Subgroups — #2295×C₁₆₇ #36320655

Order 21376.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	#391×C₁₆₇	C₂	1	2	yes	maximal
3	#389×C₁₆₇	C₂	1	2	yes	maximal
4	#387×C₁₆₇	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₁₆₇	C₂	1	2	yes	maximal
7	#376×C₁₆₇	C₂	1	2	yes	maximal
8	C₄³×C₁₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₆₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
22	#95×C₁₆₇	C₂²	1	4	yes	—
23	#95×C₁₆₇	C₂²	1	4	yes	—
24	#95×C₁₆₇	C₂²	1	2	yes	—
25	#95×C₁₆₇	C₂²	1	4	yes	—
26	C₂×Q₈×C₁₆₇	—	2	4	no	—
27	C₂×Q₈×C₁₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₆₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₆₇	—	2	8	no	—
37	C₄⋊C₄×C₁₆₇	—	2	4	no	—
38	C₄⋊C₄×C₁₆₇	C₂×C₄	1	8	yes	—
39	C₄⋊C₄×C₁₆₇	C₂×C₄	1	4	yes	—
40	C₄²×C₁₆₇	—	2	4	no	—
41	C₄²×C₁₆₇	—	2	4	no	—
42	C₄²×C₁₆₇	—	2	2	no	—
43	C₄²×C₁₆₇	—	2	4	no	—
44	C₄²×C₁₆₇	C₂×C₄	1	8	yes	—
45	C₄²×C₁₆₇	C₂×C₄	1	4	yes	—
46	C₄²×C₁₆₇	Q₈	1	4	yes	complemented
47	C₂³×C₁₆₇	C₂⁴	1	1	yes	characteristic
48	Q₈×C₁₆₇	—	2	8	no	—
49	C₂×C₄×C₁₆₇	—	2	2	no	—
50	C₂×C₄×C₁₆₇	—	2	2	no	—
51	C₂×C₄×C₁₆₇	—	2	2	no	—
52	C₂×C₄×C₁₆₇	—	2	2	no	—
53	C₂×C₄×C₁₆₇	—	2	2	no	—
54	C₂×C₄×C₁₆₇	—	2	4	no	—
55	C₂×C₄×C₁₆₇	—	2	4	no	—
56	C₂×C₄×C₁₆₇	—	2	4	no	—
57	C₂×C₄×C₁₆₇	—	2	4	no	—
58	C₂×C₄×C₁₆₇	—	2	4	no	—
59	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₆₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₁₆₇	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₆₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₁₆₇	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₁₆₇	#41	1	4	yes	—
67	C₂×C₄×C₁₆₇	#41	1	4	yes	—
68	C₂×C₄×C₁₆₇	#41	1	2	yes	—
69	C₂²×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₁₆₇	C₂×#41	1	2	yes	—
72	C₂²×C₁₆₇	C₂×#41	1	1	yes	characteristic
73	C₂²×C₁₆₇	C₂×#41	1	2	yes	—
74	C₄×C₁₆₇	—	2	4	no	—
75	C₄×C₁₆₇	—	2	2	no	—
76	C₄×C₁₆₇	—	2	4	no	—
77	C₄×C₁₆₇	—	2	4	no	—
78	C₄×C₁₆₇	—	2	4	no	—
79	C₄×C₁₆₇	—	4	4	no	—
80	C₄×C₁₆₇	#117	1	4	yes	—
81	C₄×C₁₆₇	C₄×Q₈	1	4	yes	—
82	C₂×C₁₆₇	C₂×#117	1	1	yes	characteristic
83	C₂×C₁₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
84	C₂×C₁₆₇	C₄×#41	1	1	yes	characteristic
85	C₂×C₁₆₇	#529	1	2	yes	—
86	C₂×C₁₆₇	#533	1	2	yes	—
87	C₁₆₇	#2295	1	1	yes	characteristic, direct product factor, Sylow
88	#2295	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₄×Q₈	C₂×C₁₆₇	1	1	yes	characteristic
90	#391	C₂×C₁₆₇	1	2	yes	—
91	#389	C₂×C₁₆₇	1	2	yes	—
92	#387	C₂×C₁₆₇	1	4	yes	—
93	C₄×C₄⋊C₄	C₂×C₁₆₇	1	1	yes	characteristic
94	C₄×C₄⋊C₄	C₂×C₁₆₇	1	2	yes	—
95	#376	C₂×C₁₆₇	1	2	yes	—
96	C₄³	C₂×C₁₆₇	1	1	yes	characteristic
97	C₂²×Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
98	C₄×Q₈	C₄×C₁₆₇	1	8	yes	complemented
