Subgroups — C₂²×#16863159 #30853227

Order 17152.

3006 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
2	C₂×#16863159	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₆₇	—	2	2	no	—
5	C₂²×C₄⋊Dic₆₇	—	2	6	no	—
6	C₂²×C₄⋊Dic₆₇	—	2	8	no	—
7	C₂²×C₄⋊Dic₆₇	C₄	1	16	yes	complemented
8	C₂²×C₄⋊Dic₆₇	C₂²	1	12	yes	—
9	#16863159	C₂²	1	16	yes	direct product factor
10	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	—	2	8	no	—
12	C₂³×C₈×C₆₇	C₄	1	8	yes	complemented
13	C₂³×C₈×C₆₇	C₂²	1	6	yes	—
14	C₂³×Dic₆₇	—	2	16	no	—
15	C₂³×Dic₆₇	—	2	12	no	—
16	C₂³×Dic₆₇	—	4	8	no	—
17	C₂³×Dic₆₇	—	4	8	no	—
18	C₂×C₄⋊Dic₆₇	—	2	48	no	—
19	C₂×C₄⋊Dic₆₇	—	2	24	no	—
20	C₂×C₄⋊Dic₆₇	—	2	12	no	—
21	C₂×C₄⋊Dic₆₇	C₂×C₄	1	48	yes	complemented
22	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
23	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₆₇	D₄	1	3	yes	—
27	C₂³×C₄×C₆₇	D₄	1	8	yes	—
28	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
30	C₂²×C₈×C₆₇	—	2	32	no	—
31	C₂²×C₈×C₆₇	—	2	24	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	24	yes	complemented
33	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
34	C₂²×Dic₆₇	—	2	48	no	—
35	C₂²×Dic₆₇	—	2	8	no	—
36	C₂²×Dic₆₇	—	4	64	no	—
37	C₂²×Dic₆₇	—	4	24	no	—
38	C₂²×Dic₆₇	—	4	24	no	—
39	C₄⋊Dic₆₇	—	2	48	no	—
40	C₄⋊Dic₆₇	—	2	8	no	—
41	C₄⋊Dic₆₇	C₂²×C₄	1	16	yes	complemented
42	C₂⁴×C₆₇	C₂²⋊C₄	1	8	yes	—
43	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
44	C₂⁴×C₆₇	D₈	1	4	yes	—
45	C₂⁴×C₆₇	SD₁₆	1	8	yes	—
46	C₂⁴×C₆₇	Q₁₆	1	4	yes	—
47	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₆₇	C₄²	1	16	yes	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	24	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	16	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	32	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	48	no	—
60	C₂×C₈×C₆₇	—	2	8	no	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	complemented
62	C₂³×C₆₇	#95	1	16	yes	—
63	C₂³×C₆₇	#102	1	32	yes	—
64	C₂³×C₆₇	#103	1	32	yes	—
65	C₂³×C₆₇	#106	1	16	yes	—
66	C₂³×C₆₇	#107	1	16	yes	—
67	C₂³×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
68	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
69	C₂³×C₆₇	C₂×D₈	1	6	yes	—
70	C₂³×C₆₇	C₂×SD₁₆	1	12	yes	—
71	C₂³×C₆₇	C₂×Q₁₆	1	6	yes	—
72	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#95	1	48	yes	—
74	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
75	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
76	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
79	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×Dic₆₇	—	2	16	no	—
84	C₂×Dic₆₇	—	4	8	no	—
85	C₂×Dic₆₇	—	4	8	no	—
86	C₂×Dic₆₇	—	4	96	no	—
87	C₈×C₆₇	—	2	8	no	—
88	C₂²×C₆₇	#340	1	64	yes	complemented
89	C₂²×C₆₇	C₂×#95	1	12	yes	—
