Subgroups — C₂²×C₄×#11761612 #31719219

Order 17152.

3950 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761612	C₂	1	2	yes	maximal
2	C₂×C₄×#11761612	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523954	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₄	1	32	yes	direct product factor
7	C₂²×#11761612	C₂²	1	24	yes	—
8	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
9	C₄×#11761612	C₂²	1	64	yes	direct product factor
10	C₂×#14523954	C₂²	1	24	yes	complemented
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	24	yes	complemented
12	C₂⁴×C₄×C₆₇	C₄	1	2	yes	—
13	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₄	1	12	yes	—
15	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	12	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	C₂×C₄	1	192	yes	direct product factor
20	C₂×#11761612	C₂³	1	32	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	32	yes	complemented
22	C₂²×C₆₇⋊C₈	C₂³	1	24	yes	—
23	#14523954	C₂³	1	32	yes	complemented
24	C₄×C₆₇⋊C₈	C₂³	1	32	yes	complemented
25	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	—	2	16	no	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	16	yes	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	24	yes	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	48	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
42	#11761612	C₂²×C₄	1	128	yes	direct product factor
43	C₂×C₆₇⋊C₈	C₂²×C₄	1	96	yes	complemented
44	C₂×C₆₇⋊C₈	C₂⁴	1	16	yes	—
45	C₂⁴×C₆₇	—	2	8	no	—
46	C₂⁴×C₆₇	C₄²	1	8	yes	—
47	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
48	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	—	2	96	no	—
51	C₂²×C₄×C₆₇	C₄²	1	48	yes	—
52	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
53	C₂²×C₄×C₆₇	M₁₆	1	16	yes	complemented
54	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	48	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	32	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
64	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	32	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
67	C₄²×C₆₇	C₂⁴	1	4	yes	—
68	C₄²×C₆₇	C₂⁴	1	4	yes	—
69	C₂³×C₆₇	—	2	24	no	—
70	C₂³×C₆₇	—	2	32	no	—
71	C₂³×C₆₇	C₂×C₄²	1	24	yes	—
72	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
73	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
74	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
76	C₂×C₄×C₆₇	—	2	24	no	—
77	C₂×C₄×C₆₇	—	2	64	no	—
78	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
79	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
80	C₂×C₄×C₆₇	C₂×C₄²	1	24	yes	—
81	C₂×C₄×C₆₇	C₂×M₁₆	1	48	yes	complemented
82	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
84	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
85	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
87	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
89	C₆₇⋊C₈	C₂³×C₄	1	32	yes	complemented
90	C₂²×C₆₇	—	2	48	no	—
91	C₂²×C₆₇	—	2	8	no	—
92	C₂²×C₆₇	C₄×M₁₆	1	16	yes	complemented
93	C₂²×C₆₇	C₂²×C₄²	1	8	yes	—
94	C₂²×C₆₇	C₂²×C₄²	1	6	yes	—
95	C₂²×C₆₇	C₂²×M₁₆	1	12	yes	—
96	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
97	C₄×C₆₇	—	2	8	no	—
98	C₄×C₆₇	C₂²×C₄²	1	6	yes	—
99	C₄×C₆₇	C₂²×C₄²	1	8	yes	—
100	C₄×C₆₇	C₂²×M₁₆	1	16	yes	complemented
101	C₄×C₆₇	C₂⁴×C₄	1	2	yes	—
102	C₂²×C₄×M₁₆	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	8	no	—
104	C₂×C₆₇	C₂×C₄×M₁₆	1	12	yes	complemented
105	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
106	C₂×C₆₇	C₂³×M₁₆	1	2	yes	—
107	C₂³×M₁₆	—	67	2	no	—
108	C₂³×C₄²	D₆₇	1	1	yes	characteristic
109	C₂×C₄×M₁₆	—	67	24	no	—
110	C₂²×#97	—	67	2	no	—
111	C₂²×C₄×C₈	—	67	2	no	—
112	C₆₇	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂⁴×C₄	Dic₆₇	1	2	yes	—
114	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂²×M₁₆	—	67	32	no	—
116	C₂²×M₁₆	—	67	24	no	—
117	C₂³×C₈	—	67	4	no	—
118	C₂²×C₄²	Dic₆₇	1	2	yes	—
119	C₂²×C₄²	Dic₆₇	1	12	yes	—
120	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
121	C₂²×C₄²	C₂×D₆₇	1	12	yes	—
122	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
123	C₄×M₁₆	—	67	64	no	—
124	C₂×#97	—	67	24	no	—
125	C₂×C₄×C₈	—	67	24	no	—
126	C₂⁵	C₂×Dic₆₇	1	1	yes	characteristic
127	C₂³×C₄	—	2	16	no	—
128	C₂³×C₄	C₄×D₆₇	1	16	yes	—
129	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
130	C₂³×C₄	C₂×Dic₆₇	1	24	yes	—
