Subgroups — C₂²×#16863800 #31725428

Order 17152.

2398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524042	C₂	1	2	yes	maximal
2	C₂×#16863800	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14523976	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₄	—	2	2	no	—
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524042	—	2	12	no	—
10	C₂×#14524042	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂²×Dic₆₇⋊C₄	—	2	2	no	—
13	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
14	#16863800	C₂²	1	64	yes	direct product factor
15	C₂×#14523977	C₂²	1	12	yes	—
16	C₂×#14523976	C₂²	1	24	yes	—
17	C₂×C₈×Dic₆₇	C₂²	1	12	yes	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂×#107×C₆₇	C₂²	1	12	yes	—
21	C₂²×Dic₁₃₄	—	2	24	no	—
22	C₂²×Dic₁₃₄	—	4	16	no	—
23	C₂³×Dic₆₇	—	2	2	no	—
24	C₂³×Dic₆₇	D₄	1	1	yes	characteristic
25	C₂³×Dic₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	complemented
27	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
28	#14524042	—	2	48	no	—
29	#14524042	C₂³	1	32	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	6	no	—
31	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
32	C₂×Dic₆₇⋊C₄	—	2	24	no	—
33	C₂×C₄×Dic₆₇	D₄	1	6	yes	—
34	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
35	#14523977	C₂³	1	16	yes	—
36	#14523976	C₂³	1	32	yes	—
37	C₈×Dic₆₇	C₂³	1	16	yes	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	Q₈	1	8	yes	complemented
41	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
42	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
44	#107×C₆₇	C₂³	1	16	yes	—
45	C₂×Dic₁₃₄	—	2	32	no	—
46	C₂×Dic₁₃₄	—	4	96	no	—
47	C₂²×Dic₆₇	—	2	12	no	—
48	C₂²×Dic₆₇	—	4	8	no	—
49	C₂²×Dic₆₇	D₈	1	8	yes	—
50	C₂²×Dic₆₇	Q₁₆	1	8	yes	—
51	C₂²×Dic₆₇	C₂×D₄	1	6	yes	—
52	C₂²×Dic₆₇	C₂×D₄	1	6	yes	—
53	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	complemented
54	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
55	C₄⋊Dic₆₇	—	2	12	no	—
56	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
57	Dic₆₇⋊C₄	—	2	32	no	—
58	C₄×Dic₆₇	C₂×D₄	1	12	yes	—
59	C₄×Dic₆₇	C₂⁴	1	4	yes	—
60	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	12	no	—
62	C₂²×C₄×C₆₇	—	4	8	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
67	C₂×C₈×C₆₇	C₂×Q₈	1	24	yes	complemented
68	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
69	C₄⋊C₄×C₆₇	—	2	12	no	—
70	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
71	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₆₇	C₂×D₈	1	4	yes	—
73	C₂³×C₆₇	C₂×Q₁₆	1	4	yes	—
74	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₆₇	—	2	8	no	—
76	C₂×C₄×C₆₇	—	4	24	no	—
77	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
78	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
79	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
80	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
81	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
82	C₂×Dic₆₇	—	2	8	no	—
83	C₂×Dic₆₇	—	4	24	no	—
84	C₂×Dic₆₇	C₂×D₈	1	24	yes	—
85	C₂×Dic₆₇	C₂×Q₁₆	1	24	yes	—
86	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
87	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
88	Dic₁₃₄	—	4	64	no	—
89	C₈×C₆₇	C₂²×Q₈	1	8	yes	complemented
90	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	complemented
91	C₂²×C₆₇	#500	1	16	yes	complemented
92	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	6	yes	—
94	C₂²×C₆₇	C₂²×Q₁₆	1	6	yes	—
95	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
96	C₄×C₆₇	—	4	8	no	—
97	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
98	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
99	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
100	Dic₆₇	—	4	8	no	—
101	Dic₆₇	C₂²×D₈	1	8	yes	—
102	Dic₆₇	C₂²×Q₁₆	1	8	yes	—
103	C₂²×#500	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	C₂×#500	1	12	yes	complemented
105	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
107	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
108	C₂²×C₄⋊Q₈	—	67	2	no	—
109	C₂×#500	—	67	24	no	—
110	C₂²×#107	—	67	1	no	—
111	C₂²×#107	—	67	2	no	—
112	C₂²×#107	D₆₇	1	1	yes	characteristic
