Subgroups — #30754697

Order 17152.

486 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×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×#16863117	C₂	1	1	yes	maximal, characteristic
5	C₂×#334×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	#16863117	—	2	2	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	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	2	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	Dic₆₇⋊C₄	—	4	2	no	—
63	Dic₆₇⋊C₄	—	4	2	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	1	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	4	4	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	#41	1	1	yes	characteristic
81	C₂³×C₆₇	C₄×D₄	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	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#137	1	4	yes	—
94	C₂×Dic₆₇	—	2	4	no	—
95	C₂×Dic₆₇	—	4	2	no	—
96	C₂×Dic₆₇	—	4	2	no	—
97	C₂×Dic₆₇	—	4	2	no	—
98	C₂×Dic₆₇	—	4	4	no	—
99	Dic₁₃₄	—	2	4	no	—
100	Dic₁₃₄	—	2	4	no	—
101	Dic₁₃₄	—	4	2	no	—
102	Dic₁₃₄	—	4	2	no	—
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	—	8	2	no	—
105	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
106	C₆₇⋊C₈	—	4	2	no	—
107	C₆₇⋊C₈	—	4	2	no	—
108	C₂²×C₆₇	#390	1	1	yes	characteristic
109	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
110	C₂²×C₆₇	#436	1	1	yes	characteristic
111	C₂²×C₆₇	#441	1	2	yes	—
112	C₂²×C₆₇	#444	1	2	yes	—
113	C₄×C₆₇	—	2	1	no	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₄×C₆₇	#408	1	1	yes	characteristic
117	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
118	C₄×C₆₇	#436	1	1	yes	characteristic
119	C₄×C₆₇	#470	1	2	yes	—
120	C₄×C₆₇	#497	1	2	yes	—
121	Dic₆₇	—	4	4	no	—
122	Dic₆₇	—	8	2	no	—
123	#9658	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	#1579	1	2	yes	—
125	C₂×C₆₇	#1593	1	2	yes	—
126	C₂×C₆₇	#1954	1	1	yes	characteristic
127	C₂×C₆₇	#1971	1	1	yes	characteristic
128	C₂×C₆₇	#1975	1	1	yes	characteristic
129	C₂×C₄×Q₁₆	—	67	1	no	—
130	#1984	—	67	2	no	—
131	C₂×#334	D₆₇	1	1	yes	characteristic
132	C₂×Q₈⋊C₈	—	67	2	no	—
133	C₂×#322	—	67	1	no	—
134	C₆₇	#9658	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₁₆	—	67	1	no	—
136	C₂×C₄×Q₈	—	67	2	no	—
137	C₄×Q₁₆	—	134	2	no	—
138	C₄×Q₁₆	—	134	2	no	—
139	C₂×#107	—	67	1	no	—
140	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
141	C₂×#103	—	67	2	no	—
142	C₂×C₄×C₈	—	67	1	no	—
143	C₂×C₄×C₈	—	67	1	no	—
144	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
145	#336	—	134	4	no	—
146	#334	—	2	2	no	—
147	Q₈⋊C₈	—	134	4	no	—
148	#322	—	134	2	no	—
149	C₂²×Q₈	—	67	2	no	—
150	C₂×Q₁₆	—	67	4	no	—
151	C₂×Q₁₆	—	134	2	no	—
152	C₂×Q₁₆	—	134	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	67	2	no	—
155	C₂²×C₈	—	134	1	no	—
156	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
158	C₄×Q₈	—	134	2	no	—
159	C₄×Q₈	—	134	2	no	—
160	C₄×Q₈	—	134	2	no	—
161	C₄×Q₈	—	134	2	no	—
162	C₂×C₄⋊C₄	—	67	2	no	—
163	C₂×C₄⋊C₄	—	134	2	no	—
164	C₂×C₄²	—	134	2	no	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	#107	—	134	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	#103	—	134	4	no	—
170	C₄×C₈	—	67	2	no	—
171	C₄×C₈	—	134	1	no	—
172	C₄×C₈	—	134	1	no	—
173	C₄×C₈	—	134	1	no	—
174	C₄×C₈	D₂₆₈	1	2	yes	—
175	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
176	C₂×Q₈	—	67	4	no	—
177	C₂×Q₈	—	134	2	no	—
178	C₂×Q₈	—	134	2	no	—
179	C₂×Q₈	—	134	2	no	—
180	C₂×Q₈	—	134	2	no	—
181	C₂×Q₈	—	134	2	no	—
182	C₂²×C₄	—	134	2	no	—
183	C₂²×C₄	—	134	2	no	—
184	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₄	#11761629	1	1	yes	characteristic
187	Q₁₆	—	134	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	268	2	no	—
193	C₂×C₈	—	134	2	no	—
194	C₂×C₈	—	134	2	no	—
195	C₂×C₈	—	134	4	no	—
196	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
197	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
198	C₂×C₈	C₈×D₆₇	1	2	yes	—
199	C₂×C₈	#11761611	1	2	yes	—
200	C₄⋊C₄	—	268	2	no	—
201	C₄⋊C₄	—	268	2	no	—
202	C₄⋊C₄	—	134	2	no	—
203	C₄⋊C₄	—	134	2	no	—
204	C₄²	—	268	2	no	—
205	C₄²	—	268	2	no	—
206	C₄²	D₄×D₆₇	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
208	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
209	C₄²	#11761629	1	1	yes	characteristic
210	C₂³	#14524005	1	1	yes	characteristic
211	Q₈	—	268	2	no	—
212	Q₈	—	268	2	no	—
213	Q₈	—	134	4	no	—
214	Q₈	—	134	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	268	2	no	—
217	C₂×C₄	—	268	2	no	—
218	C₂×C₄	—	268	4	no	—
219	C₂×C₄	—	268	2	no	—
220	C₂×C₄	—	134	4	no	—
221	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#14524005	1	1	yes	characteristic
223	C₂×C₄	#14524018	1	1	yes	characteristic
224	C₂×C₄	#14524018	1	1	yes	characteristic
225	C₂×C₄	#14524032	1	1	yes	characteristic
226	C₂×C₄	#14524039	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
228	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
229	C₂×C₄	#14524051	1	1	yes	characteristic
230	C₂×C₄	#14524051	1	1	yes	characteristic
231	C₂×C₄	#14524072	1	4	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	8	2	no	—
234	C₈	—	268	2	no	—
235	C₈	—	268	2	no	—
236	C₈	(C₂×D₆₇)⋊C₈	1	4	yes	—
237	C₂²	#16863297	1	1	yes	characteristic
238	C₂²	#16863461	1	1	yes	characteristic
239	C₂²	#16863473	1	1	yes	characteristic
240	C₂²	#16863502	1	2	yes	—
241	C₂²	#16863517	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	268	4	no	—
245	C₄	—	536	2	no	—
246	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
247	C₄	#16863364	1	1	yes	characteristic
248	C₄	#16863461	1	1	yes	characteristic
249	C₄	#16863473	1	1	yes	characteristic
250	C₄	#16863655	1	2	yes	—
251	C₄	#16863787	1	2	yes	—
252	C₂	#22559027	1	2	yes	—
253	C₂	#22559117	1	2	yes	—
254	C₂	#22561002	1	1	yes	characteristic
255	C₂	#22561078	1	1	yes	characteristic
256	C₂	#22561100	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×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#9658
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₆₈	C₈	semidirect
134. C₆₇	#9658	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×#16863117	2
C₂×#334×C₆₇	2
#9658	67