99	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	4	yes	—
100	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₁₆₇	1	2	yes	—
105	C₂×C₄²	C₂²×C₁₆₇	1	2	yes	—
106	C₂×C₄²	C₂²×C₁₆₇	1	2	yes	—
107	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
108	C₂×C₄²	C₂²×C₁₆₇	1	2	yes	—
109	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
110	#95	C₂²×C₁₆₇	1	4	yes	—
111	#95	C₂²×C₁₆₇	1	4	yes	—
112	#95	C₂²×C₁₆₇	1	2	yes	—
113	#95	C₂²×C₁₆₇	1	4	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×C₄×C₁₆₇	1	4	yes	—
116	C₂²×C₄	C₂³×C₁₆₇	1	4	yes	—
117	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
118	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
120	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
121	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
122	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
124	C₄⋊C₄	—	2	8	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	C₂×C₄×C₁₆₇	1	8	yes	—
127	C₄⋊C₄	C₂×C₄×C₁₆₇	1	4	yes	—
128	C₄²	—	2	4	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	4	no	—
132	C₄²	C₂×C₄×C₁₆₇	1	8	yes	—
133	C₄²	C₂×C₄×C₁₆₇	1	4	yes	—
134	C₄²	Q₈×C₁₆₇	1	4	yes	complemented
135	C₂³	C₂⁴×C₁₆₇	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂²×C₄×C₁₆₇	1	2	yes	—
148	C₂×C₄	C₂²×C₄×C₁₆₇	1	4	yes	—
149	C₂×C₄	C₂²×C₄×C₁₆₇	1	4	yes	—
150	C₂×C₄	C₂²×C₄×C₁₆₇	1	2	yes	—
151	C₂×C₄	C₂²×C₄×C₁₆₇	1	2	yes	—
152	C₂×C₄	C₂×Q₈×C₁₆₇	1	4	yes	—
153	C₂×C₄	C₂×Q₈×C₁₆₇	1	2	yes	—
154	C₂×C₄	#41×C₁₆₇	1	4	yes	—
155	C₂×C₄	#41×C₁₆₇	1	4	yes	—
156	C₂×C₄	#41×C₁₆₇	1	2	yes	—
157	C₂²	C₂³×C₄×C₁₆₇	1	1	yes	characteristic
158	C₂²	C₂²×Q₈×C₁₆₇	1	1	yes	characteristic
159	C₂²	C₂×#41×C₁₆₇	1	2	yes	—
160	C₂²	C₂×#41×C₁₆₇	1	1	yes	characteristic
161	C₂²	C₂×#41×C₁₆₇	1	2	yes	—
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	2	4	no	—
166	C₄	—	2	4	no	—
167	C₄	—	4	4	no	—
168	C₄	#117×C₁₆₇	1	4	yes	—
169	C₄	C₄×Q₈×C₁₆₇	1	4	yes	—
170	C₂	C₂×#117×C₁₆₇	1	1	yes	characteristic
171	C₂	C₂×C₄×Q₈×C₁₆₇	1	1	yes	characteristic
172	C₂	C₄×#41×C₁₆₇	1	1	yes	characteristic
173	C₂	#529×C₁₆₇	1	2	yes	—
174	C₂	#533×C₁₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₆₇	C₂³
commutator	C₂²	C₂³×C₄×C₁₆₇
Frattini, cosocle	C₂³	C₂⁴×C₁₆₇
socle, prime radical	C₂³×C₁₆₇	C₂⁴
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
10. C₄×Q₈×C₁₆₇	C₄	semidirect
46. C₄²×C₁₆₇	Q₈	semidirect
87. C₁₆₇	#2295	direct (N × H)
88. #2295	C₁₆₇	direct (N × H)
98. C₄×Q₈	C₄×C₁₆₇	semidirect
134. C₄²	Q₈×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×Q₈×C₁₆₇	2
#391×C₁₆₇	2
#389×C₁₆₇	2
#387×C₁₆₇	2
C₄×C₄⋊C₄×C₁₆₇	2
#376×C₁₆₇	2
C₄³×C₁₆₇	2
#2295	167

Subgroups — #2295×C167 #36320655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2295×C₁₆₇ #36320655