Subgroups — C₂³×#14523965 #31719340

Order 17152.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14523965	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₆₇	—	2	1	no	—
6	C₂³×D₂₆₈	—	2	7	no	—
7	C₂³×D₂₆₈	—	2	8	no	—
8	C₂³×D₂₆₈	C₄	1	16	yes	complemented
9	C₂³×D₂₆₈	C₂²	1	14	yes	—
10	C₂⁴×Dic₆₇	—	2	1	no	—
11	C₂²×C₄⋊Dic₆₇	—	2	7	no	—
12	C₂²×C₄⋊Dic₆₇	C₂²	1	14	yes	complemented
13	C₂×#14523965	C₂²	1	112	yes	direct product factor
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	—	2	8	no	—
16	C₂³×C₈×C₆₇	C₂²	1	14	yes	complemented
17	C₂⁴×D₆₇	—	2	16	no	—
18	C₂⁴×D₆₇	—	2	14	no	—
19	C₂⁴×D₆₇	—	4	8	no	—
20	C₂⁴×D₆₇	—	4	8	no	—
21	C₂²×D₂₆₈	—	2	112	no	—
22	C₂²×D₂₆₈	—	2	56	no	—
23	C₂²×D₂₆₈	—	2	42	no	—
24	C₂²×D₂₆₈	C₂×C₄	1	112	yes	complemented
25	C₂²×D₂₆₈	C₂³	1	28	yes	—
26	C₂³×Dic₆₇	—	2	14	no	—
27	C₂³×Dic₆₇	—	4	8	no	—
28	C₂×C₄⋊Dic₆₇	—	2	42	no	—
29	C₂×C₄⋊Dic₆₇	C₂³	1	28	yes	complemented
30	#14523965	C₂³	1	64	yes	direct product factor
31	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₆₇	D₄	1	7	yes	—
34	C₂³×C₄×C₆₇	D₄	1	8	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
36	C₂²×C₈×C₆₇	—	2	56	no	—
37	C₂²×C₈×C₆₇	C₂³	1	28	yes	complemented
38	C₂³×D₆₇	—	2	112	no	—
39	C₂³×D₆₇	—	2	28	no	—
40	C₂³×D₆₇	—	4	128	no	—
41	C₂³×D₆₇	—	4	56	no	—
42	C₂³×D₆₇	—	4	56	no	—
43	C₂×D₂₆₈	—	2	336	no	—
44	C₂×D₂₆₈	—	2	56	no	—
45	C₂×D₂₆₈	—	2	28	no	—
46	C₂×D₂₆₈	C₂²×C₄	1	112	yes	complemented
47	C₂×D₂₆₈	C₂⁴	1	8	yes	—
48	C₂²×Dic₆₇	—	2	28	no	—
49	C₂²×Dic₆₇	—	4	56	no	—
50	C₄⋊Dic₆₇	—	2	28	no	—
51	C₄⋊Dic₆₇	C₂⁴	1	8	yes	complemented
52	C₂⁴×C₆₇	C₂²⋊C₄	1	8	yes	—
53	C₂⁴×C₆₇	D₈	1	8	yes	—
54	C₂⁴×C₆₇	SD₁₆	1	8	yes	—
55	C₂⁴×C₆₇	C₂×D₄	1	7	yes	—
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	56	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	28	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
61	C₂×C₈×C₆₇	—	2	56	no	—
62	C₂×C₈×C₆₇	C₂⁴	1	8	yes	complemented
63	C₂³×C₆₇	#102	1	64	yes	complemented
64	C₂³×C₆₇	C₂×C₂²⋊C₄	1	28	yes	—
65	C₂³×C₆₇	C₂×D₈	1	28	yes	—
66	C₂³×C₆₇	C₂×SD₁₆	1	28	yes	—
67	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
68	C₂²×D₆₇	—	2	112	no	—
69	C₂²×D₆₇	—	2	8	no	—
70	C₂²×D₆₇	—	4	56	no	—
71	C₂²×D₆₇	—	4	56	no	—
72	C₂²×D₆₇	—	4	448	no	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	84	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
75	C₂×C₄×C₆₇	C₂²×D₄	1	14	yes	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
77	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₆₇	—	2	8	no	—
79	C₂×Dic₆₇	—	4	56	no	—
80	D₂₆₈	—	2	112	no	—
81	D₂₆₈	—	2	8	no	—
82	D₂₆₈	C₂³×C₄	1	16	yes	complemented
83	C₈×C₆₇	—	2	8	no	—
