Subgroups — C₂²×#16863173 #31008484

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863173	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₆₇	C₄	1	1	yes	characteristic
7	C₂×#14523983	C₄	1	6	yes	complemented
8	C₂×#14523983	C₂²	1	6	yes	—
9	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
10	C₂×#14523953	C₄	1	6	yes	—
11	C₂×#14523953	C₂²	1	6	yes	—
12	#16863173	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₆₇	C₄	1	6	yes	complemented
15	C₂×M₃₂×C₆₇	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
17	C₂³×Dic₆₇	—	2	1	no	—
18	C₂²×C₆₇⋊C₈	—	2	3	no	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₆₇	—	2	3	no	—
22	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
23	#14523983	C₂×C₄	1	12	yes	complemented
24	#14523983	C₂³	1	4	yes	—
25	C₂×C₆₇⋊C₁₆	C₂×C₄	1	12	yes	complemented
26	#14523953	C₂×C₄	1	12	yes	—
27	#14523953	C₂³	1	4	yes	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
35	M₃₂×C₆₇	C₂×C₄	1	12	yes	complemented
36	M₃₂×C₆₇	C₂³	1	4	yes	—
37	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	complemented
38	C₂²×Dic₆₇	—	2	6	no	—
39	C₂²×Dic₆₇	—	4	4	no	—
40	C₂×C₆₇⋊C₈	—	2	6	no	—
41	C₂×C₆₇⋊C₈	—	2	6	no	—
42	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
43	C₄×Dic₆₇	—	2	6	no	—
44	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
45	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
46	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₄²	1	3	yes	—
48	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	M₁₆	1	3	yes	—
50	C₂²×C₄×C₆₇	M₁₆	1	3	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₆₇	C₄²	1	3	yes	—
54	C₂×C₈×C₆₇	C₄²	1	3	yes	—
55	C₂×C₈×C₆₇	C₄²	1	6	yes	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
61	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
63	C₂³×C₆₇	—	2	4	no	—
64	C₂³×C₆₇	#97	1	3	yes	—
65	C₂³×C₆₇	#97	1	1	yes	characteristic
66	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
67	C₂×C₄×C₆₇	#97	1	6	yes	—
68	C₂×C₄×C₆₇	#97	1	3	yes	—
69	C₂×C₄×C₆₇	#97	1	3	yes	—
70	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
71	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
72	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
73	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₆₇	—	2	4	no	—
77	C₂×Dic₆₇	—	4	12	no	—
78	C₈×C₆₇	C₂×C₄²	1	3	yes	—
79	C₈×C₆₇	C₂×C₄²	1	3	yes	—
80	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
82	C₆₇⋊C₈	—	2	4	no	—
83	C₂²×C₆₇	—	2	6	no	—
84	C₂²×C₆₇	—	2	6	no	—
85	C₂²×C₆₇	#347	1	4	yes	complemented
86	C₂²×C₆₇	C₂×#97	1	3	yes	—
87	C₂²×C₆₇	C₂×#97	1	3	yes	—
88	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
89	C₄×C₆₇	C₂×#97	1	3	yes	—
90	C₄×C₆₇	C₂×#97	1	3	yes	—
91	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
92	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
93	Dic₆₇	—	4	4	no	—
94	C₂²×#347	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	—	2	1	no	—
96	C₂×C₆₇	—	2	3	no	—
97	C₂×C₆₇	C₂×#347	1	6	yes	complemented
98	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
99	C₂²×M₃₂	—	67	1	no	—
100	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
101	C₂²×#97	—	67	1	no	—
102	C₂×#347	—	67	12	no	—
103	C₆₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₈	—	67	1	no	—
105	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
106	C₂²×C₄²	—	67	1	no	—
107	C₂×M₃₂	—	67	6	no	—
108	C₂×M₃₂	—	67	6	no	—
109	C₂×M₃₂	Dic₆₇	1	6	yes	complemented
110	C₂×M₃₂	C₂×D₆₇	1	6	yes	—
111	C₂²×C₁₆	—	67	2	no	—
112	C₂²×C₁₆	Dic₆₇	1	2	yes	complemented
113	C₂×#97	—	67	6	no	—
114	C₂×#97	—	67	6	no	—
115	#347	—	67	16	no	—
116	C₂³×C₄	—	134	1	no	—
117	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
118	C₂²×C₈	—	67	6	no	—
119	C₂²×C₈	—	134	1	no	—
