C₂	1	1	yes	maximal, characteristic
4	#22558083	C₂	1	2	yes	maximal
5	C₆₇×#1566	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	—	2	1	no	—
10	#16863526	—	2	1	no	—
11	#16863420	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
14	C₂×#14523961	—	2	2	no	—
15	C₂×#14523953	C₄	1	2	yes	—
16	#16863164	—	2	2	no	—
17	#16863149	—	2	1	no	—
18	#16863149	—	2	1	no	—
19	#423×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#334×C₆₇	—	2	1	no	—
22	#334×C₆₇	—	2	1	no	—
23	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
24	C₂×#11761612	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₃₄	—	2	1	no	—
27	C₄×Dic₁₃₄	—	2	1	no	—
28	C₄×Dic₁₃₄	—	2	1	no	—
29	C₄×Dic₁₃₄	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
31	C₂×Dic₆₇⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	Dic₆₇⋊C₈	—	2	2	no	—
39	#14523961	—	4	4	no	—
40	#14523953	—	2	2	no	—
41	#14523953	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄⋊C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	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	—	2	1	no	—
54	C₂×Dic₁₃₄	—	2	2	no	—
55	C₂×Dic₁₃₄	—	2	2	no	—
56	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
58	#11761612	—	4	2	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₄⋊Dic₆₇	—	2	1	no	—
64	C₄⋊Dic₆₇	—	2	1	no	—
65	Dic₆₇⋊C₄	—	2	2	no	—
66	Dic₆₇⋊C₄	—	2	2	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	M₁₆×C₆₇	—	4	2	no	—
73	M₁₆×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	4	1	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊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	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₆₇	#137	1	2	yes	—
97	C₂×Dic₆₇	—	2	2	no	—
98	C₂×Dic₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	2	4	no	—
100	C₂×Dic₆₇	—	4	2	no	—
101	Dic₁₃₄	—	4	2	no	—
102	Dic₁₃₄	—	4	1	no	—
103	Dic₁₃₄	—	4	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	8	2	no	—
108	C₈×C₆₇	#101	1	2	yes	complemented
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	#428	1	2	yes	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
119	C₄×C₆₇	#413	1	1	yes	characteristic
120	C₄×C₆₇	#468	1	2	yes	—
121	C₄×C₆₇	#469	1	1	yes	characteristic
122	C₄×C₆₇	#470	1	1	yes	characteristic
123	Dic₆₇	—	4	2	no	—
124	Dic₆₇	—	4	2	no	—
125	#8619	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	2	2	no	—
127	C₂×C₆₇	#1801	1	1	yes	characteristic
128	C₂×C₆₇	#1809	1	1	yes	characteristic
129	C₂×C₆₇	#1813	1	1	yes	characteristic
130	C₂×#446	—	67	1	no	—
131	#1801	—	67	2	no	—
132	#1567	—	67	1	no	—
133	#1566	D₆₇	1	1	yes	characteristic
134	#1292	—	67	2	no	—
135	C₆₇	#8619	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄×Q₈	—	67	1	no	—
137	#446	—	134	1	no	—
138	#446	—	134	1	no	—
139	#446	—	134	1	no	—
140	#446	—	134	1	no	—
141	#423	—	67	2	no	—
142	#423	C₂×D₆₇	1	2	yes	—
143	C₂×C₄⋊C₈	—	67	2	no	—
144	C₂×C₄⋊C₈	—	67	1	no	—
145	C₂×#101	—	134	2	no	—
146	C₂×#97	—	67	2	no	—
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	#343	—	134	2	no	—
149	#335	—	134	1	no	—
150	#335	—	134	1	no	—
151	#334	—	2	1	no	—
152	#334	—	2	1	no	—
153	C₂²×Q₈	—	67	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	134	2	no	—
156	C₂²×C₈	—	67	2	no	—
157	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
159	C₄×Q₈	—	134	1	no	—
160	C₄×Q₈	—	134	1	no	—
161	C₄×Q₈	—	134	1	no	—
162	C₄×Q₈	—	134	1	no	—
163	C₂×C₄⋊C₄	—	67	1	no	—
164	C₂×C₄⋊C₄	—	67	2	no	—
165	C₂×C₄²	—	67	2	no	—
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	134	2	no	—
170	C₄⋊C₈	—	134	2	no	—
171	C₄⋊C₈	—	134	1	no	—
172	C₄⋊C₈	—	134	2	no	—
173	C₄⋊C₈	—	134	1	no	—
174	C₄⋊C₈	—	134	2	no	—
175	#101	—	268	4	no	—
176	#97	—	134	2	no	—
177	#97	—	134	2	no	—
178	C₄×C₈	D₂₆₈	1	1	yes	characteristic
179	C₄×C₈	D₂₆₈	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×Q₈	—	134	1	no	—
183	C₂×Q₈	—	134	1	no	—
184	C₂×Q₈	—	134	2	no	—
185	C₂×Q₈	—	134	2	no	—
186	C₂²×C₄	—	67	2	no	—
187	C₂²×C₄	—	67	2	no	—
188	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
191	M₁₆	—	4	2	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	268	2	no	—
194	M₁₆	—	268	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	268	2	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	C₄⋊C₄	—	134	1	no	—
206	C₄⋊C₄	—	134	1	no	—
207	C₄⋊C₄	—	134	2	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₄²	—	134	2	no	—
210	C₄²	—	134	2	no	—
211	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
213	C₄²	#11761627	1	1	yes	characteristic
214	C₄²	#11761629	1	1	yes	characteristic
215	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
216	Q₈	—	268	2	no	—
217	Q₈	—	268	1	no	—
218	Q₈	—	268	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	134	2	no	—
224	C₂×C₄	—	134	2	no	—
225	C₂×C₄	—	134	4	no	—
226	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	#14524001	1	1	yes	characteristic
228	C₂×C₄	#14524014	1	1	yes	characteristic
229	C₂×C₄	#14524018	1	1	yes	characteristic
230	C₂×C₄	#14524019	1	1	yes	characteristic
231	C₂×C₄	#14524023	1	1	yes	characteristic
232	C₂×C₄	#14524070	1	2	yes	—
233	C₂×C₄	#14524072	1	2	yes	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	8	2	no	—
238	C₈	—	268	2	no	—
239	C₈	—	536	2	no	—
240	C₈	#14523961	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#16863253	1	1	yes	characteristic
244	C₂²	#16863435	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	268	2	no	—
249	C₄	—	268	2	no	—
250	C₄	C₂×#14523961	1	1	yes	characteristic
251	C₄	#16863378	1	1	yes	characteristic
252	C₄	#16863639	1	2	yes	—
253	C₄	#16863650	1	1	yes	characteristic
254	C₄	#16863655	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#22560202	1	1	yes	characteristic
257	C₂	#22560239	1	1	yes	characteristic
258	C₂	#22560256	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#8619
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
108. C₈×C₆₇	#101	semidirect
135. C₆₇	#8619	semidirect
240. C₈	#14523961	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
#22560202	2
#22558968	2
#22558083	2
C₆₇×#1566	2
#8619	67