Subgroups — #30748911

Order 17152.

438 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	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×#335×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	#335×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	1	no	—
47	C₂×Dic₁₃₄	—	2	1	no	—
48	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₁₃₄	D₈	1	2	yes	—
50	C₂×Dic₁₃₄	Q₁₆	1	2	yes	—
51	C₂²×Dic₆₇	—	2	1	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₄⋊Dic₆₇	—	2	1	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	8	2	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×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₆₇	#41	1	1	yes	characteristic
81	C₂³×C₆₇	#121	1	1	yes	characteristic
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#135	1	2	yes	—
93	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#137	1	2	yes	—
95	C₂×Dic₆₇	—	2	2	no	—
96	C₂×Dic₆₇	—	2	2	no	—
97	C₂×Dic₆₇	—	2	2	no	—
98	C₂×Dic₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	4	2	no	—
100	C₂×Dic₆₇	—	8	2	no	—
101	C₂×Dic₆₇	—	8	2	no	—
102	C₂×Dic₆₇	—	8	2	no	—
103	Dic₁₃₄	—	2	2	no	—
104	Dic₁₃₄	—	2	2	no	—
105	Dic₁₃₄	—	2	2	no	—
106	Dic₁₃₄	#107	1	4	yes	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	—	8	2	no	—
110	C₆₇⋊C₈	—	4	2	no	—
111	C₂²×C₆₇	#380	1	1	yes	characteristic
112	C₂²×C₆₇	#464	1	2	yes	—
113	C₂²×C₆₇	#466	1	1	yes	characteristic
114	C₂²×C₆₇	#467	1	1	yes	characteristic
115	C₂²×C₆₇	#471	1	2	yes	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	#416	1	1	yes	characteristic
119	C₄×C₆₇	#419	1	1	yes	characteristic
120	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
121	C₄×C₆₇	#427	1	1	yes	characteristic
122	C₄×C₆₇	#466	1	1	yes	characteristic
123	C₄×C₆₇	#467	1	1	yes	characteristic
124	C₄×C₆₇	#479	1	2	yes	—
125	Dic₆₇	—	2	4	no	—
126	Dic₆₇	—	4	2	no	—
127	Dic₆₇	—	16	2	no	—
128	#8649	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	#1682	1	2	yes	—
130	C₂×C₆₇	#1684	1	2	yes	—
131	C₂×C₆₇	#1805	1	1	yes	characteristic
132	C₂×C₆₇	#1808	1	1	yes	characteristic
133	C₂×C₆₇	#1815	1	1	yes	characteristic
134	C₂×C₈×Q₈	—	67	1	no	—
135	#1834	—	67	1	no	—
136	#1803	—	67	2	no	—
137	C₂×#335	D₆₇	1	1	yes	characteristic
138	#1296	—	67	2	no	—
139	C₆₇	#8649	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄×Q₈	—	67	1	no	—
141	C₈×Q₈	—	67	4	no	—
142	C₈×Q₈	—	134	2	no	—
143	C₂×C₄⋊C₈	—	67	1	no	—
144	C₂×C₄⋊C₈	—	134	1	no	—
145	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
146	C₂×#103	—	134	2	no	—
147	C₂×C₄×C₈	—	134	1	no	—
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	#384	—	67	2	no	—
150	#340	—	134	2	no	—
151	#340	—	134	1	no	—
152	#340	—	134	1	no	—
153	#335	—	2	2	no	—
154	C₂²×Q₈	—	67	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	134	1	no	—
157	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
158	C₂²×C₈	D₂₆₈	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	2	no	—
165	C₂×C₄⋊C₄	—	134	2	no	—
166	C₂×C₄⋊C₄	—	134	2	no	—
167	C₂×C₄⋊C₄	—	134	1	no	—
168	C₂×C₄²	—	134	1	no	—
169	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	67	2	no	—
173	C₄⋊C₈	—	134	4	no	—
174	C₄⋊C₈	—	134	1	no	—
175	#103	—	268	4	no	—
176	C₄×C₈	—	134	4	no	—
177	C₄×C₈	C₄×D₆₇	1	2	yes	—
178	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
179	C₂×Q₈	—	67	2	no	—
180	C₂×Q₈	—	67	2	no	—
181	C₂×Q₈	—	67	2	no	—
182	C₂×Q₈	—	134	1	no	—
183	C₂×Q₈	—	134	1	no	—
184	C₂×Q₈	—	134	1	no	—
185	C₂²×C₄	—	268	2	no	—
186	C₂²×C₄	—	134	1	no	—
187	C₂²×C₄	—	134	1	no	—
188	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
190	C₂²×C₄	#11761629	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	268	2	no	—
197	C₂×C₈	—	134	2	no	—
198	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
199	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
200	C₄⋊C₄	—	268	2	no	—
201	C₄⋊C₄	—	268	2	no	—
202	C₄⋊C₄	—	268	2	no	—
203	C₄⋊C₄	—	268	2	no	—
204	C₄⋊C₄	—	536	2	no	—
205	C₄⋊C₄	—	67	2	no	—
206	C₄⋊C₄	—	134	2	no	—
207	C₄⋊C₄	—	134	1	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₄²	—	134	2	no	—
210	C₄²	—	134	2	no	—
211	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₄²	Q₈×D₆₇	1	1	yes	characteristic
214	C₄²	#11761627	1	1	yes	characteristic
215	C₂³	#14524018	1	1	yes	characteristic
216	Q₈	—	67	4	no	—
217	Q₈	—	134	2	no	—
218	Q₈	—	134	2	no	—
219	Q₈	—	134	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	268	2	no	—
222	C₂×C₄	—	536	2	no	—
223	C₂×C₄	—	536	2	no	—
224	C₂×C₄	—	536	2	no	—
225	C₂×C₄	—	134	2	no	—
226	C₂×C₄	—	134	2	no	—
227	C₂×C₄	—	134	2	no	—
228	C₂×C₄	—	134	2	no	—
229	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
231	C₂×C₄	#14524001	1	1	yes	characteristic
232	C₂×C₄	#14524014	1	1	yes	characteristic
233	C₂×C₄	#14524019	1	1	yes	characteristic
234	C₂×C₄	#14524023	1	1	yes	characteristic
235	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
236	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
237	C₂×C₄	#14524062	1	1	yes	characteristic
238	C₂×C₄	#14524065	1	2	yes	—
239	C₂×C₄	#14524066	1	1	yes	characteristic
240	C₂×C₄	#14524076	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	2	no	—
244	C₈	—	268	2	no	—
245	C₂²	#16863253	1	1	yes	characteristic
246	C₂²	#16863619	1	2	yes	—
247	C₂²	#16863633	1	1	yes	characteristic
248	C₂²	#16863638	1	1	yes	characteristic
249	C₂²	#16863660	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	268	2	no	—
253	C₄	—	1072	2	no	—
254	C₄	—	134	4	no	—
255	C₄	#16863393	1	1	yes	characteristic
256	C₄	#16863406	1	1	yes	characteristic
257	C₄	#107×D₆₇	1	1	yes	characteristic
258	C₄	#16863433	1	1	yes	characteristic
259	C₄	#16863629	1	1	yes	characteristic
260	C₄	#16863634	1	1	yes	characteristic
261	C₄	#16863704	1	2	yes	—
262	C₂	#22559654	1	2	yes	—
263	C₂	#22559668	1	2	yes	—
264	C₂	#22560225	1	1	yes	characteristic
265	C₂	#22560235	1	1	yes	characteristic
266	C₂	#22560265	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#8649
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₃₄	C₄	semidirect
139. C₆₇	#8649	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×#335×C₆₇	2
#8649	67