Subgroups — C₄×#16863776 #30864268

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×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆₈	C₂	1	4	yes	maximal
3	C₄×#14524043	C₂	1	2	yes	maximal
4	#22561070	C₂	1	4	yes	maximal
5	#22561011	C₂	1	2	yes	maximal
6	#22560369	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₆₈	C₂²	1	4	yes	—
9	C₂×#14524043	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₃₄	C₂²	1	4	yes	—
11	#16863776	C₄	1	8	yes	direct product factor
12	C₄×Dic₂₆₈	—	2	16	no	—
13	C₂×#14523977	C₂²	1	4	yes	—
14	C₂×#14523968	C₂²	1	8	yes	—
15	#16863302	C₂²	1	4	yes	—
16	#16863294	—	2	4	no	—
17	#16863266	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₆₇	—	2	4	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₃₄	C₂³	1	4	yes	—
25	C₂×Dic₂₆₈	—	2	8	no	—
26	C₂×Dic₂₆₈	—	2	8	no	—
27	C₂×Dic₂₆₈	C₂×C₄	1	16	yes	—
28	#14524043	—	2	4	no	—
29	#14524043	C₂×C₄	1	8	yes	—
30	C₄×Dic₁₃₄	—	2	16	no	—
31	C₂×C₄⋊Dic₆₇	—	2	4	no	—
32	C₂×C₄⋊Dic₆₇	—	2	4	no	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	C₂×Dic₆₇⋊C₄	—	2	4	no	—
35	C₂×C₄×Dic₆₇	—	2	4	no	—
36	#14523977	—	2	8	no	—
37	#14523968	—	2	16	no	—
38	C₆₇⋊#95	—	2	8	no	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₆₇	—	2	4	no	—
47	C₄×C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₆₇	D₄	1	8	yes	—
50	C₂×Dic₁₃₄	—	2	8	no	—
51	C₂×Dic₁₃₄	—	2	8	no	—
52	C₂×Dic₁₃₄	—	2	4	no	—
53	C₂×Dic₁₃₄	C₂²×C₄	1	8	yes	—
54	C₂²×Dic₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	4	no	—
56	Dic₂₆₈	—	2	32	no	—
57	C₄⋊Dic₆₇	—	2	16	no	—
58	C₄⋊Dic₆₇	—	2	8	no	—
59	C₄⋊Dic₆₇	—	4	8	no	—
60	Dic₆₇⋊C₄	—	4	8	no	—
61	C₄×Dic₆₇	—	4	8	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₆₇	#41	1	4	yes	—
73	C₄²×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	Q₁₆	1	8	yes	complemented
76	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₆₇	C₂×D₄	1	4	yes	—
78	C₄²×C₆₇	#41	1	2	yes	—
79	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₆₇	#127	1	2	yes	—
85	C₂×C₄×C₆₇	#127	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
88	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
89	C₂×C₄×C₆₇	#135	1	4	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
94	C₂×Dic₆₇	—	2	8	no	—
95	C₂×Dic₆₇	—	4	4	no	—
96	C₂×Dic₆₇	—	4	4	no	—
97	C₂×Dic₆₇	—	4	4	no	—
98	C₂×Dic₆₇	—	4	8	no	—
99	Dic₁₃₄	—	2	16	no	—
100	Dic₁₃₄	—	4	8	no	—
101	C₈×C₆₇	—	2	4	no	—
102	C₈×C₆₇	C₄×D₄	1	8	yes	—
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	—	2	2	no	—
109	C₄×C₆₇	—	2	4	no	—
110	C₄×C₆₇	C₄×Q₁₆	1	8	yes	complemented
111	C₄×C₆₇	#494	1	4	yes	complemented
112	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
113	C₄×C₆₇	#532	1	2	yes	—
114	Dic₆₇	—	4	8	no	—
115	Dic₆₇	—	8	4	no	—
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	—	67	1	no	—
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	D₆₇	1	1	yes	characteristic
129	C₆₇	C₄×#494	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	67	4	no	—
131	C₂×C₄⋊Q₈	—	67	2	no	—
132	C₂×C₄×Q₈	—	67	4	no	—
133	#494	—	67	8	no	—
134	C₄×Q₁₆	—	134	16	no	—
135	C₂×#107	—	67	4	no	—
136	C₂×#103	—	67	8	no	—
137	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
141	#391	—	67	4	no	—
142	#389	—	134	4	no	—
143	#384	—	67	2	no	—
144	C₄×C₄⋊C₄	—	134	4	no	—
145	C₄³	C₂×D₆₇	1	1	yes	characteristic
146	C₂²×Q₈	—	67	4	no	—
147	C₂×Q₁₆	—	67	16	no	—
148	C₂×Q₁₆	—	134	8	no	—
149	C₂×Q₁₆	—	134	8	no	—