113	C₂²×C₄×C₈	—	67	1	no	—
114	C₆₇	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
115	C₂³×Q₈	—	134	2	no	—
116	C₂³×C₈	—	67	1	no	—
117	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	67	24	no	—
119	C₂×C₄⋊Q₈	—	134	12	no	—
120	C₂²×C₄⋊C₄	—	67	2	no	—
121	C₂²×C₄⋊C₄	—	134	2	no	—
122	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
123	C₂²×C₄²	—	67	1	no	—
124	#500	—	67	64	no	—
125	C₂×#107	—	67	12	no	—
126	C₂×#107	—	67	24	no	—
127	C₂×#107	C₂×D₆₇	1	12	yes	—
128	C₂×C₄×C₈	—	67	12	no	—
129	C₂²×Q₈	—	268	16	no	—
130	C₂²×Q₈	—	134	24	no	—
131	C₂³×C₄	—	2	2	no	—
132	C₂³×C₄	—	67	1	no	—
133	C₂³×C₄	—	67	1	no	—
134	C₂³×C₄	—	134	2	no	—
135	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
136	C₂²×C₈	—	67	8	no	—
137	C₂²×C₈	—	67	6	no	—
138	C₂²×C₈	Dic₁₃₄	1	8	yes	complemented
139	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
140	C₄⋊Q₈	—	67	32	no	—
141	C₄⋊Q₈	—	134	48	no	—
142	C₂×C₄⋊C₄	—	2	6	no	—
143	C₂×C₄⋊C₄	—	67	12	no	—
144	C₂×C₄⋊C₄	—	134	24	no	—
145	C₂×C₄⋊C₄	—	134	6	no	—
146	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
147	C₂×C₄²	—	67	6	no	—
148	C₂×C₄²	—	67	6	no	—
149	#107	—	67	16	no	—
150	#107	—	67	32	no	—
151	#107	C₂²×D₆₇	1	16	yes	—
152	C₄×C₈	—	67	16	no	—
153	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
154	C₂×Q₈	—	268	96	no	—
155	C₂×Q₈	—	134	32	no	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	4	8	no	—
158	C₂²×C₄	—	268	8	no	—
159	C₂²×C₄	—	67	8	no	—
160	C₂²×C₄	—	67	8	no	—
161	C₂²×C₄	—	67	6	no	—
162	C₂²×C₄	—	67	6	no	—
163	C₂²×C₄	—	134	12	no	—
164	C₂²×C₄	D₄×D₆₇	1	3	yes	—
165	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
166	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
167	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
168	C₂×C₈	—	67	24	no	—
169	C₂×C₈	—	67	4	no	—
170	C₂×C₈	C₂×Dic₁₃₄	1	24	yes	complemented
171	C₂×C₈	C₂³×D₆₇	1	4	yes	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	—	67	8	no	—
174	C₄⋊C₄	—	134	32	no	—
175	C₄⋊C₄	—	134	12	no	—
176	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
177	C₄²	—	67	12	no	—
178	C₄²	—	67	4	no	—
179	C₂³	#14524042	1	4	yes	—
180	C₂³	D₈×D₆₇	1	4	yes	—
181	C₂³	Q₁₆×D₆₇	1	4	yes	—
182	C₂³	C₂×D₄×D₆₇	1	3	yes	—
183	Q₈	—	268	64	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	4	24	no	—
186	C₂×C₄	—	268	24	no	—
187	C₂×C₄	—	67	24	no	—
188	C₂×C₄	—	67	24	no	—
189	C₂×C₄	—	67	4	no	—
190	C₂×C₄	—	67	4	no	—
191	C₂×C₄	—	134	8	no	—
192	C₂×C₄	#14524042	1	12	yes	—
193	C₂×C₄	C₂×D₄×D₆₇	1	3	yes	—
194	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
195	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
196	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
197	C₈	—	67	8	no	—
198	C₈	C₂²×Dic₁₃₄	1	8	yes	complemented
199	C₂²	#16863800	1	16	yes	direct product factor
200	C₂²	C₂×#14524042	1	6	yes	—
201	C₂²	C₂×D₈×D₆₇	1	6	yes	—
202	C₂²	C₂×Q₁₆×D₆₇	1	6	yes	—
203	C₂²	C₂²×D₄×D₆₇	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	268	8	no	—
206	C₄	—	67	8	no	—
207	C₄	—	67	8	no	—
208	C₄	C₂×#14524042	1	6	yes	—
209	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
211	C₂	C₂×#16863800	1	12	yes	direct product factor
212	C₂	C₂²×#14524042	1	1	yes	characteristic
213	C₂	C₂²×D₈×D₆₇	1	1	yes	characteristic
214	C₂	C₂²×Q₁₆×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	C₂²×#500
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#16863800	C₂	direct (N × H)
14. #16863800	C₂²	direct (N × H)
26. C₂²×C₆₇⋊C₈	Q₈	semidirect
40. C₂²×C₈×C₆₇	Q₈	semidirect
53. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
67. C₂×C₈×C₆₇	C₂×Q₈	semidirect
89. C₈×C₆₇	C₂²×Q₈	semidirect
90. C₆₇⋊C₈	C₂²×Q₈	semidirect
91. C₂²×C₆₇	#500	semidirect
104. C₂×C₆₇	C₂×#500	semidirect
114. C₆₇	C₂²×#500	semidirect
138. C₂²×C₈	Dic₁₃₄	semidirect
170. C₂×C₈	C₂×Dic₁₃₄	semidirect
198. C₈	C₂²×Dic₁₃₄	semidirect
199. C₂²	#16863800	direct (N × H)
211. C₂	C₂×#16863800	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14524042	2
C₂×#16863800	2
C₂²×#14523977	2
C₂²×#14523976	2
C₂²×C₈×Dic₆₇	2
C₂²×#107×C₆₇	2
C₂²×#500	67

Subgroups — C22×#16863800 #31725428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863800 #31725428