Subgroups — C₆₇×#13283 #30692534

Order 17152.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1380	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
6	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
7	#500×C₆₇	—	2	2	no	—
8	#500×C₆₇	C₄	1	4	yes	complemented
9	C₂×#107×C₆₇	—	2	1	no	—
10	C₂×#107×C₆₇	C₂²	1	2	yes	—
11	C₂×#107×C₆₇	C₂²	1	2	yes	—
12	C₂×#106×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#382×C₆₇	—	2	1	no	—
15	#346×C₆₇	—	2	2	no	—
16	C₂²×Q₈×C₆₇	—	4	1	no	—
17	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₇	—	2	4	no	—
20	C₄⋊Q₈×C₆₇	—	4	2	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
24	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₇	—	2	2	no	—
26	C₂×C₁₆×C₆₇	—	2	2	no	—
27	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
28	#107×C₆₇	—	2	4	no	—
29	#107×C₆₇	—	2	2	no	—
30	#107×C₆₇	—	2	4	no	—
31	#107×C₆₇	C₂×C₄	1	4	yes	—
32	#106×C₆₇	—	4	2	no	—
33	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₇	D₄	1	2	yes	—
35	#95×C₆₇	—	4	1	no	—
36	#95×C₆₇	—	4	1	no	—
37	C₂×Q₈×C₆₇	—	4	4	no	—
38	C₂×Q₈×C₆₇	—	8	4	no	—
39	C₂²×C₄×C₆₇	—	4	2	no	—
40	C₂²×C₄×C₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₆₇	#41	1	2	yes	—
51	C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₄⋊C₄×C₆₇	—	4	4	no	—
53	C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₆₇	SD₁₆	1	2	yes	—
58	C₁₆×C₆₇	—	2	4	no	—
59	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₆₇	—	8	8	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	4	4	no	—
64	C₂×C₄×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	—	8	2	no	—
66	C₂×C₄×C₆₇	—	8	2	no	—
67	C₂×C₄×C₆₇	—	8	2	no	—
68	C₂×C₄×C₆₇	#102	1	2	yes	—
69	C₂×C₄×C₆₇	#102	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₈×C₆₇	#103	1	4	yes	—
77	C₈×C₆₇	C₄×Q₈	1	2	yes	—
78	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
80	C₂²×C₆₇	#486	1	1	yes	characteristic
81	C₂²×C₆₇	#500	1	1	yes	characteristic
82	C₂²×C₆₇	#509	1	2	yes	—
83	C₂²×C₆₇	#510	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	—	8	4	no	—
87	C₄×C₆₇	—	16	2	no	—
88	C₄×C₆₇	#391	1	1	yes	characteristic
89	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
90	C₄×C₆₇	#487	1	1	yes	characteristic
91	C₄×C₆₇	#498	1	1	yes	characteristic
92	#13283	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₆₇	#1990	1	1	yes	characteristic
94	C₂×C₆₇	#2140	1	1	yes	characteristic
95	C₂×C₆₇	#2141	1	1	yes	characteristic
96	C₂×C₆₇	#2243	1	2	yes	—
97	C₂×C₆₇	#2255	1	2	yes	—
98	C₂×#500	C₂×C₆₇	1	1	yes	characteristic
99	C₂×#346	C₂×C₆₇	1	1	yes	characteristic
100	#1846	C₂×C₆₇	1	1	yes	characteristic
101	#1380	C₂×C₆₇	1	4	yes	—
102	C₆₇	#13283	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
105	#500	—	2	2	no	—
106	#500	C₄×C₆₇	1	4	yes	complemented
107	C₂×#107	—	2	1	no	—
108	C₂×#107	C₂²×C₆₇	1	2	yes	—
109	C₂×#107	C₂²×C₆₇	1	2	yes	—
110	C₂×#106	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
112	#382	—	2	1	no	—
113	#346	—	2	2	no	—
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	4	1	no	—
122	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	Q₈×C₆₇	1	4	yes	complemented
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	2	4	no	—
129	#107	C₂×C₄×C₆₇	1	4	yes	—
130	#106	—	4	2	no	—
131	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
132	C₄×C₈	D₄×C₆₇	1	2	yes	—
133	#95	—	4	1	no	—
134	#95	—	4	1	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
140	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
143	C₂×C₈	SD₁₆×C₆₇	1	2	yes	—
144	C₂×C₈	Q₁₆×C₆₇	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
147	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
148	C₂×C₈	#41×C₆₇	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
155	C₄²	SD₁₆×C₆₇	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	#102×C₆₇	1	2	yes	—
167	C₂×C₄	#102×C₆₇	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
169	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
170	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
174	C₈	#103×C₆₇	1	4	yes	—
175	C₈	C₄×Q₈×C₆₇	1	2	yes	—
176	C₈	C₂²⋊Q₈×C₆₇	1	2	yes	—
177	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
178	C₂²	#486×C₆₇	1	1	yes	characteristic
179	C₂²	#500×C₆₇	1	1	yes	characteristic
180	C₂²	#509×C₆₇	1	2	yes	—
181	C₂²	#510×C₆₇	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	16	2	no	—
186	C₄	#391×C₆₇	1	1	yes	characteristic
187	C₄	C₂×#103×C₆₇	1	1	yes	characteristic
188	C₄	#487×C₆₇	1	1	yes	characteristic
189	C₄	#498×C₆₇	1	1	yes	characteristic
190	C₂	C₆₇×#1990	1	1	yes	characteristic
191	C₂	C₆₇×#2140	1	1	yes	characteristic
192	C₂	C₆₇×#2141	1	1	yes	characteristic
193	C₂	C₆₇×#2243	1	2	yes	—
194	C₂	C₆₇×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #500×C₆₇	C₄	semidirect
27. C₂×C₁₆×C₆₇	Q₈	semidirect
92. #13283	C₆₇	direct (N × H)
102. C₆₇	#13283	direct (N × H)
106. #500	C₄×C₆₇	semidirect
125. C₂×C₁₆	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#346×C₆₇	2
C₆₇×#1846	2
C₆₇×#1380	2
#13283	67

Subgroups — C67×#13283 #30692534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13283 #30692534