Subgroups — #30754687

Order 17152.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#22561144	C₂	1	2	yes	maximal
3	C₂×#16863126	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
8	C₄×Dic₂₆₈	—	2	2	no	—
9	C₄×Dic₂₆₈	—	2	2	no	—
10	C₂×#14523977	C₄	1	1	yes	characteristic
11	C₂×#14523968	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16863150	—	2	4	no	—
15	#16863126	—	2	4	no	—
16	C₈×C₆₇⋊C₈	—	2	2	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	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	C₈	1	4	yes	complemented
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
26	C₄×Dic₁₃₄	—	2	2	no	—
27	C₄×Dic₁₃₄	—	2	2	no	—
28	C₄×Dic₁₃₄	—	2	2	no	—
29	C₄×Dic₁₃₄	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
31	C₂×Dic₆₇⋊C₄	—	2	2	no	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	#14523977	—	2	2	no	—
34	#14523968	—	2	4	no	—
35	C₄×C₆₇⋊C₈	—	2	1	no	—
36	C₄×C₆₇⋊C₈	—	2	1	no	—
37	C₄×C₆₇⋊C₈	—	2	1	no	—
38	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂×Dic₁₃₄	—	2	2	no	—
48	C₂×Dic₁₃₄	—	2	2	no	—
49	C₂×Dic₁₃₄	—	2	2	no	—
50	C₂×Dic₁₃₄	—	2	2	no	—
51	C₂×Dic₁₃₄	—	2	2	no	—
52	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
53	C₂²×Dic₆₇	—	2	2	no	—
54	C₂²×Dic₆₇	—	2	2	no	—
55	Dic₂₆₈	—	2	8	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	—	2	2	no	—
63	Dic₆₇⋊C₄	—	4	2	no	—
64	Dic₆₇⋊C₄	—	4	2	no	—
65	C₄×Dic₆₇	—	4	2	no	—
66	C₄×Dic₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	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	1	no	—
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	4	4	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	#41	1	1	yes	characteristic
82	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₆₇	#135	1	2	yes	—
96	C₂×Dic₆₇	—	2	4	no	—
97	C₂×Dic₆₇	—	4	2	no	—
98	C₂×Dic₆₇	—	4	2	no	—
99	C₂×Dic₆₇	—	4	2	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	Dic₁₃₄	—	2	4	no	—
102	Dic₁₃₄	—	2	4	no	—
103	Dic₁₃₄	—	4	2	no	—
104	Dic₁₃₄	—	4	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	—	8	2	no	—
107	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
108	C₆₇⋊C₈	—	2	4	no	—
109	C₆₇⋊C₈	—	2	4	no	—
110	C₂²×C₆₇	#390	1	1	yes	characteristic
111	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
112	C₂²×C₆₇	#436	1	1	yes	characteristic
113	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
114	C₂²×C₆₇	#443	1	2	yes	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₆₇	#408	1	1	yes	characteristic
119	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
120	C₄×C₆₇	#436	1	1	yes	characteristic
121	C₄×C₆₇	#467	1	2	yes	—
122	C₄×C₆₇	#494	1	1	yes	characteristic
123	C₄×C₆₇	#495	1	1	yes	characteristic
124	Dic₆₇	—	4	4	no	—
125	Dic₆₇	—	8	2	no	—
126	#9656	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	C₈×Q₁₆	1	2	yes	—
128	C₂×C₆₇	#1591	1	2	yes	—
129	C₂×C₆₇	#1954	1	1	yes	characteristic
130	C₂×C₆₇	#1969	1	1	yes	characteristic
131	C₂×C₆₇	#1974	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	—	67	1	no	—
133	#1984	—	67	2	no	—
134	C₂×#335	D₆₇	1	1	yes	characteristic
135	C₂×Q₈⋊C₈	—	67	2	no	—
136	C₂×C₈²	—	67	1	no	—
137	C₆₇	#9656	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₁₆	—	67	1	no	—
139	C₂×C₄×Q₈	—	67	2	no	—
140	C₄×Q₁₆	—	134	2	no	—
141	C₄×Q₁₆	—	134	2	no	—
142	C₂×#107	—	67	1	no	—
143	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
144	C₂×#103	—	67	2	no	—
145	C₂×C₄×C₈	—	67	1	no	—
146	C₂×C₄×C₈	—	67	1	no	—
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	134	4	no	—