150	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
151	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
152	C₄⋊Q₈	—	67	8	no	—
153	C₄⋊Q₈	—	134	4	no	—
154	C₄×Q₈	—	134	16	no	—
155	C₂×C₄⋊C₄	—	67	4	no	—
156	C₂×C₄⋊C₄	—	134	4	no	—
157	C₂×C₄⋊C₄	—	134	4	no	—
158	C₂×C₄⋊C₄	—	134	4	no	—
159	C₂×C₄²	—	134	4	no	—
160	C₂×C₄²	D₂₆₈	1	2	yes	—
161	C₂×C₄²	D₂₆₈	1	2	yes	—
162	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	#107	—	134	8	no	—
166	#103	—	134	16	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₄×D₆₇	1	4	yes	—
170	C₄×C₈	D₂₆₈	1	8	yes	—
171	#95	—	134	8	no	—
172	C₂×Q₈	—	67	8	no	—
173	C₂×Q₈	—	134	8	no	—
174	C₂×Q₈	—	134	8	no	—
175	C₂×Q₈	—	134	4	no	—
176	C₂²×C₄	—	134	4	no	—
177	C₂²×C₄	—	134	4	no	—
178	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
180	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
181	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	Q₁₆	—	134	32	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
187	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
188	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
189	C₂×C₈	#11761628	1	4	yes	—
190	C₄⋊C₄	—	268	8	no	—
191	C₄⋊C₄	—	268	8	no	—
192	C₄⋊C₄	—	134	8	no	—
193	C₄⋊C₄	—	134	16	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	268	8	no	—
197	C₄²	Dic₂₆₈	1	8	yes	complemented
198	C₄²	C₂×C₄×D₆₇	1	2	yes	—
199	C₄²	C₂×D₂₆₈	1	4	yes	—
200	C₄²	#11761628	1	2	yes	—
201	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
202	Q₈	—	268	8	no	—
203	Q₈	—	134	16	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	268	4	no	—
208	C₂×C₄	—	268	4	no	—
209	C₂×C₄	—	268	8	no	—
210	C₂×C₄	—	268	4	no	—
211	C₂×C₄	—	134	8	no	—
212	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
213	C₂×C₄	#14524040	1	2	yes	—
214	C₂×C₄	#14524040	1	2	yes	—
215	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
216	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
217	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
218	C₂×C₄	#14524065	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
221	C₂×C₄	C₂×#11761628	1	2	yes	—
222	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	C₄×D₂₆₈	1	8	yes	—
225	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
226	C₂²	C₂×#14524040	1	1	yes	characteristic
227	C₂²	#16863946	1	1	yes	characteristic
228	C₂²	C₂²×Dic₂₆₈	1	2	yes	—
229	C₂²	C₂×#14524065	1	2	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	268	8	no	—
233	C₄	—	536	4	no	—
234	C₄	C₄×Dic₂₆₈	1	8	yes	complemented
235	C₄	#16863776	1	4	yes	direct product factor
236	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
237	C₄	#16863946	1	2	yes	—
238	C₂	C₄×#14524040	1	1	yes	characteristic
239	C₂	C₂×C₄×Dic₂₆₈	1	2	yes	—
240	C₂	C₂×#16863776	1	1	yes	characteristic
241	C₂	#22570637	1	1	yes	characteristic
242	C₂	#22570795	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	C₄×#494
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. #16863776	C₄	direct (N × H)
75. C₄²×C₆₇	Q₁₆	semidirect
110. C₄×C₆₇	C₄×Q₁₆	semidirect
111. C₄×C₆₇	#494	semidirect
129. C₆₇	C₄×#494	semidirect
197. C₄²	Dic₂₆₈	semidirect
234. C₄	C₄×Dic₂₆₈	semidirect
235. C₄	#16863776	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
C₂×C₄×Dic₂₆₈	2
C₄×#14524043	2
#22561070	2
#22561011	2
#22560369	2
C₄²×C₈×C₆₇	2
C₄×#494	67

Subgroups — C4×#16863776 #30864268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863776 #30864268