Subgroups — C₄×#16863149 #30747016

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	2	yes	maximal
3	#22558044	C₂	1	2	yes	maximal
4	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
7	#16863151	—	2	2	no	—
8	#16863149	C₄	1	8	yes	direct product factor
9	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	—	2	4	no	—
15	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
16	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
17	C₄×C₆₇⋊C₈	—	4	4	no	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₆₇	—	2	4	no	—
27	C₄×C₈×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	C₈	1	8	yes	complemented
29	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₆₇⋊C₈	—	2	8	no	—
31	C₂×C₆₇⋊C₈	—	4	4	no	—
32	C₂×C₆₇⋊C₈	—	4	4	no	—
33	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	C₄²	1	2	yes	—
42	C₂×C₈×C₆₇	C₄²	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
45	C₄²×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	—	2	2	no	—
47	C₄²×C₆₇	C₂×C₈	1	4	yes	—
48	C₄²×C₆₇	M₁₆	1	4	yes	—
49	C₄²×C₆₇	D₈	1	2	yes	—
50	C₄²×C₆₇	Q₁₆	1	2	yes	—
51	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
52	C₄²×C₆₇	#41	1	2	yes	—
53	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₆₇	#107	1	4	yes	—
60	C₂×C₄×C₆₇	#108	1	4	yes	—
61	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#135	1	2	yes	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	—	2	4	no	—
72	C₈×C₆₇	C₄×C₈	1	4	yes	complemented
73	C₆₇⋊C₈	—	4	8	no	—
74	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
76	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
77	C₂²×C₆₇	#427	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#108	1	2	yes	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	#335	1	8	yes	complemented
82	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
83	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
84	C₄×C₆₇	#431	1	2	yes	—
85	C₄×C₆₇	C₄×D₈	1	2	yes	—
86	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
87	C₄×#335	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	C₂×#335	1	2	yes	—
89	C₂×C₆₇	#1564	1	2	yes	—
90	C₂×C₆₇	C₄×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₆₇	C₄×#107	1	1	yes	characteristic
92	C₂×C₆₇	C₄×#108	1	1	yes	characteristic
93	C₄×C₄⋊C₈	—	67	2	no	—
94	C₄²×C₈	D₆₇	1	1	yes	characteristic
95	C₂×#335	—	67	2	no	—
96	#1277	—	67	2	no	—
97	C₆₇	C₄×#335	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊C₈	—	67	4	no	—
99	C₂×C₄×C₈	—	134	2	no	—
100	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
101	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
102	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
104	C₄³	C₂×D₆₇	1	1	yes	characteristic
105	#336	—	134	2	no	—
106	#335	—	67	8	no	—
107	C₂²×C₈	—	134	4	no	—
108	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
110	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
111	C₂×C₄²	Dic₁₃₄	1	2	yes	—
112	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
113	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
115	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	C₄⋊C₈	—	67	8	no	—
117	C₄⋊C₈	—	134	4	no	—
118	C₄×C₈	—	2	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	268	4	no	—
121	C₄×C₈	C₆₇⋊C₈	1	8	yes	complemented
122	C₄×C₈	C₄×D₆₇	1	4	yes	—
123	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
127	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	268	4	no	—
132	C₂×C₈	—	268	4	no	—
133	C₂×C₈	—	134	8	no	—
134	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
135	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
136	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
137	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
141	C₄²	#11761612	1	4	yes	—
142	C₄²	D₅₃₆	1	2	yes	—
143	C₄²	Dic₂₆₈	1	2	yes	—
144	C₄²	C₂×C₄×D₆₇	1	2	yes	—
145	C₄²	#11761628	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
151	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
152	C₂×C₄	#14523977	1	4	yes	—
153	C₂×C₄	#14523979	1	4	yes	—
154	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
155	C₂×C₄	#14524002	1	1	yes	characteristic
156	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
157	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
158	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
159	C₂×C₄	C₂×#11761612	1	2	yes	—
160	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
161	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
162	C₂×C₄	#14524065	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	268	8	no	—
166	C₈	C₄×C₆₇⋊C₈	1	4	yes	complemented
167	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
168	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
169	C₂²	C₂×#14523977	1	1	yes	characteristic
170	C₂²	#16863434	1	1	yes	characteristic
171	C₂²	C₂×#14523979	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	#16863149	1	8	yes	direct product factor
175	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
176	C₄	C₄×#11761612	1	1	yes	characteristic
177	C₄	#16863449	1	2	yes	—
178	C₄	C₄×D₅₃₆	1	2	yes	—
179	C₄	C₄×Dic₂₆₈	1	2	yes	—
180	C₂	C₂×#16863149	1	2	yes	—
181	C₂	#22558957	1	2	yes	—
182	C₂	C₄×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
183	C₂	C₄×#14523977	1	1	yes	characteristic
184	C₂	C₄×#14523979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₂₆₈
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	C₄×#335
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #16863149	C₄	direct (N × H)
28. C₄×C₈×C₆₇	C₈	semidirect
72. C₈×C₆₇	C₄×C₈	semidirect
81. C₄×C₆₇	#335	semidirect
97. C₆₇	C₄×#335	semidirect
121. C₄×C₈	C₆₇⋊C₈	semidirect
166. C₈	C₄×C₆₇⋊C₈	semidirect
174. C₄	#16863149	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863149	2
#22558044	2
C₄²×C₈×C₆₇	2
C₄×#335	67

Subgroups — C4×#16863149 #30747016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863149 #30747016