Subgroups — #30748901

Order 17152.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	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	C₈×Dic₁₃₄	—	2	2	no	—
8	C₈×Dic₁₃₄	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×C₈×Dic₆₇	—	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	C₈×Dic₆₇	—	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	1	no	—
46	C₂×Dic₁₃₄	—	2	2	no	—
47	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
48	C₂×Dic₁₃₄	SD₁₆	1	4	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	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₆₇	#135	1	2	yes	—
90	C₂×C₄×C₆₇	#135	1	2	yes	—
91	C₂×C₄×C₆₇	#137	1	2	yes	—
92	C₂×Dic₆₇	—	2	2	no	—
93	C₂×Dic₆₇	—	2	4	no	—
94	C₂×Dic₆₇	—	2	2	no	—
95	C₂×Dic₆₇	—	4	1	no	—
96	C₂×Dic₆₇	—	4	1	no	—
97	C₂×Dic₆₇	—	8	2	no	—
98	C₂×Dic₆₇	—	8	2	no	—
99	C₂×Dic₆₇	—	8	2	no	—
100	Dic₁₃₄	—	2	2	no	—
101	Dic₁₃₄	—	2	4	no	—
102	Dic₁₃₄	#106	1	4	yes	—
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	8	2	no	—
106	C₆₇⋊C₈	—	4	2	no	—
107	C₂²×C₆₇	#380	1	1	yes	characteristic
108	C₂²×C₆₇	#464	1	2	yes	—
109	C₂²×C₆₇	#465	1	2	yes	—
110	C₂²×C₆₇	#472	1	1	yes	characteristic
111	C₂²×C₆₇	#473	1	1	yes	characteristic
112	C₄×C₆₇	—	2	1	no	—
113	C₄×C₆₇	—	2	1	no	—
114	C₄×C₆₇	#416	1	1	yes	characteristic
115	C₄×C₆₇	#419	1	1	yes	characteristic
116	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
117	C₄×C₆₇	#427	1	1	yes	characteristic
118	C₄×C₆₇	#465	1	2	yes	—
119	C₄×C₆₇	#479	1	2	yes	—
120	Dic₆₇	—	2	4	no	—
121	Dic₆₇	—	4	2	no	—
122	Dic₆₇	—	16	2	no	—
123	#8647	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	#1678	1	2	yes	—
125	C₂×C₆₇	#1680	1	2	yes	—
126	C₂×C₆₇	#1805	1	1	yes	characteristic
127	C₂×C₆₇	#1807	1	1	yes	characteristic
128	C₂×C₆₇	#1816	1	1	yes	characteristic
129	C₂×C₈×Q₈	—	67	1	no	—
130	#1834	—	67	1	no	—
131	#1803	—	67	2	no	—
132	C₂×#334	D₆₇	1	1	yes	characteristic
133	#1296	—	67	2	no	—
134	C₆₇	#8647	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄×Q₈	—	67	1	no	—
136	C₈×Q₈	—	67	4	no	—
137	C₈×Q₈	—	134	2	no	—
138	C₂×C₄⋊C₈	—	67	1	no	—
139	C₂×C₄⋊C₈	—	134	1	no	—
140	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
141	C₂×#103	—	134	2	no	—
142	C₂×C₄×C₈	—	134	1	no	—
143	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
144	#384	—	67	2	no	—
145	#340	—	134	2	no	—
146	#340	—	134	1	no	—
147	#340	—	134	1	no	—
148	#334	—	2	2	no	—
149	C₂²×Q₈	—	67	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	134	1	no	—
152	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
153	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
154	C₄×Q₈	—	67	2	no	—
155	C₄×Q₈	—	67	2	no	—
156	C₄×Q₈	—	134	1	no	—
157	C₄×Q₈	—	134	1	no	—
158	C₂×C₄⋊C₄	—	67	1	no	—
159	C₂×C₄⋊C₄	—	134	2	no	—
160	C₂×C₄⋊C₄	—	134	2	no	—
161	C₂×C₄⋊C₄	—	134	2	no	—
162	C₂×C₄⋊C₄	—	134	1	no	—
