Subgroups — C₂×#22558285 #30767424

Order 17152.

230 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	2	yes	maximal
2	#22558285	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	2	no	—
5	#16863217	—	2	2	no	—
6	#16863217	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
12	#14523983	—	2	4	no	—
13	#14523983	—	4	2	no	—
14	C₂×C₆₇⋊C₁₆	—	4	2	no	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	—	2	1	no	—
19	C₂×C₁₆×C₆₇	—	2	1	no	—
20	C₂×C₁₆×C₆₇	C₈	1	2	yes	—
21	C₂×C₁₆×C₆₇	C₈	1	2	yes	—
22	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₆₇⋊C₁₆	—	4	4	no	—
29	C₂²×C₄×C₆₇	—	2	1	no	—
30	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
38	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₆₇	—	2	1	no	—
44	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₁₆×C₆₇	—	2	2	no	—
47	C₁₆×C₆₇	C₂×C₈	1	4	yes	—
48	C₂³×C₆₇	#108	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#108	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₆₇	#107	1	1	yes	characteristic
61	C₈×C₆₇	#107	1	1	yes	characteristic
62	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²×C₆₇	—	2	2	no	—
67	C₂²×C₆₇	#335	1	1	yes	characteristic
68	C₂²×C₆₇	#335	1	1	yes	characteristic
69	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	#335	1	1	yes	characteristic
73	C₄×C₆₇	#335	1	1	yes	characteristic
74	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
76	C₂×#1373	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	1	no	—
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	#1373	1	2	yes	complemented
80	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
81	C₂×#364	—	67	2	no	—
82	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
83	#1373	—	67	4	no	—
84	C₆₇	C₂×#1373	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	134	2	no	—
86	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
87	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	#364	—	67	4	no	—
90	#364	—	134	2	no	—
91	C₄×C₁₆	Dic₆₇	1	2	yes	—
92	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
93	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
95	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
96	M₃₂	—	268	2	no	—
97	M₃₂	—	134	4	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	268	2	no	—
101	C₂×C₁₆	C₆₇⋊C₈	1	2	yes	—
102	C₂×C₁₆	C₆₇⋊C₈	1	2	yes	—
103	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
104	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
105	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
106	C₄×C₈	D₂₆₈	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
111	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
113	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
114	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
115	C₂×C₈	#11761612	1	1	yes	characteristic
116	C₂×C₈	#11761612	1	1	yes	characteristic
117	C₂×C₈	D₅₃₆	1	1	yes	characteristic
118	C₂×C₈	D₅₃₆	1	1	yes	characteristic
119	C₂×C₈	Dic₂₆₈	1	1	yes	characteristic
120	C₂×C₈	Dic₂₆₈	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
122	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	—	268	4	no	—
128	C₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
129	C₂³	#14523979	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
134	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
135	C₂×C₄	#14523977	1	1	yes	characteristic
136	C₂×C₄	#14523977	1	1	yes	characteristic
137	C₂×C₄	#14523979	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
139	C₈	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
140	C₈	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
141	C₈	#14523977	1	1	yes	characteristic
142	C₈	#14523977	1	1	yes	characteristic
143	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
144	C₈	C₂×#11761612	1	1	yes	characteristic
145	C₈	C₂×D₅₃₆	1	1	yes	characteristic
146	C₈	C₂×Dic₂₆₈	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	#16863149	1	1	yes	characteristic
149	C₂²	#16863149	1	1	yes	characteristic
150	C₂²	C₂×#14523979	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	#16863149	1	1	yes	characteristic
154	C₄	#16863149	1	1	yes	characteristic
155	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
156	C₄	C₂×#14523977	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#22558285	1	2	yes	direct product factor
160	C₂	C₂×#16863149	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₅₃₆
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#335
prime radical	C₂³×C₆₇	#108
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1373
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22558285	C₂	direct (N × H)
79. C₂×C₆₇	#1373	semidirect
84. C₆₇	C₂×#1373	semidirect
159. C₂	#22558285	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22558285	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#1373	67

Subgroups — C2×#22558285 #30767424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558285 #30767424