Subgroups — #30754689

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	#22561145	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863128	C₂	1	2	yes	maximal
5	C₂×C₈²×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×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#14523968	C₄	1	2	yes	—
13	#16863150	—	2	4	no	—
14	#16863149	—	2	2	no	—
15	#16863128	—	2	4	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₈²×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	2	no	—
25	C₄×Dic₁₃₄	—	2	2	no	—
26	C₄×Dic₁₃₄	—	2	2	no	—
27	C₄×Dic₁₃₄	—	2	2	no	—
28	C₄×Dic₁₃₄	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₆₇⋊C₄	—	2	2	no	—
31	C₂×C₄×Dic₆₇	—	2	2	no	—
32	#14523977	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	#14523968	—	2	4	no	—
36	C₂²×C₈×C₆₇	—	2	1	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₆₇	—	2	1	no	—
42	C₄×C₈×C₆₇	—	2	1	no	—
43	C₄×C₈×C₆₇	—	2	1	no	—
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
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	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	4	no	—
58	C₄⋊Dic₆₇	—	2	2	no	—
59	C₄⋊Dic₆₇	—	2	2	no	—
60	Dic₆₇⋊C₄	—	4	2	no	—
61	Dic₆₇⋊C₄	—	4	2	no	—
62	C₄×Dic₆₇	—	4	2	no	—
63	C₄×Dic₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	2	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	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
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	4	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	—	2	4	no	—
108	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
109	C₆₇⋊C₈	—	8	2	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	—	67	1	no	—
135	C₂×Q₈⋊C₈	—	67	2	no	—
136	C₂×C₈²	D₆₇	1	1	yes	characteristic
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	—	67	2	no	—
144	C₂×#103	—	67	2	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	134	4	no	—
149	#335	—	134	2	no	—
150	Q₈⋊C₈	—	134	4	no	—
151	C₈²	—	2	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	1	no	—
157	C₂²×C₈	—	134	2	no	—
158	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₂²×C₈	D₂₆₈	1	2	yes	—
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	—	134	2	no	—
171	C₄⋊C₈	—	134	2	no	—
172	#103	—	134	4	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	D₂₆₈	1	2	yes	—
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	C₂×D₂₆₈	1	1	yes	characteristic
189	C₂²×C₄	#11761628	1	1	yes	characteristic
190	Q₁₆	—	134	8	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	—	268	4	no	—
199	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂×C₈	C₈×D₆₇	1	2	yes	—
202	C₂×C₈	#11761611	1	2	yes	—
203	C₂×C₈	Dic₂₆₈	1	4	yes	complemented
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	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
212	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
213	C₄²	#11761628	1	1	yes	characteristic
214	C₂³	C₄×D₂₆₈	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	C₄×D₂₆₈	1	1	yes	characteristic
227	C₂×C₄	#14524017	1	1	yes	characteristic
228	C₂×C₄	#14524017	1	1	yes	characteristic
229	C₂×C₄	#14524031	1	1	yes	characteristic
230	C₂×C₄	#14524040	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	#14524049	1	1	yes	characteristic
234	C₂×C₄	#14524049	1	1	yes	characteristic
235	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
236	C₂×C₄	#14524065	1	2	yes	—
237	C₈	—	2	4	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	4	no	—
240	C₈	—	536	2	no	—
241	C₈	(C₂×D₆₇)⋊C₈	1	4	yes	—
242	C₂²	#16863298	1	1	yes	characteristic
243	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
244	C₂²	#16863475	1	1	yes	characteristic
245	C₂²	C₄×Dic₂₆₈	1	2	yes	—
246	C₂²	#16863514	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	#16863362	1	1	yes	characteristic
253	C₄	C₈×D₂₆₈	1	1	yes	characteristic
254	C₄	#16863475	1	1	yes	characteristic
255	C₄	#16863637	1	2	yes	—
256	C₄	#16863776	1	1	yes	characteristic
257	C₄	#16863780	1	1	yes	characteristic
258	C₂	C₈×Dic₂₆₈	1	2	yes	—
259	C₂	#22559107	1	2	yes	—
260	C₂	#22561003	1	1	yes	characteristic
261	C₂	#22561070	1	1	yes	characteristic
262	C₂	#22561097	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#9656
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₆₈	C₈	semidirect
77. C₂×C₈×C₆₇	Q₁₆	semidirect
137. C₆₇	#9656	semidirect
203. C₂×C₈	Dic₂₆₈	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
#22561145	2
C₂×#16863149	2
C₂×#16863128	2
C₂×C₈²×C₆₇	2
#9656	67