90	C₂²×C₆₇	C₂×#102	1	24	yes	—
91	C₂²×C₆₇	C₂×#103	1	24	yes	—
92	C₂²×C₆₇	C₂×#106	1	12	yes	—
93	C₂²×C₆₇	C₂×#107	1	12	yes	—
94	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
95	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
98	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
99	C₄×C₆₇	C₂×#95	1	12	yes	—
100	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
103	Dic₆₇	—	4	16	no	—
104	C₂²×#340	—	67	1	no	maximal, Sylow
105	C₂×C₆₇	C₂×#340	1	24	yes	complemented
106	C₂×C₆₇	C₂²×#95	1	1	yes	characteristic
107	C₂×C₆₇	C₂²×#102	1	2	yes	—
108	C₂×C₆₇	C₂²×#103	1	2	yes	—
109	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
110	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
111	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
112	C₂³×C₄⋊C₄	—	67	2	no	—
113	C₂×#340	—	67	12	no	—
114	C₆₇	C₂²×#340	1	1	yes	characteristic, complemented, Sylow
115	C₂⁴×C₄	—	134	2	no	—
116	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
117	C₂³×C₈	—	2	8	no	—
118	C₂³×C₈	Dic₆₇	1	8	yes	complemented
119	C₂³×C₈	C₂×D₆₇	1	6	yes	—
120	C₂²×C₄⋊C₄	—	67	16	no	—
121	C₂²×C₄⋊C₄	—	67	12	no	—
122	C₂²×C₄⋊C₄	—	134	6	no	—
123	C₂²×C₄⋊C₄	—	134	8	no	—
124	#340	—	67	16	no	—
125	C₂⁵	D₂₆₈	1	1	yes	characteristic
126	C₂³×C₄	—	268	8	no	—
127	C₂³×C₄	—	268	8	no	—
128	C₂³×C₄	—	134	16	no	—
129	C₂³×C₄	—	134	12	no	—
130	C₂³×C₄	Dic₁₃₄	1	4	yes	—
131	C₂³×C₄	C₄×D₆₇	1	8	yes	—
132	C₂³×C₄	D₂₆₈	1	3	yes	—
133	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
134	C₂³×C₄	C₂²⋊D₆₇	1	8	yes	—
135	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
136	C₂²×C₈	—	2	32	no	—
137	C₂²×C₈	—	2	24	no	—
138	C₂²×C₈	C₂×Dic₆₇	1	24	yes	complemented
139	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
140	C₂×C₄⋊C₄	—	67	48	no	—
141	C₂×C₄⋊C₄	—	67	8	no	—
142	C₂×C₄⋊C₄	—	134	48	no	—
143	C₂×C₄⋊C₄	—	134	24	no	—
144	C₂×C₄⋊C₄	—	134	12	no	—
145	C₂⁴	(C₂×D₆₇)⋊C₄	1	8	yes	—
146	C₂⁴	C₄⋊Dic₆₇	1	4	yes	—
147	C₂⁴	D₅₃₆	1	4	yes	—
148	C₂⁴	#11761617	1	8	yes	—
149	C₂⁴	Dic₂₆₈	1	4	yes	—
150	C₂⁴	C₂×D₂₆₈	1	3	yes	—
151	C₂²×C₄	—	268	64	no	—
152	C₂²×C₄	—	268	24	no	—
153	C₂²×C₄	—	268	24	no	—
154	C₂²×C₄	—	134	48	no	—
155	C₂²×C₄	—	134	8	no	—
156	C₂²×C₄	C₄×Dic₆₇	1	16	yes	—
157	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	24	yes	—
158	C₂²×C₄	C₂²⋊Dic₆₇	1	16	yes	—
159	C₂²×C₄	Dic₆₇⋊C₄	1	32	yes	—
160	C₂²×C₄	C₄⋊Dic₆₇	1	12	yes	—
161	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
162	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
164	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
165	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
166	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
167	C₂×C₈	—	2	48	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	C₂²×Dic₆₇	1	8	yes	complemented