84	C₂²×C₆₇	C₂×#102	1	112	yes	complemented
85	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
86	C₂²×C₆₇	C₂²×D₈	1	14	yes	—
87	C₂²×C₆₇	C₂²×SD₁₆	1	14	yes	—
88	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
89	C₂×D₆₇	—	2	16	no	—
90	C₂×D₆₇	—	4	224	no	—
91	C₂×D₆₇	—	4	8	no	—
92	C₂×D₆₇	—	4	8	no	—
93	C₄×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
94	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
95	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
96	Dic₆₇	—	4	8	no	—
97	C₂³×#102	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	C₂²×#102	1	28	yes	complemented
99	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
101	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
102	D₆₇	—	4	16	no	—
103	C₂⁴×D₄	—	67	1	no	—
104	C₂⁴×C₈	D₆₇	1	1	yes	characteristic, complemented
105	C₂³×C₄⋊C₄	—	67	1	no	—
106	C₂²×#102	—	67	28	no	—
107	C₆₇	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
108	C₂⁶	—	134	1	no	—
109	C₂³×D₄	—	67	16	no	—
110	C₂³×D₄	—	67	14	no	—
111	C₂³×D₄	—	134	7	no	—
112	C₂³×D₄	—	134	8	no	—
113	C₂⁴×C₄	—	134	1	no	—
114	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂³×C₈	—	2	8	no	—
116	C₂³×C₈	C₂×D₆₇	1	14	yes	complemented
117	C₂²×C₄⋊C₄	—	67	14	no	—
118	C₂²×C₄⋊C₄	—	134	7	no	—
119	C₂×#102	—	67	112	no	—
120	C₂⁵	—	268	8	no	—
121	C₂⁵	—	268	8	no	—
122	C₂⁵	—	134	16	no	—
123	C₂⁵	—	134	14	no	—
124	C₂⁵	D₂₆₈	1	1	yes	characteristic
125	C₂²×D₄	—	67	112	no	—
126	C₂²×D₄	—	67	28	no	—
127	C₂²×D₄	—	134	112	no	—
128	C₂²×D₄	—	134	56	no	—
129	C₂²×D₄	—	134	42	no	—
130	C₂³×C₄	—	268	8	no	—
131	C₂³×C₄	—	134	14	no	—
132	C₂³×C₄	C₄×D₆₇	1	8	yes	—
133	C₂³×C₄	D₂₆₈	1	7	yes	—
134	C₂³×C₄	C₂²⋊D₆₇	1	8	yes	—
135	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
136	C₂²×C₈	—	2	56	no	—
137	C₂²×C₈	C₂²×D₆₇	1	28	yes	complemented
138	C₂×C₄⋊C₄	—	67	28	no	—
139	C₂×C₄⋊C₄	—	134	42	no	—
140	#102	—	67	64	no	—
141	C₂⁴	—	268	128	no	—
142	C₂⁴	—	268	56	no	—
143	C₂⁴	—	268	56	no	—
144	C₂⁴	—	134	112	no	—
145	C₂⁴	—	134	28	no	—
146	C₂⁴	(C₂×D₆₇)⋊C₄	1	8	yes	—
147	C₂⁴	D₅₃₆	1	8	yes	—
148	C₂⁴	#11761617	1	8	yes	—
149	C₂⁴	C₂×D₂₆₈	1	7	yes	—
150	C₂×D₄	—	67	112	no	—
151	C₂×D₄	—	67	8	no	—
152	C₂×D₄	—	134	336	no	—
153	C₂×D₄	—	134	56	no	—
154	C₂×D₄	—	134	28	no	—
155	C₂²×C₄	—	268	56	no	—
156	C₂²×C₄	—	134	28	no	—
157	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	56	yes	—
158	C₂²×C₄	C₂×C₄×D₆₇	1	28	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₆₇	1	28	yes	—
160	C₂²×C₄	C₂×D₂₆₈	1	21	yes	—
161	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
162	C₂×C₈	—	2	56	no	—
163	C₂×C₈	C₂³×D₆₇	1	8	yes	complemented
164	C₄⋊C₄	—	67	8	no	—
165	C₄⋊C₄	—	134	28	no	—
