Subgroups — #30748951

Order 17152.

418 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#22560337	C₂	1	1	yes	maximal, characteristic
3	#22560215	C₂	1	2	yes	maximal
4	#22558093	C₂	1	2	yes	maximal
5	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	#16863526	—	2	2	no	—
8	#16863526	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₇⋊C₈	—	2	1	no	—
11	C₂×#14523968	—	2	2	no	—
12	C₂×#14523953	—	2	1	no	—
13	#16863266	C₂²	1	2	yes	—
14	#16863159	—	2	2	no	—
15	#16863159	—	2	1	no	—
16	#16863159	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#334×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₃₄	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
30	C₂×Dic₆₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₈	—	2	4	no	—
35	#14523968	—	4	4	no	—
36	#14523953	—	2	4	no	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
45	C₂×Dic₁₃₄	—	2	2	no	—
46	C₂×Dic₁₃₄	—	2	2	no	—
47	C₂×Dic₁₃₄	—	2	1	no	—
48	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
49	C₂²×Dic₆₇	—	2	1	no	—
50	C₂²×Dic₆₇	—	2	1	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	1	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	8	2	no	—
60	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
61	Dic₆₇⋊C₄	—	2	2	no	—
62	Dic₆₇⋊C₄	—	2	2	no	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₄×Dic₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	4	2	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₆₇	#41	1	1	yes	characteristic
79	C₂³×C₆₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#135	1	2	yes	—
88	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₆₇	#137	1	2	yes	—
90	C₂×C₄×C₆₇	#137	1	2	yes	—
91	C₂×Dic₆₇	—	2	2	no	—
92	C₂×Dic₆₇	—	2	2	no	—
93	C₂×Dic₆₇	—	4	1	no	—
94	C₂×Dic₆₇	—	4	1	no	—
95	C₂×Dic₆₇	—	4	2	no	—
96	C₂×Dic₆₇	—	8	2	no	—
97	C₂×Dic₆₇	—	8	2	no	—
98	C₂×Dic₆₇	—	8	2	no	—
99	Dic₁₃₄	—	2	4	no	—
100	Dic₁₃₄	—	4	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	2	2	no	—
103	C₈×C₆₇	—	8	2	no	—
104	C₆₇⋊C₈	—	4	2	no	—
105	C₂²×C₆₇	#380	1	1	yes	characteristic
106	C₂²×C₆₇	#464	1	2	yes	—
107	C₂²×C₆₇	#469	1	1	yes	characteristic
108	C₂²×C₆₇	#470	1	1	yes	characteristic
109	C₂²×C₆₇	#471	1	2	yes	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	#416	1	1	yes	characteristic
113	C₄×C₆₇	#419	1	1	yes	characteristic
114	C₄×C₆₇	#428	1	2	yes	—
115	C₄×C₆₇	#468	1	2	yes	—
116	C₄×C₆₇	#479	1	2	yes	—
117	Dic₆₇	—	4	2	no	—
118	Dic₆₇	—	4	2	no	—
119	Dic₆₇	—	16	2	no	—
120	#8657	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	#1702	1	2	yes	—
122	C₂×C₆₇	#1704	1	2	yes	—
123	C₂×C₆₇	#1805	1	1	yes	characteristic
124	C₂×C₆₇	#1809	1	1	yes	characteristic
125	C₂×C₆₇	#1815	1	1	yes	characteristic
126	C₂×#446	—	67	1	no	—
127	#1834	—	67	1	no	—
128	#1803	—	67	2	no	—
129	C₂×#334	D₆₇	1	1	yes	characteristic
130	#1296	—	67	2	no	—
131	C₆₇	#8657	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄×Q₈	—	67	1	no	—
133	#446	—	67	4	no	—
134	#446	—	134	2	no	—
135	C₂×C₄⋊C₈	—	67	1	no	—
136	C₂×C₄⋊C₈	—	134	1	no	—
137	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
138	C₂×#103	—	134	2	no	—
139	C₂×#97	—	134	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#384	—	67	2	no	—
142	#340	—	134	2	no	—
143	#340	—	134	1	no	—
