Subgroups — C₆₇×#21917 #30701168

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1917	C₂	1	8	yes	maximal, complemented
3	C₆₇×#1818	C₂	1	2	yes	maximal
4	C₆₇×#1798	C₂	1	2	yes	maximal
5	C₆₇×#1749	C₂	1	2	yes	maximal
6	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
11	C₂×#97×C₆₇	C₂²	1	4	yes	—
12	#389×C₆₇	C₄	1	4	yes	—
13	#385×C₆₇	C₄	1	4	yes	—
14	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
15	#336×C₆₇	C₂²	1	4	yes	—
16	#336×C₆₇	C₂²	1	4	yes	—
17	#336×C₆₇	C₂²	1	8	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
21	#117×C₆₇	—	2	4	no	—
22	#117×C₆₇	—	4	2	no	—
23	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	—	4	2	no	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
33	C₄⋊C₈×C₆₇	—	4	4	no	—
34	C₂²⋊C₈×C₆₇	—	2	8	no	—
35	#97×C₆₇	—	4	4	no	—
36	#95×C₆₇	C₂×C₄	1	8	yes	—
37	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	—	4	4	no	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	#41	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₆₇	—	2	8	no	—
49	C₂×C₈×C₆₇	—	2	8	no	—
50	C₂×C₈×C₆₇	—	2	8	no	—
51	C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₄⋊C₄×C₆₇	—	4	8	no	—
53	C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₂²⋊C₄×C₆₇	—	2	4	no	—
55	C₂²⋊C₄×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	—	2	4	no	—
57	C₄²×C₆₇	—	2	4	no	—
58	C₄²×C₆₇	—	2	4	no	—
59	C₄²×C₆₇	—	4	2	no	—
60	C₂³×C₆₇	—	2	1	no	—
61	C₂³×C₆₇	—	2	1	no	—
62	C₂³×C₆₇	—	2	4	no	—
63	C₂³×C₆₇	#117	1	2	yes	—
64	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	8	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	8	no	—
73	C₂×C₄×C₆₇	—	2	8	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	4	8	no	—
76	C₂×C₄×C₆₇	#117	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
78	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
79	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
80	C₈×C₆₇	—	4	8	no	—
81	C₂²×C₆₇	—	2	2	no	—
82	C₂²×C₆₇	—	2	4	no	—
83	C₂²×C₆₇	—	4	2	no	—
84	C₂²×C₆₇	—	4	2	no	—
85	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
86	C₂²×C₆₇	#518	1	1	yes	characteristic
87	C₂²×C₆₇	#519	1	1	yes	characteristic
88	C₂²×C₆₇	#568	1	4	yes	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	4	4	no	—
91	C₄×C₆₇	—	4	4	no	—
92	C₄×C₆₇	—	4	4	no	—
93	C₄×C₆₇	#539	1	4	yes	—
94	#21917	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₆₇	—	4	2	no	—
96	C₂×C₆₇	#2313	1	1	yes	characteristic
97	C₂×C₆₇	#2923	1	2	yes	—
98	C₂×C₆₇	#2978	1	2	yes	—
99	C₂×C₆₇	#2987	1	2	yes	—
100	#2296	C₂×C₆₇	1	1	yes	characteristic
101	#1917	C₂×C₆₇	1	8	yes	complemented
102	#1818	C₂×C₆₇	1	2	yes	—
103	#1798	C₂×C₆₇	1	2	yes	—
104	#1749	C₂×C₆₇	1	2	yes	—
105	C₆₇	#21917	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
107	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
108	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
109	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
110	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
111	C₂×#97	C₂²×C₆₇	1	4	yes	—
112	#389	C₄×C₆₇	1	4	yes	—
113	#385	C₄×C₆₇	1	4	yes	—
114	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
115	#336	C₂²×C₆₇	1	4	yes	—
116	#336	C₂²×C₆₇	1	8	yes	—
117	#336	C₂²×C₆₇	1	4	yes	—
118	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
119	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
121	#117	—	2	4	no	—
122	#117	—	4	2	no	—
123	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
124	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
125	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
126	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
128	C₂×C₄²	—	4	2	no	—
129	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
130	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
131	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
132	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
133	C₄⋊C₈	—	4	4	no	—
134	C₂²⋊C₈	—	2	8	no	—
135	#97	—	4	4	no	—
136	#95	C₂×C₄×C₆₇	1	8	yes	—
137	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	4	no	—
142	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
143	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
146	C₂²×C₄	#41×C₆₇	1	4	yes	—
147	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	4	2	no	—
160	C₂³	—	2	1	no	—
161	C₂³	—	2	1	no	—
162	C₂³	—	2	4	no	—
163	C₂³	#117×C₆₇	1	2	yes	—
164	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	#117×C₆₇	1	2	yes	—
177	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
178	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
179	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
180	C₈	—	4	8	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
186	C₂²	#518×C₆₇	1	1	yes	characteristic
187	C₂²	#519×C₆₇	1	1	yes	characteristic
188	C₂²	#568×C₆₇	1	4	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	4	4	no	—
191	C₄	—	4	4	no	—
192	C₄	—	4	4	no	—
193	C₄	#539×C₆₇	1	4	yes	—
194	C₂	—	4	2	no	—
195	C₂	C₆₇×#2313	1	1	yes	characteristic
196	C₂	C₆₇×#2923	1	2	yes	—
197	C₂	C₆₇×#2978	1	2	yes	—
198	C₂	C₆₇×#2987	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1917	C₂	semidirect
94. #21917	C₆₇	direct (N × H)
101. #1917	C₂×C₆₇	semidirect
105. C₆₇	#21917	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2296	2
C₆₇×#1917	2
C₆₇×#1818	2
C₆₇×#1798	2
C₆₇×#1749	2
#21917	67

Subgroups — C67×#21917 #30701168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21917 #30701168