120	C₂²×C₈	—	134	3	no	—
121	C₂²×C₈	C₄×D₆₇	1	3	yes	—
122	C₂²×C₈	C₄×D₆₇	1	3	yes	—
123	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
125	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
127	C₂×C₄²	—	67	6	no	—
128	C₂×C₄²	—	134	3	no	—
129	M₃₂	—	67	12	no	—
130	M₃₂	—	67	4	no	—
131	M₃₂	C₂×Dic₆₇	1	12	yes	complemented
132	M₃₂	C₂²×D₆₇	1	4	yes	—
133	C₂×C₁₆	—	67	12	no	—
134	C₂×C₁₆	C₂×Dic₆₇	1	12	yes	complemented
135	#97	—	67	12	no	—
136	#97	—	67	4	no	—
137	C₂⁴	#11761611	1	1	yes	characteristic
138	C₂²×C₄	—	268	4	no	—
139	C₂²×C₄	—	134	6	no	—
140	C₂²×C₄	C₄×Dic₆₇	1	3	yes	—
141	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
142	C₂²×C₄	#11761611	1	3	yes	—
143	C₂²×C₄	#11761611	1	1	yes	characteristic
144	C₂²×C₄	#11761611	1	3	yes	—
145	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
146	C₂×C₈	—	67	4	no	—
147	C₂×C₈	—	134	6	no	—
148	C₂×C₈	—	134	6	no	—
149	C₂×C₈	C₄×Dic₆₇	1	6	yes	—
150	C₂×C₈	C₄×Dic₆₇	1	3	yes	—
151	C₂×C₈	C₄×Dic₆₇	1	3	yes	—
152	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
153	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
154	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
155	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
156	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
157	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
158	C₄²	—	67	4	no	—
159	C₄²	—	134	6	no	—
160	C₁₆	—	67	8	no	—
161	C₁₆	C₂²×Dic₆₇	1	8	yes	complemented
162	C₂³	—	2	4	no	—
163	C₂³	#14523953	1	3	yes	—
164	C₂³	#14523953	1	1	yes	characteristic
165	C₂³	C₂×#11761611	1	3	yes	—
166	C₂×C₄	—	268	12	no	—
167	C₂×C₄	—	134	4	no	—
168	C₂×C₄	#14523953	1	6	yes	—
169	C₂×C₄	#14523953	1	3	yes	—
170	C₂×C₄	#14523953	1	3	yes	—
171	C₂×C₄	C₂×C₄×Dic₆₇	1	3	yes	—
172	C₂×C₄	C₂×C₄×Dic₆₇	1	3	yes	—
173	C₂×C₄	C₂×#11761611	1	3	yes	—
174	C₂×C₄	C₂×#11761611	1	3	yes	—
175	C₂×C₄	C₂×#11761611	1	3	yes	—
176	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
177	C₈	—	134	4	no	—
178	C₈	C₂×C₄×Dic₆₇	1	3	yes	—
179	C₈	C₂×C₄×Dic₆₇	1	3	yes	—
180	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
181	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#16863173	1	4	yes	direct product factor
185	C₂²	C₂×#14523953	1	3	yes	—
186	C₂²	C₂×#14523953	1	3	yes	—
187	C₂²	C₂²×#11761611	1	1	yes	characteristic
188	C₄	—	268	4	no	—
189	C₄	C₂×#14523953	1	3	yes	—
190	C₄	C₂×#14523953	1	3	yes	—
191	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
192	C₄	C₂²×#11761611	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#16863173	1	6	yes	direct product factor
196	C₂	C₂²×#14523953	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂³×C₆₇	#97
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂²×#347
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#16863173	C₂	direct (N × H)
7. C₂×#14523983	C₄	semidirect
9. C₂²×C₆₇⋊C₁₆	C₄	semidirect
12. #16863173	C₂²	direct (N × H)
14. C₂×M₃₂×C₆₇	C₄	semidirect
16. C₂²×C₁₆×C₆₇	C₄	semidirect
23. #14523983	C₂×C₄	semidirect
25. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
35. M₃₂×C₆₇	C₂×C₄	semidirect
37. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
45. C₆₇⋊C₁₆	C₂²×C₄	semidirect
62. C₁₆×C₆₇	C₂²×C₄	semidirect
85. C₂²×C₆₇	#347	semidirect
97. C₂×C₆₇	C₂×#347	semidirect
103. C₆₇	C₂²×#347	semidirect
109. C₂×M₃₂	Dic₆₇	semidirect
112. C₂²×C₁₆	Dic₆₇	semidirect
131. M₃₂	C₂×Dic₆₇	semidirect
134. C₂×C₁₆	C₂×Dic₆₇	semidirect
161. C₁₆	C₂²×Dic₆₇	semidirect
184. C₂²	#16863173	direct (N × H)
195. C₂	C₂×#16863173	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×#14523983	2
C₂²×#14523953	2
C₂×#16863173	2
C₂²×M₃₂×C₆₇	2
C₂²×#347	67

Subgroups — C22×#16863173 #31008484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863173 #31008484