131	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
132	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
134	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
136	C₂×M₁₆	—	67	192	no	—
137	C₂×M₁₆	—	67	32	no	—
138	C₂²×C₈	—	67	32	no	—
139	C₂²×C₈	—	67	24	no	—
140	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
141	C₂×C₄²	C₂×Dic₆₇	1	48	yes	—
142	C₂×C₄²	C₂×Dic₆₇	1	24	yes	—
143	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
144	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
145	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
146	C₂×C₄²	C₂²×D₆₇	1	16	yes	—
147	#97	—	67	32	no	—
148	C₄×C₈	—	67	32	no	—
149	C₂⁴	—	2	8	no	—
150	C₂⁴	C₄×Dic₆₇	1	8	yes	—
151	C₂⁴	C₂²×Dic₆₇	1	6	yes	—
152	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	96	no	—
155	C₂²×C₄	C₄×Dic₆₇	1	48	yes	—
156	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
157	C₂²×C₄	#11761612	1	16	yes	direct product factor
158	C₂²×C₄	C₂×C₄×D₆₇	1	8	yes	—
159	C₂²×C₄	C₂×C₄×D₆₇	1	48	yes	—
160	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
161	C₂²×C₄	C₂²×Dic₆₇	1	32	yes	—
162	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
163	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
164	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
165	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
166	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
167	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
168	M₁₆	—	67	128	no	—
169	C₂×C₈	—	67	96	no	—
170	C₂×C₈	—	67	16	no	—
171	C₄²	C₂²×Dic₆₇	1	12	yes	—
172	C₄²	C₂²×Dic₆₇	1	32	yes	—
173	C₄²	C₂²×Dic₆₇	1	12	yes	—
174	C₄²	C₂³×D₆₇	1	4	yes	—
175	C₄²	C₂³×D₆₇	1	4	yes	—
176	C₂³	—	2	32	no	—
177	C₂³	—	2	24	no	—
178	C₂³	C₂×C₄×Dic₆₇	1	24	yes	—
179	C₂³	C₂×C₄×Dic₆₇	1	4	yes	—
180	C₂³	C₂×#11761612	1	8	yes	—
181	C₂³	C₂³×Dic₆₇	1	4	yes	—
182	C₂³	C₂³×Dic₆₇	1	3	yes	—
183	C₂×C₄	—	2	24	no	—
184	C₂×C₄	—	2	64	no	—
185	C₂×C₄	C₂×C₄×Dic₆₇	1	48	yes	—
186	C₂×C₄	C₂×C₄×Dic₆₇	1	12	yes	—
187	C₂×C₄	C₂×C₄×Dic₆₇	1	24	yes	—
188	C₂×C₄	C₂×#11761612	1	48	yes	direct product factor
189	C₂×C₄	C₂²×C₄×D₆₇	1	12	yes	—
190	C₂×C₄	C₂²×C₄×D₆₇	1	16	yes	—
191	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
192	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
193	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
194	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
195	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
196	C₈	—	67	32	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	48	no	—
199	C₂²	C₄×#11761612	1	16	yes	direct product factor
200	C₂²	C₂²×C₄×Dic₆₇	1	8	yes	—
201	C₂²	C₂²×C₄×Dic₆₇	1	6	yes	—
202	C₂²	C₂²×#11761612	1	12	yes	—
203	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	C₂²×C₄×Dic₆₇	1	6	yes	—
206	C₄	C₂²×C₄×Dic₆₇	1	8	yes	—
207	C₄	C₂²×#11761612	1	16	yes	direct product factor
208	C₄	C₂³×C₄×D₆₇	1	2	yes	—
209	C₂	—	2	8	no	—
210	C₂	C₂×C₄×#11761612	1	12	yes	direct product factor
211	C₂	C₂³×C₄×Dic₆₇	1	1	yes	characteristic
212	C₂	C₂³×#11761612	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₆₇
commutator	C₂×C₆₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×C₄²×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×C₄²	D₆₇
hypocenter	C₆₇	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11761612	C₂	direct (N × H)
3. C₂²×#14523954	C₂	semidirect
4. C₂²×C₄×C₆₇⋊C₈	C₂	semidirect
6. C₂²×#11761612	C₄	direct (N × H)
9. C₄×#11761612	C₂²	direct (N × H)
10. C₂×#14523954	C₂²	semidirect
11. C₂×C₄×C₆₇⋊C₈	C₂²	semidirect
19. C₂×#11761612	C₂×C₄	direct (N × H)
21. C₂²×C₆₇⋊C₈	C₂×C₄	semidirect
23. #14523954	C₂³	semidirect
24. C₄×C₆₇⋊C₈	C₂³	semidirect
42. #11761612	C₂²×C₄	direct (N × H)
43. C₂×C₆₇⋊C₈	C₂²×C₄	semidirect
53. C₂²×C₄×C₆₇	M₁₆	semidirect
81. C₂×C₄×C₆₇	C₂×M₁₆	semidirect
89. C₆₇⋊C₈	C₂³×C₄	semidirect
92. C₂²×C₆₇	C₄×M₁₆	semidirect
100. C₄×C₆₇	C₂²×M₁₆	semidirect
104. C₂×C₆₇	C₂×C₄×M₁₆	semidirect
112. C₆₇	C₂²×C₄×M₁₆	semidirect
157. C₂²×C₄	#11761612	direct (N × H)
188. C₂×C₄	C₂×#11761612	direct (N × H)
199. C₂²	C₄×#11761612	direct (N × H)
207. C₄	C₂²×#11761612	direct (N × H)
210. C₂	C₂×C₄×#11761612	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×#11761612	2
C₂×C₄×#11761612	2
C₂²×#14523954	2
C₂²×C₄×C₆₇⋊C₈	2
C₂³×C₄²×C₆₇	2
C₂²×C₄×M₁₆	67

Subgroups — C22×C4×#11761612 #31719219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11761612 #31719219