170	C₄⋊C₄	—	67	16	no	—
171	C₄⋊C₄	—	134	48	no	—
172	C₄⋊C₄	—	134	8	no	—
173	C₂³	C₆₇⋊#95	1	16	yes	—
174	C₂³	#14523965	1	32	yes	—
175	C₂³	#14523968	1	32	yes	—
176	C₂³	#14523975	1	16	yes	—
177	C₂³	#14523977	1	16	yes	—
178	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	12	yes	—
179	C₂³	C₂×C₄⋊Dic₆₇	1	6	yes	—
180	C₂³	C₂×D₅₃₆	1	6	yes	—
181	C₂³	C₂×#11761617	1	12	yes	—
182	C₂³	C₂×Dic₂₆₈	1	6	yes	—
183	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
184	C₂×C₄	—	268	96	no	—
185	C₂×C₄	—	268	8	no	—
186	C₂×C₄	—	268	8	no	—
187	C₂×C₄	—	134	16	no	—
188	C₂×C₄	C₆₇⋊#95	1	48	yes	—
189	C₂×C₄	C₂×C₄×Dic₆₇	1	12	yes	—
190	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	12	yes	—
191	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	12	yes	—
192	C₂×C₄	C₂×Dic₆₇⋊C₄	1	24	yes	—
193	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
194	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
195	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
197	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
198	C₈	—	2	8	no	—
199	C₂²	#16863159	1	64	yes	direct product factor
200	C₂²	C₂×C₆₇⋊#95	1	12	yes	—
201	C₂²	C₂×#14523965	1	24	yes	—
202	C₂²	C₂×#14523968	1	24	yes	—
203	C₂²	C₂×#14523975	1	12	yes	—
204	C₂²	C₂×#14523977	1	12	yes	—
205	C₂²	C₂²×(C₂×D₆₇)⋊C₄	1	2	yes	—
206	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
207	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
208	C₂²	C₂²×#11761617	1	2	yes	—
209	C₂²	C₂²×Dic₂₆₈	1	1	yes	characteristic
210	C₄	—	268	16	no	—
211	C₄	C₂×C₆₇⋊#95	1	12	yes	—
212	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
213	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
214	C₄	C₂²×Dic₆₇⋊C₄	1	2	yes	—
215	C₂	C₂×#16863159	1	24	yes	direct product factor
216	C₂	C₂²×C₆₇⋊#95	1	1	yes	characteristic
217	C₂	C₂²×#14523965	1	2	yes	—
218	C₂	C₂²×#14523968	1	2	yes	—
219	C₂	C₂²×#14523975	1	1	yes	characteristic
220	C₂	C₂²×#14523977	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂²×#340
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16863159	C₂	direct (N × H)
7. C₂²×C₄⋊Dic₆₇	C₄	semidirect
9. #16863159	C₂²	direct (N × H)
12. C₂³×C₈×C₆₇	C₄	semidirect
21. C₂×C₄⋊Dic₆₇	C₂×C₄	semidirect
32. C₂²×C₈×C₆₇	C₂×C₄	semidirect
41. C₄⋊Dic₆₇	C₂²×C₄	semidirect
61. C₂×C₈×C₆₇	C₂²×C₄	semidirect
88. C₂²×C₆₇	#340	semidirect
105. C₂×C₆₇	C₂×#340	semidirect
114. C₆₇	C₂²×#340	semidirect
118. C₂³×C₈	Dic₆₇	semidirect
138. C₂²×C₈	C₂×Dic₆₇	semidirect
169. C₂×C₈	C₂²×Dic₆₇	semidirect
199. C₂²	#16863159	direct (N × H)
215. C₂	C₂×#16863159	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₆₇	2
C₂×#16863159	2
C₂⁴×C₈×C₆₇	2
C₂²×#340	67

Subgroups — C22×#16863159 #30853227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863159 #30853227