Subgroups — C₂²×#16863424 #31719448

Order 17152.

3006 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	C₂⁴×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14524002	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863424	C₂	1	24	yes	maximal, direct product factor
4	C₂²×Dic₆₇⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	—	2	8	no	—
7	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
8	C₂³×C₆₇⋊C₈	C₂²	1	12	yes	—
9	C₂×#14524002	C₄	1	12	yes	—
10	C₂×#14524002	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₆₇	C₄	1	2	yes	—
12	C₂²×C₂²⋊Dic₆₇	C₄	1	2	yes	—
13	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
14	#16863424	C₂²	1	64	yes	direct product factor
15	C₂×Dic₆₇⋊C₈	C₂²	1	48	yes	complemented
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₆₇	—	2	8	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	12	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
21	C₂²×C₆₇⋊C₈	—	2	8	no	—
22	C₂²×C₆₇⋊C₈	—	2	48	no	—
23	C₂²×C₆₇⋊C₈	C₂³	1	12	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	16	yes	—
25	#14524002	C₂×C₄	1	48	yes	—
26	#14524002	C₂³	1	16	yes	—
27	C₂×C₄⋊Dic₆₇	C₂×C₄	1	24	yes	—
28	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	24	yes	—
29	C₂×C₄×Dic₆₇	C₂×C₄	1	12	yes	—
30	C₂×C₄×Dic₆₇	C₂³	1	12	yes	—
31	Dic₆₇⋊C₈	C₂³	1	64	yes	complemented
32	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
35	C₂³×C₄×C₆₇	D₄	1	8	yes	—
36	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₆₇	—	2	48	no	—
40	C₂×M₁₆×C₆₇	C₂³	1	16	yes	—
41	C₂²×C₈×C₆₇	—	2	8	no	—
42	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
43	C₂²×Dic₆₇	—	2	16	no	—
44	C₂²×Dic₆₇	C₂²×C₄	1	24	yes	—
45	C₂×C₆₇⋊C₈	—	2	24	no	—
46	C₂×C₆₇⋊C₈	—	2	32	no	—
47	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
48	C₄⋊Dic₆₇	C₂²×C₄	1	32	yes	—
49	C₂²⋊Dic₆₇	C₂²×C₄	1	32	yes	—
50	C₄×Dic₆₇	C₂²×C₄	1	24	yes	—
51	C₄×Dic₆₇	C₂⁴	1	8	yes	—
52	C₂⁴×C₆₇	—	2	8	no	—
53	C₂⁴×C₆₇	C₄⋊C₄	1	8	yes	—
54	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
55	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	48	yes	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
63	C₂²×C₄×C₆₇	C₂×Q₈	1	24	yes	—
64	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
67	M₁₆×C₆₇	—	2	32	no	—
68	C₂×C₈×C₆₇	—	2	24	no	—
69	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
70	C₂³×C₆₇	—	2	24	no	—
71	C₂³×C₆₇	—	2	32	no	—
72	C₂³×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
73	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂³×C₆₇	#131	1	8	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
76	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	48	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
79	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
82	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
83	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
84	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
85	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
86	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₆₇	—	2	48	no	—
88	C₂×Dic₆₇	C₂³×C₄	1	16	yes	—
89	C₈×C₆₇	—	2	8	no	—
90	C₆₇⋊C₈	—	2	8	no	—
91	C₂²×C₆₇	—	2	48	no	—
92	C₂²×C₆₇	—	2	8	no	—
93	C₂²×C₆₇	#424	1	16	yes	complemented
94	C₂²×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
95	C₂²×C₆₇	C₂²×C₄⋊C₄	1	6	yes	—
96	C₂²×C₆₇	C₂×#131	1	12	yes	—
97	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
98	C₄×C₆₇	C₂²×C₄⋊C₄	1	6	yes	—
99	C₄×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
100	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
102	Dic₆₇	—	2	16	no	—
103	C₂²×#424	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	—	2	8	no	—
105	C₂×C₆₇	C₂×#424	1	12	yes	complemented
106	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₆₇	C₂²×#131	1	2	yes	—
108	C₂³×M₁₆	D₆₇	1	1	yes	characteristic
109	C₂⁴×C₈	—	67	1	no	—
110	C₂²×#117	—	67	1	no	—
111	C₂×#424	—	67	24	no	—
112	C₂²×C₄⋊C₈	—	67	4	no	—
113	C₆₇	C₂²×#424	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂²×M₁₆	—	2	8	no	—