166	C₂³	—	268	448	no	—
167	C₂³	—	268	56	no	—
168	C₂³	—	268	56	no	—
169	C₂³	—	134	112	no	—
170	C₂³	—	134	8	no	—
171	C₂³	#14523965	1	64	yes	direct product factor
172	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	28	yes	—
173	C₂³	C₂×D₅₃₆	1	28	yes	—
174	C₂³	C₂×#11761617	1	28	yes	—
175	C₂³	C₂²×D₂₆₈	1	7	yes	—
176	D₄	—	67	16	no	—
177	D₄	—	134	112	no	—
178	D₄	—	134	8	no	—
179	C₂×C₄	—	268	56	no	—
180	C₂×C₄	—	134	8	no	—
181	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	84	yes	—
182	C₂×C₄	C₂²×C₄×D₆₇	1	14	yes	—
183	C₂×C₄	C₂²×C₂²⋊D₆₇	1	14	yes	—
184	C₂×C₄	C₂²×D₂₆₈	1	7	yes	—
185	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
186	C₈	—	2	8	no	—
187	C₂²	—	268	8	no	—
188	C₂²	—	268	8	no	—
189	C₂²	—	268	224	no	—
190	C₂²	—	134	16	no	—
191	C₂²	C₂×#14523965	1	112	yes	direct product factor
192	C₂²	C₂²×(C₂×D₆₇)⋊C₄	1	14	yes	—
193	C₂²	C₂²×D₅₃₆	1	14	yes	—
194	C₂²	C₂²×#11761617	1	14	yes	—
195	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
196	C₄	—	268	8	no	—
197	C₄	C₂²×(C₂×D₆₇)⋊C₄	1	14	yes	—
198	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
199	C₄	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₂	—	268	16	no	—
201	C₂	C₂²×#14523965	1	28	yes	direct product factor
202	C₂	C₂³×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂	C₂³×D₅₃₆	1	1	yes	characteristic
204	C₂	C₂³×#11761617	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂⁵×C₆₇	D₄
prime radical	C₂⁴×D₂₆₈	C₂
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂³×#102
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. C₂³×C₄⋊Dic₆₇	C₂	semidirect
3. C₂²×#14523965	C₂	direct (N × H)
4. C₂⁴×C₈×C₆₇	C₂	semidirect
8. C₂³×D₂₆₈	C₄	semidirect
12. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
13. C₂×#14523965	C₂²	direct (N × H)
16. C₂³×C₈×C₆₇	C₂²	semidirect
24. C₂²×D₂₆₈	C₂×C₄	semidirect
29. C₂×C₄⋊Dic₆₇	C₂³	semidirect
30. #14523965	C₂³	direct (N × H)
37. C₂²×C₈×C₆₇	C₂³	semidirect
46. C₂×D₂₆₈	C₂²×C₄	semidirect
51. C₄⋊Dic₆₇	C₂⁴	semidirect
62. C₂×C₈×C₆₇	C₂⁴	semidirect
63. C₂³×C₆₇	#102	semidirect
82. D₂₆₈	C₂³×C₄	semidirect
84. C₂²×C₆₇	C₂×#102	semidirect
98. C₂×C₆₇	C₂²×#102	semidirect
104. C₂⁴×C₈	D₆₇	semidirect
107. C₆₇	C₂³×#102	semidirect
116. C₂³×C₈	C₂×D₆₇	semidirect
137. C₂²×C₈	C₂²×D₆₇	semidirect
163. C₂×C₈	C₂³×D₆₇	semidirect
171. C₂³	#14523965	direct (N × H)
191. C₂²	C₂×#14523965	direct (N × H)
201. C₂	C₂²×#14523965	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₆₈	2
C₂³×C₄⋊Dic₆₇	2
C₂²×#14523965	2
C₂⁴×C₈×C₆₇	2
C₂³×#102	67

Subgroups — C23×#14523965 #31719340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523965 #31719340