Subgroups — #494×Dic₆₇ #30864269

Order 17152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₆₇	C₂	1	4	yes	maximal
2	C₄⋊Q₈×Dic₆₇	C₂	1	2	yes	maximal
3	#22561071	C₂	1	4	yes	maximal
4	#22561012	C₂	1	2	yes	maximal
5	#22560369	C₂	1	1	yes	maximal, characteristic
6	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₆₇	C₂²	1	4	yes	—
9	Q₁₆×Dic₆₇	—	2	16	no	—
10	C₂×#14523977	C₂²	1	4	yes	—
11	C₂×#14523967	C₂²	1	8	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
14	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
15	#16863303	C₂²	1	4	yes	—
16	#16863295	—	2	4	no	—
17	#16863266	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₆₇	—	2	4	no	—
19	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#494×C₆₇	C₄	1	8	yes	complemented
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
25	Q₈×Dic₆₇	—	2	16	no	—
26	C₂×C₄⋊Dic₆₇	—	2	4	no	—
27	C₂×C₄⋊Dic₆₇	—	2	4	no	—
28	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
29	C₂×C₄×Dic₆₇	—	2	4	no	—
30	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
31	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
34	#14523977	—	2	8	no	—
35	#14523967	—	2	16	no	—
36	C₄×C₆₇⋊C₈	—	2	2	no	—
37	C₈×Dic₆₇	—	2	4	no	—
38	C₈×Dic₆₇	D₄	1	8	yes	—
39	C₆₇⋊#95	—	2	8	no	—
40	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
41	C₂×Q₁₆×C₆₇	—	2	8	no	—
42	C₂×Q₁₆×C₆₇	—	2	8	no	—
43	C₂×Q₁₆×C₆₇	C₂×C₄	1	16	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×Dic₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
53	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
57	C₄⋊Dic₆₇	—	2	8	no	—
58	C₄⋊Dic₆₇	—	4	8	no	—
59	C₄⋊Dic₆₇	—	4	8	no	—
60	C₄×Dic₆₇	—	2	4	no	—
61	C₄×Dic₆₇	—	2	2	no	—
62	C₄×Dic₆₇	—	4	8	no	—
63	C₄×Dic₆₇	Q₁₆	1	8	yes	complemented
64	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
65	C₂×Q₈×C₆₇	—	2	4	no	—
66	C₂×Q₈×C₆₇	—	2	8	no	—
67	C₂×Q₈×C₆₇	—	2	8	no	—
68	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₆₇	—	2	32	no	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₆₇	#41	1	4	yes	—
76	C₄⋊C₄×C₆₇	—	2	16	no	—
77	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₆₇	#41	1	2	yes	—
79	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
80	Q₈×C₆₇	—	2	16	no	—
81	Q₈×C₆₇	—	4	8	no	—
82	C₂×C₄×C₆₇	—	2	8	no	—
83	C₂×C₄×C₆₇	—	4	4	no	—
84	C₂×C₄×C₆₇	—	4	4	no	—
85	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₆₇	#135	1	4	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
92	C₂×Dic₆₇	—	2	2	no	—
93	C₂×Dic₆₇	—	2	2	no	—
94	C₂×Dic₆₇	—	2	2	no	—
95	C₂×Dic₆₇	—	4	4	no	—
96	C₂×Dic₆₇	—	4	8	no	—
97	C₂×Dic₆₇	#127	1	2	yes	—
98	C₂×Dic₆₇	#127	1	2	yes	—
99	C₂×Dic₆₇	C₂×Q₁₆	1	4	yes	—
100	C₂×Dic₆₇	C₂×Q₁₆	1	4	yes	—
101	C₈×C₆₇	C₄×D₄	1	8	yes	—
102	C₆₇⋊C₈	—	2	4	no	—
103	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
105	C₂²×C₆₇	#532	1	1	yes	characteristic
106	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
107	C₂²×C₆₇	C₂×#135	1	2	yes	—
108	C₄×C₆₇	—	4	8	no	—
109	C₄×C₆₇	C₄×Q₁₆	1	8	yes	complemented
110	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
111	C₄×C₆₇	#532	1	2	yes	—
112	Dic₆₇	—	2	2	no	—
113	Dic₆₇	—	2	4	no	—
114	Dic₆₇	—	8	4	no	—
115	Dic₆₇	#494	1	4	yes	direct product factor
116	C₄×#494	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
118	C₂×C₆₇	C₂×C₄×Q₁₆	1	2	yes	—
119	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
120	C₂×C₆₇	#3147	1	1	yes	characteristic
121	C₂×C₆₇	#3170	1	2	yes	—
122	C₂×#494	D₆₇	1	1	yes	characteristic
123	C₂×C₄×Q₁₆	—	67	4	no	—
124	C₄×C₄⋊Q₈	—	67	2	no	—
125	#1969	—	67	4	no	—
126	#1956	—	67	2	no	—
127	#1845	—	67	1	no	—
128	C₄²×C₈	—	67	1	no	—
129	C₆₇	C₄×#494	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	C₂×D₆₇	1	4	yes	—