116	C₂²×M₁₆	C₂×D₆₇	1	12	yes	—
117	C₂³×C₈	—	67	12	no	—
118	C₂³×C₈	—	67	2	no	—
119	C₂³×C₈	—	134	8	no	—
120	C₂³×C₈	C₂×D₆₇	1	2	yes	—
121	C₂×#117	—	67	12	no	—
122	C₂×#117	—	67	12	no	—
123	C₂²×C₄⋊C₄	—	67	2	no	—
124	C₂²×C₂²⋊C₄	—	67	2	no	—
125	C₂²×C₄²	—	67	2	no	—
126	#424	—	67	64	no	—
127	C₂×C₄⋊C₈	—	67	48	no	—
128	C₂⁵	C₄×D₆₇	1	1	yes	characteristic
129	C₂³×C₄	—	67	4	no	—
130	C₂³×C₄	Dic₁₃₄	1	8	yes	—
131	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
132	C₂³×C₄	C₄×D₆₇	1	6	yes	—
133	C₂³×C₄	C₂²⋊D₆₇	1	8	yes	—
134	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
135	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
136	C₂×M₁₆	—	2	48	no	—
137	C₂×M₁₆	C₂²×D₆₇	1	16	yes	—
138	C₂²×C₈	—	2	8	no	—
139	C₂²×C₈	—	67	12	no	—
140	C₂²×C₈	—	67	16	no	—
141	C₂²×C₈	—	134	8	no	—
142	C₂²×C₈	—	134	48	no	—
143	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
144	#117	—	67	48	no	—
145	#117	—	67	16	no	—
146	C₂×C₄⋊C₄	—	67	24	no	—
147	C₂×C₂²⋊C₄	—	67	24	no	—
148	C₂×C₄²	—	67	12	no	—
149	C₂×C₄²	—	67	12	no	—
150	C₄⋊C₈	—	67	64	no	—
151	C₂⁴	—	2	8	no	—
152	C₂⁴	Dic₆₇⋊C₄	1	8	yes	—
153	C₂⁴	C₂×C₄×D₆₇	1	6	yes	—
154	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₄	—	67	24	no	—
156	C₂²×C₄	—	134	16	no	—
157	C₂²×C₄	Dic₆₇⋊C₄	1	48	yes	—
158	C₂²×C₄	Dic₆₇⋊C₄	1	8	yes	—
159	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
160	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
161	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	—
164	C₂²×C₄	C₂×Dic₁₃₄	1	24	yes	—
165	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
166	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
167	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
168	M₁₆	—	2	32	no	—
169	C₂×C₈	—	2	24	no	—
170	C₂×C₈	—	67	8	no	—
171	C₂×C₈	—	134	24	no	—
172	C₂×C₈	—	134	32	no	—
173	C₂×C₈	C₂³×D₆₇	1	8	yes	—
174	C₄⋊C₄	—	67	32	no	—
175	C₂²⋊C₄	—	67	32	no	—
176	C₄²	—	67	24	no	—
177	C₄²	—	67	8	no	—
178	C₂³	—	2	24	no	—
179	C₂³	—	2	32	no	—
180	C₂³	C₂×Dic₆₇⋊C₄	1	24	yes	—
181	C₂³	C₂×Dic₆₇⋊C₄	1	4	yes	—
182	C₂³	#14524051	1	8	yes	—
183	C₂³	C₂²×C₄×D₆₇	1	4	yes	—
184	C₂³	C₂²×C₄×D₆₇	1	3	yes	—
185	C₂×C₄	—	67	16	no	—
186	C₂×C₄	—	134	48	no	—
187	C₂×C₄	C₂×Dic₆₇⋊C₄	1	48	yes	—
188	C₂×C₄	C₂×Dic₆₇⋊C₄	1	12	yes	—
189	C₂×C₄	C₂×Dic₆₇⋊C₄	1	24	yes	—
190	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
191	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₆₇	1	6	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₆₇	1	8	yes	—
194	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
195	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
196	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
197	C₈	—	2	8	no	—
198	C₈	—	134	8	no	—
199	C₂²	—	2	48	no	—
200	C₂²	—	2	8	no	—
201	C₂²	#16863424	1	16	yes	direct product factor
202	C₂²	C₂²×Dic₆₇⋊C₄	1	8	yes	—
203	C₂²	C₂²×Dic₆₇⋊C₄	1	6	yes	—
204	C₂²	C₂×#14524051	1	12	yes	—
205	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
206	C₄	—	134	16	no	—
207	C₄	C₂²×Dic₆₇⋊C₄	1	6	yes	—
208	C₄	C₂²×Dic₆₇⋊C₄	1	8	yes	—
209	C₄	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#16863424	1	12	yes	direct product factor
213	C₂	C₂³×Dic₆₇⋊C₄	1	1	yes	characteristic
214	C₂	C₂²×#14524051	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×M₁₆×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×M₁₆	D₆₇
hypocenter	C₆₇	C₂²×#424
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16863424	C₂	direct (N × H)
4. C₂²×Dic₆₇⋊C₈	C₂	semidirect
14. #16863424	C₂²	direct (N × H)
15. C₂×Dic₆₇⋊C₈	C₂²	semidirect
31. Dic₆₇⋊C₈	C₂³	semidirect
93. C₂²×C₆₇	#424	semidirect
105. C₂×C₆₇	C₂×#424	semidirect
113. C₆₇	C₂²×#424	semidirect
201. C₂²	#16863424	direct (N × H)
212. C₂	C₂×#16863424	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×C₆₇⋊C₈	2
C₂²×#14524002	2
C₂×#16863424	2
C₂²×Dic₆₇⋊C₈	2
C₂³×M₁₆×C₆₇	2
C₂²×#424	67

Subgroups — C22×#16863424 #31719448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863424 #31719448