163	C₂×C₄²	—	134	1	no	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	67	2	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	—	134	1	no	—
170	#103	—	268	4	no	—
171	C₄×C₈	—	134	4	no	—
172	C₄×C₈	C₄×D₆₇	1	2	yes	—
173	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
174	C₂×Q₈	—	67	2	no	—
175	C₂×Q₈	—	67	4	no	—
176	C₂×Q₈	—	134	1	no	—
177	C₂×Q₈	—	134	2	no	—
178	C₂²×C₄	—	268	2	no	—
179	C₂²×C₄	—	134	1	no	—
180	C₂²×C₄	—	134	1	no	—
181	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
183	C₂²×C₄	#11761629	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	268	2	no	—
190	C₂×C₈	—	134	2	no	—
191	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
192	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
193	C₄⋊C₄	—	268	2	no	—
194	C₄⋊C₄	—	268	2	no	—
195	C₄⋊C₄	—	268	2	no	—
196	C₄⋊C₄	—	268	2	no	—
197	C₄⋊C₄	—	536	2	no	—
198	C₄⋊C₄	—	67	2	no	—
199	C₄⋊C₄	—	134	2	no	—
200	C₄⋊C₄	—	134	1	no	—
201	C₄⋊C₄	—	134	2	no	—
202	C₄²	—	134	2	no	—
203	C₄²	—	134	2	no	—
204	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
205	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₄²	Q₈×D₆₇	1	1	yes	characteristic
207	C₄²	#11761627	1	1	yes	characteristic
208	C₂³	#14524018	1	1	yes	characteristic
209	Q₈	—	67	4	no	—
210	Q₈	—	134	4	no	—
211	Q₈	—	134	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	268	1	no	—
215	C₂×C₄	—	268	1	no	—
216	C₂×C₄	—	536	2	no	—
217	C₂×C₄	—	536	2	no	—
218	C₂×C₄	—	536	2	no	—
219	C₂×C₄	—	134	2	no	—
220	C₂×C₄	—	134	2	no	—
221	C₂×C₄	—	134	4	no	—
222	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
224	C₂×C₄	#14524001	1	1	yes	characteristic
225	C₂×C₄	#14524014	1	1	yes	characteristic
226	C₂×C₄	#14524019	1	1	yes	characteristic
227	C₂×C₄	#14524023	1	1	yes	characteristic
228	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
229	C₂×C₄	#14524064	1	2	yes	—
230	C₂×C₄	#14524065	1	2	yes	—
231	C₂×C₄	#14524076	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	8	2	no	—
235	C₈	—	268	2	no	—
236	C₂²	#16863253	1	1	yes	characteristic
237	C₂²	#16863619	1	2	yes	—
238	C₂²	#16863626	1	2	yes	—
239	C₂²	#16863667	1	1	yes	characteristic
240	C₂²	#16863672	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	268	2	no	—
244	C₄	—	1072	2	no	—
245	C₄	—	134	4	no	—
246	C₄	#16863393	1	1	yes	characteristic
247	C₄	#16863406	1	1	yes	characteristic
248	C₄	#106×D₆₇	1	1	yes	characteristic
249	C₄	#16863431	1	1	yes	characteristic
250	C₄	#16863622	1	2	yes	—
251	C₄	#16863704	1	2	yes	—
252	C₂	#22559626	1	2	yes	—
253	C₂	#22559640	1	2	yes	—
254	C₂	#22560225	1	1	yes	characteristic
255	C₂	#22560231	1	1	yes	characteristic
256	C₂	#22560270	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	#8647
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. C₈×Dic₁₃₄	C₄	semidirect
134. C₆₇	#8647	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22560337	2
#22560215	2
#22558093	2
C₂×#334×C₆₇	2
#8647	67