131	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
132	C₂×C₄×Q₈	—	67	4	no	—
133	#494	Dic₆₇	1	8	yes	direct product factor
134	C₄×Q₁₆	—	134	16	no	—
135	C₂×#107	—	67	4	no	—
136	C₂×#103	—	67	8	no	—
137	C₂×C₄×C₈	—	67	2	no	—
138	C₂×C₄×C₈	—	67	2	no	—
139	C₂×C₄×C₈	—	67	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#391	—	67	4	no	—
142	#389	—	134	4	no	—
143	#384	—	67	2	no	—
144	C₄×C₄⋊C₄	—	134	4	no	—
145	C₄³	—	67	1	no	—
146	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	2	8	no	—
149	C₂×Q₁₆	C₂×Dic₆₇	1	16	yes	—
150	C₂²×C₈	—	67	2	no	—
151	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
154	C₄×Q₈	—	134	16	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	67	4	no	—
157	C₂×C₄⋊C₄	—	134	4	no	—
158	C₂×C₄⋊C₄	—	134	4	no	—
159	C₂×C₄²	—	67	2	no	—
160	C₂×C₄²	—	67	1	no	—
161	C₂×C₄²	—	67	2	no	—
162	C₂×C₄²	—	67	1	no	—
163	C₂×C₄²	—	134	4	no	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	#107	—	134	8	no	—
166	#103	—	134	16	no	—
167	C₄×C₈	—	67	8	no	—
168	C₄×C₈	—	134	4	no	—
169	C₄×C₈	—	134	2	no	—
170	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
171	#95	—	134	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	C₂²×Dic₆₇	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	67	1	no	—
178	C₂²×C₄	—	67	2	no	—
179	C₂²×C₄	—	67	1	no	—
180	C₂²×C₄	—	134	4	no	—
181	C₂²×C₄	D₄×D₆₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	Q₁₆	—	2	32	no	—
184	C₂×C₈	—	67	4	no	—
185	C₂×C₈	—	134	2	no	—
186	C₂×C₈	—	134	2	no	—
187	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
188	C₂×C₈	D₄×D₆₇	1	4	yes	—
189	C₂×C₈	#11761627	1	4	yes	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	268	8	no	—
192	C₄⋊C₄	—	268	8	no	—
193	C₄⋊C₄	—	134	8	no	—
194	C₄²	—	268	8	no	—
195	C₄²	—	67	8	no	—
196	C₄²	—	67	4	no	—
197	C₄²	—	134	4	no	—
198	C₄²	—	134	2	no	—
199	C₄²	C₂²×Dic₆₇	1	2	yes	—
200	C₄²	#11761627	1	2	yes	—
201	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
202	Q₈	—	2	16	no	—
203	Q₈	—	4	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	268	8	no	—
208	C₂×C₄	—	268	4	no	—
209	C₂×C₄	—	67	4	no	—
210	C₂×C₄	—	67	2	no	—
211	C₂×C₄	—	67	4	no	—
212	C₂×C₄	—	67	2	no	—
213	C₂×C₄	—	134	2	no	—
214	C₂×C₄	—	134	2	no	—
215	C₂×C₄	—	134	2	no	—
216	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
217	C₂×C₄	Q₁₆×D₆₇	1	4	yes	—
218	C₂×C₄	#14524066	1	4	yes	—
219	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
221	C₂×C₄	C₂×#11761627	1	2	yes	—
222	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
223	C₈	—	134	4	no	—
224	C₈	D₄×Dic₆₇	1	8	yes	—
225	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
226	C₂²	#127×D₆₇	1	1	yes	characteristic
227	C₂²	#16863944	1	1	yes	characteristic
228	C₂²	C₂×Q₁₆×D₆₇	1	2	yes	—
229	C₂²	C₂×#14524066	1	2	yes	—
230	C₄	—	4	8	no	—
231	C₄	—	536	4	no	—
232	C₄	—	67	4	no	—
233	C₄	—	134	4	no	—
234	C₄	—	134	2	no	—
235	C₄	Q₁₆×Dic₆₇	1	8	yes	complemented
236	C₄	C₂×D₄×Dic₆₇	1	2	yes	—
237	C₄	#16863944	1	2	yes	—
238	C₂	#127×Dic₆₇	1	1	yes	characteristic
239	C₂	C₂×Q₁₆×Dic₆₇	1	2	yes	—
240	C₂	#494×D₆₇	1	1	yes	characteristic
241	C₂	#22570638	1	1	yes	characteristic
242	C₂	#22570793	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	C₄×#494
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
22. #494×C₆₇	C₄	semidirect
63. C₄×Dic₆₇	Q₁₆	semidirect
109. C₄×C₆₇	C₄×Q₁₆	semidirect
115. Dic₆₇	#494	direct (N × H)
129. C₆₇	C₄×#494	semidirect
133. #494	Dic₆₇	direct (N × H)
235. C₄	Q₁₆×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₆₇	2
C₄⋊Q₈×Dic₆₇	2
#22561071	2
#22561012	2
#22560369	2
C₄×C₈×Dic₆₇	2
C₂×#494×C₆₇	2
C₄×#494	67

Subgroups — #494×Dic67 #30864269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #494×Dic₆₇ #30864269