149	#335	—	2	2	no	—
150	Q₈⋊C₈	—	134	4	no	—
151	C₈²	—	134	2	no	—
152	C₂²×Q₈	—	67	2	no	—
153	C₂×Q₁₆	—	67	4	no	—
154	C₂×Q₁₆	—	134	2	no	—
155	C₂×Q₁₆	—	134	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	67	2	no	—
158	C₂²×C₈	—	134	1	no	—
159	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
161	C₄×Q₈	—	134	2	no	—
162	C₄×Q₈	—	134	2	no	—
163	C₄×Q₈	—	134	2	no	—
164	C₄×Q₈	—	134	2	no	—
165	C₂×C₄⋊C₄	—	67	2	no	—
166	C₂×C₄⋊C₄	—	134	2	no	—
167	C₂×C₄²	—	134	2	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	#107	—	134	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	#103	—	134	4	no	—
173	C₄×C₈	—	67	2	no	—
174	C₄×C₈	—	134	1	no	—
175	C₄×C₈	—	134	1	no	—
176	C₄×C₈	—	134	1	no	—
177	C₄×C₈	D₂₆₈	1	2	yes	—
178	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
179	C₂×Q₈	—	67	4	no	—
180	C₂×Q₈	—	134	2	no	—
181	C₂×Q₈	—	134	2	no	—
182	C₂×Q₈	—	134	2	no	—
183	C₂×Q₈	—	134	2	no	—
184	C₂×Q₈	—	134	2	no	—
185	C₂²×C₄	—	134	2	no	—
186	C₂²×C₄	—	134	2	no	—
187	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	#11761629	1	1	yes	characteristic
190	Q₁₆	—	134	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	67	4	no	—
196	C₂×C₈	—	134	4	no	—
197	C₂×C₈	—	134	2	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
202	C₂×C₈	C₈×D₆₇	1	2	yes	—
203	C₂×C₈	#11761611	1	2	yes	—
204	C₄⋊C₄	—	268	2	no	—
205	C₄⋊C₄	—	268	2	no	—
206	C₄⋊C₄	—	134	2	no	—
207	C₄⋊C₄	—	134	2	no	—
208	C₄²	—	268	2	no	—
209	C₄²	—	268	2	no	—
210	C₄²	D₄×D₆₇	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
213	C₄²	#11761629	1	1	yes	characteristic
214	C₂³	#14524005	1	1	yes	characteristic
215	Q₈	—	268	2	no	—
216	Q₈	—	268	2	no	—
217	Q₈	—	134	4	no	—
218	Q₈	—	134	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	268	2	no	—
221	C₂×C₄	—	268	2	no	—
222	C₂×C₄	—	268	4	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	134	4	no	—
225	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
226	C₂×C₄	#14524005	1	1	yes	characteristic
227	C₂×C₄	#14524018	1	1	yes	characteristic
228	C₂×C₄	#14524018	1	1	yes	characteristic
229	C₂×C₄	#14524032	1	1	yes	characteristic
230	C₂×C₄	#14524039	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
233	C₂×C₄	#14524051	1	1	yes	characteristic
234	C₂×C₄	#14524051	1	1	yes	characteristic
235	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
236	C₂×C₄	#14524062	1	2	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	8	2	no	—
239	C₈	—	134	4	no	—
240	C₈	—	134	4	no	—
241	C₈	(C₂×D₆₇)⋊C₈	1	4	yes	—
242	C₂²	#16863297	1	1	yes	characteristic
243	C₂²	#16863461	1	1	yes	characteristic
244	C₂²	#16863473	1	1	yes	characteristic
245	C₂²	#16863490	1	2	yes	—
246	C₂²	#16863512	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	268	4	no	—
250	C₄	—	536	2	no	—
251	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
252	C₄	#16863364	1	1	yes	characteristic
253	C₄	#16863461	1	1	yes	characteristic
254	C₄	#16863473	1	1	yes	characteristic
255	C₄	#16863638	1	2	yes	—
256	C₄	#16863774	1	1	yes	characteristic
257	C₄	#16863779	1	1	yes	characteristic
258	C₂	#22559015	1	2	yes	—
259	C₂	#22559105	1	2	yes	—
260	C₂	#22561002	1	1	yes	characteristic
261	C₂	#22561068	1	1	yes	characteristic
262	C₂	#22561095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#9656
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₆₈	C₈	semidirect
60. C₂×C₆₇⋊C₈	Q₁₆	semidirect
137. C₆₇	#9656	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆₈	2
#22561144	2
C₂×#16863126	2
C₂×C₈×C₆₇⋊C₈	2
C₂×#335×C₆₇	2
#9656	67