144	#340	—	134	1	no	—
145	#334	—	2	2	no	—
146	C₂²×Q₈	—	67	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	134	1	no	—
149	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
150	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
151	C₄×Q₈	—	67	2	no	—
152	C₄×Q₈	—	67	2	no	—
153	C₄×Q₈	—	134	1	no	—
154	C₄×Q₈	—	134	1	no	—
155	C₂×C₄⋊C₄	—	67	1	no	—
156	C₂×C₄⋊C₄	—	134	2	no	—
157	C₂×C₄⋊C₄	—	134	2	no	—
158	C₂×C₄⋊C₄	—	134	2	no	—
159	C₂×C₄⋊C₄	—	134	1	no	—
160	C₂×C₄²	—	134	1	no	—
161	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	67	2	no	—
165	C₄⋊C₈	—	134	4	no	—
166	C₄⋊C₈	—	134	1	no	—
167	#103	—	268	4	no	—
168	#97	—	134	4	no	—
169	C₄×C₈	C₄×D₆₇	1	2	yes	—
170	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
171	C₂×Q₈	—	67	2	no	—
172	C₂×Q₈	—	134	2	no	—
173	C₂×Q₈	—	134	2	no	—
174	C₂×Q₈	—	134	1	no	—
175	C₂²×C₄	—	268	2	no	—
176	C₂²×C₄	—	134	1	no	—
177	C₂²×C₄	—	134	1	no	—
178	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
180	C₂²×C₄	#11761629	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	268	2	no	—
187	C₂×C₈	—	134	2	no	—
188	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
189	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
190	C₄⋊C₄	—	268	2	no	—
191	C₄⋊C₄	—	268	2	no	—
192	C₄⋊C₄	—	268	2	no	—
193	C₄⋊C₄	—	268	2	no	—
194	C₄⋊C₄	—	536	2	no	—
195	C₄⋊C₄	—	67	2	no	—
196	C₄⋊C₄	—	134	2	no	—
197	C₄⋊C₄	—	134	1	no	—
198	C₄⋊C₄	—	134	2	no	—
199	C₄²	—	134	2	no	—
200	C₄²	—	134	2	no	—
201	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
203	C₄²	Q₈×D₆₇	1	1	yes	characteristic
204	C₄²	#11761627	1	1	yes	characteristic
205	C₂³	#14524018	1	1	yes	characteristic
206	Q₈	—	268	2	no	—
207	Q₈	—	134	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	268	1	no	—
210	C₂×C₄	—	268	1	no	—
211	C₂×C₄	—	268	2	no	—
212	C₂×C₄	—	536	2	no	—
213	C₂×C₄	—	536	2	no	—
214	C₂×C₄	—	536	2	no	—
215	C₂×C₄	—	134	2	no	—
216	C₂×C₄	—	134	2	no	—
217	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
219	C₂×C₄	#14524001	1	1	yes	characteristic
220	C₂×C₄	#14524014	1	1	yes	characteristic
221	C₂×C₄	#14524019	1	1	yes	characteristic
222	C₂×C₄	#14524023	1	1	yes	characteristic
223	C₂×C₄	#14524065	1	2	yes	—
224	C₂×C₄	#14524070	1	2	yes	—
225	C₂×C₄	#14524072	1	2	yes	—
226	C₂×C₄	#14524076	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	8	2	no	—
230	C₈	—	268	2	no	—
231	C₂²	#16863253	1	1	yes	characteristic
232	C₂²	#16863619	1	2	yes	—
233	C₂²	#16863650	1	1	yes	characteristic
234	C₂²	#16863655	1	1	yes	characteristic
235	C₂²	#16863660	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	268	2	no	—
239	C₄	—	268	2	no	—
240	C₄	—	1072	2	no	—
241	C₄	#16863393	1	1	yes	characteristic
242	C₄	#16863406	1	1	yes	characteristic
243	C₄	#16863435	1	2	yes	—
244	C₄	#16863639	1	2	yes	—
245	C₄	#16863704	1	2	yes	—
246	C₂	#22559778	1	2	yes	—
247	C₂	#22559792	1	2	yes	—
248	C₂	#22560225	1	1	yes	characteristic
249	C₂	#22560239	1	1	yes	characteristic
250	C₂	#22560265	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#8657
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16863526	C₄	semidirect
131. C₆₇	#8657	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
#22560337	2
#22560215	2
#22558093	2
C₂×#334×C₆₇	2
#8657	67