Subgroups — #1939×C₁₆₇ #36320299

Order 21376.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×C₁₆₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₆₇	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₆₇	—	2	4	no	—
14	C₄⋊C₈×C₁₆₇	—	2	2	no	—
15	C₄×C₈×C₁₆₇	—	2	2	no	—
16	#95×C₁₆₇	C₄	1	2	yes	—
17	C₂²×C₄×C₁₆₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₆₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₆₇	—	2	1	no	—
23	C₂×C₈×C₁₆₇	—	2	1	no	—
24	C₂×C₈×C₁₆₇	—	2	1	no	—
25	C₂×C₈×C₁₆₇	—	2	1	no	—
26	C₂×C₈×C₁₆₇	—	2	2	no	—
27	C₂×C₈×C₁₆₇	—	2	2	no	—
28	C₂×C₈×C₁₆₇	—	2	2	no	—
29	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
30	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
31	C₄⋊C₄×C₁₆₇	—	2	2	no	—
32	C₄⋊C₄×C₁₆₇	C₈	1	4	yes	complemented
33	C₄²×C₁₆₇	—	2	2	no	—
34	C₄²×C₁₆₇	—	2	2	no	—
35	C₄²×C₁₆₇	D₄	1	2	yes	—
36	C₄²×C₁₆₇	Q₈	1	2	yes	—
37	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	—	2	2	no	—
39	C₂×C₄×C₁₆₇	—	2	1	no	—
40	C₂×C₄×C₁₆₇	—	2	1	no	—
41	C₂×C₄×C₁₆₇	—	2	2	no	—
42	C₂×C₄×C₁₆₇	—	2	2	no	—
43	C₂×C₄×C₁₆₇	—	2	2	no	—
44	C₂×C₄×C₁₆₇	—	2	2	no	—
45	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₆₇	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₁₆₇	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₁₆₇	M₁₆	1	2	yes	—
50	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
56	C₈×C₁₆₇	—	2	2	no	—
57	C₈×C₁₆₇	—	2	2	no	—
58	C₈×C₁₆₇	—	4	2	no	—
59	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₆₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₆₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₆₇	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₁₆₇	#131	1	1	yes	characteristic
65	C₂²×C₁₆₇	#131	1	1	yes	characteristic
66	C₄×C₁₆₇	—	2	2	no	—
67	C₄×C₁₆₇	—	2	4	no	—
68	C₄×C₁₆₇	—	4	2	no	—
69	C₄×C₁₆₇	C₄⋊C₈	1	4	yes	complemented
70	C₄×C₁₆₇	#121	1	1	yes	characteristic
71	C₄×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₁₆₇	#125	1	1	yes	characteristic
73	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₆₇	#389	1	1	yes	characteristic
75	C₂×C₁₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₁₆₇	#424	1	1	yes	characteristic
77	C₂×C₁₆₇	C₈×D₄	1	1	yes	characteristic
78	C₂×C₁₆₇	#436	1	1	yes	characteristic
79	C₂×C₁₆₇	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₁₆₇	#446	1	1	yes	characteristic
81	C₁₆₇	#1939	1	1	yes	characteristic, direct product factor, Sylow
82	#1939	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₄⋊C₈	C₂×C₁₆₇	1	2	yes	—
84	C₂×C₄⋊C₈	C₂×C₁₆₇	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×C₁₆₇	1	1	yes	characteristic
86	C₄×C₄⋊C₄	C₂×C₁₆₇	1	1	yes	characteristic
87	#336	C₂×C₁₆₇	1	2	yes	—
88	C₂²×C₈	C₂²×C₁₆₇	1	2	yes	—
89	C₂²×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
90	C₂²×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₄×C₁₆₇	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₄×C₁₆₇	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₁₆₇	1	2	yes	—
94	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄×C₈	—	2	2	no	—
98	#95	C₄×C₁₆₇	1	2	yes	—
99	C₂²×C₄	C₂×C₄×C₁₆₇	1	2	yes	—
100	C₂²×C₄	C₂×C₄×C₁₆₇	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₄×C₁₆₇	1	2	yes	—
102	C₂²×C₄	C₂×C₄×C₁₆₇	1	1	yes	characteristic
103	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	D₄×C₁₆₇	1	2	yes	—
112	C₂×C₈	Q₈×C₁₆₇	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₈×C₁₆₇	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	D₄×C₁₆₇	1	2	yes	—
118	C₄²	Q₈×C₁₆₇	1	2	yes	—
119	C₂³	C₂²×C₄×C₁₆₇	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊C₄×C₁₆₇	1	2	yes	—
128	C₂×C₄	C₄⋊C₄×C₁₆₇	1	2	yes	—
129	C₂×C₄	C₂×C₈×C₁₆₇	1	4	yes	—
130	C₂×C₄	C₂×C₈×C₁₆₇	1	2	yes	—
131	C₂×C₄	M₁₆×C₁₆₇	1	2	yes	—
132	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
134	C₂×C₄	C₂×Q₈×C₁₆₇	1	1	yes	characteristic
135	C₂×C₄	C₂×Q₈×C₁₆₇	1	1	yes	characteristic
136	C₂×C₄	#41×C₁₆₇	1	1	yes	characteristic
137	C₂×C₄	#41×C₁₆₇	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₂²	C₂×C₄⋊C₄×C₁₆₇	1	1	yes	characteristic
142	C₂²	C₄×D₄×C₁₆₇	1	1	yes	characteristic
143	C₂²	C₄×Q₈×C₁₆₇	1	1	yes	characteristic
144	C₂²	C₂²×C₈×C₁₆₇	1	1	yes	characteristic
145	C₂²	C₂×M₁₆×C₁₆₇	1	1	yes	characteristic
146	C₂²	#131×C₁₆₇	1	1	yes	characteristic
147	C₂²	#131×C₁₆₇	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊C₈×C₁₆₇	1	4	yes	complemented
152	C₄	#121×C₁₆₇	1	1	yes	characteristic
153	C₄	C₂²⋊Q₈×C₁₆₇	1	1	yes	characteristic
154	C₄	#125×C₁₆₇	1	1	yes	characteristic
155	C₄	C₄⋊Q₈×C₁₆₇	1	1	yes	characteristic
156	C₂	#389×C₁₆₇	1	1	yes	characteristic
157	C₂	C₂×C₄⋊C₈×C₁₆₇	1	1	yes	characteristic
158	C₂	#424×C₁₆₇	1	1	yes	characteristic
159	C₂	C₈×D₄×C₁₆₇	1	1	yes	characteristic
160	C₂	#436×C₁₆₇	1	1	yes	characteristic
161	C₂	C₈×Q₈×C₁₆₇	1	1	yes	characteristic
162	C₂	#446×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₆₇	C₂³
commutator	C₂²	C₂²×C₈×C₁₆₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₆₇
socle, prime radical	C₂³×C₁₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
32. C₄⋊C₄×C₁₆₇	C₈	semidirect
69. C₄×C₁₆₇	C₄⋊C₈	semidirect
81. C₁₆₇	#1939	direct (N × H)
82. #1939	C₁₆₇	direct (N × H)
114. C₄⋊C₄	C₈×C₁₆₇	semidirect
151. C₄	C₄⋊C₈×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×C₁₆₇	2
C₂×C₄×C₈×C₁₆₇	2
C₄×C₄⋊C₄×C₁₆₇	2
#336×C₁₆₇	2
#1939	167

Subgroups — #1939×C167 #36320299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1939×C₁₆₇ #36320299