Subgroups — C₆₇×#32856 #30712107

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2647	C₂	1	2	yes	maximal
2	C₆₇×#2101	C₂	1	4	yes	maximal
3	C₆₇×#2087	C₂	1	2	yes	maximal
4	C₆₇×#2086	C₂	1	2	yes	maximal
5	C₆₇×#2083	C₂	1	4	yes	maximal
6	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₆₇	C₂²	1	2	yes	—
8	C₂×#107×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#400×C₆₇	—	2	4	no	—
12	#398×C₆₇	—	2	4	no	—
13	#389×C₆₇	C₂²	1	4	yes	—
14	#389×C₆₇	C₂²	1	4	yes	—
15	#384×C₆₇	C₂²	1	2	yes	—
16	#382×C₆₇	C₂²	1	2	yes	—
17	#376×C₆₇	C₂²	1	1	yes	characteristic
18	#340×C₆₇	C₂²	1	4	yes	—
19	#340×C₆₇	C₂²	1	4	yes	—
20	#340×C₆₇	C₂²	1	2	yes	—
21	#340×C₆₇	C₂²	1	2	yes	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
25	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#107×C₆₇	—	4	2	no	—
33	#107×C₆₇	—	4	2	no	—
34	C₄⋊C₈×C₆₇	—	4	1	no	—
35	C₄⋊C₈×C₆₇	—	4	1	no	—
36	#95×C₆₇	—	2	4	no	—
37	#95×C₆₇	—	2	4	no	—
38	#95×C₆₇	—	2	4	no	—
39	#95×C₆₇	D₄	1	2	yes	—
40	#95×C₆₇	D₄	1	2	yes	—
41	C₂²×C₄×C₆₇	—	2	4	no	—
42	C₂²×C₄×C₆₇	—	2	4	no	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₄⋊C₄×C₆₇	—	8	4	no	—
56	C₄⋊C₄×C₆₇	—	8	4	no	—
57	C₄⋊C₄×C₆₇	—	8	4	no	—
58	C₄²×C₆₇	—	2	1	no	—
59	C₄²×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	—	2	1	no	—
62	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	—	2	1	no	—
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	4	4	no	—
73	C₂×C₄×C₆₇	—	4	4	no	—
74	C₂×C₄×C₆₇	—	4	4	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
78	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
79	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
80	C₈×C₆₇	—	4	2	no	—
81	C₈×C₆₇	—	4	2	no	—
82	C₂²×C₆₇	#536	1	2	yes	—
83	C₂²×C₆₇	#537	1	1	yes	characteristic
84	C₂²×C₆₇	#576	1	2	yes	—
85	C₂²×C₆₇	#578	1	2	yes	—
86	C₄×C₆₇	—	4	4	no	—
87	C₄×C₆₇	—	4	1	no	—
88	C₄×C₆₇	—	4	1	no	—
89	C₄×C₆₇	—	8	4	no	—
90	C₄×C₆₇	—	8	4	no	—
91	C₄×C₆₇	#563	1	1	yes	characteristic
92	C₄×C₆₇	#563	1	1	yes	characteristic
93	C₄×C₆₇	#564	1	2	yes	—
94	#32856	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₆₇	#2822	1	1	yes	characteristic
96	C₂×C₆₇	#3243	1	4	yes	—
97	C₂×C₆₇	#3259	1	1	yes	characteristic
98	C₂×C₆₇	#3261	1	1	yes	characteristic
99	#2647	C₂×C₆₇	1	2	yes	—
100	#2101	C₂×C₆₇	1	4	yes	—
101	#2087	C₂×C₆₇	1	2	yes	—
102	#2086	C₂×C₆₇	1	2	yes	—
103	#2083	C₂×C₆₇	1	4	yes	—
104	#1832	C₂×C₆₇	1	1	yes	characteristic
105	C₆₇	#32856	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×#107	C₂²×C₆₇	1	2	yes	—
107	C₂×#107	C₂²×C₆₇	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
109	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
110	#400	—	2	4	no	—
111	#398	—	2	4	no	—
112	#389	C₂²×C₆₇	1	4	yes	—
113	#389	C₂²×C₆₇	1	4	yes	—
114	#384	C₂²×C₆₇	1	2	yes	—
115	#382	C₂²×C₆₇	1	2	yes	—
116	#376	C₂²×C₆₇	1	1	yes	characteristic
117	#340	C₂²×C₆₇	1	4	yes	—
118	#340	C₂²×C₆₇	1	4	yes	—
119	#340	C₂²×C₆₇	1	2	yes	—
120	#340	C₂²×C₆₇	1	2	yes	—
121	#336	C₂²×C₆₇	1	4	yes	—
122	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
123	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
128	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
129	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
130	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
131	#107	—	4	2	no	—
132	#107	—	4	2	no	—
133	C₄⋊C₈	—	4	1	no	—
134	C₄⋊C₈	—	4	1	no	—
135	#95	—	2	4	no	—
136	#95	—	2	4	no	—
137	#95	—	2	4	no	—
138	#95	D₄×C₆₇	1	2	yes	—
139	#95	D₄×C₆₇	1	2	yes	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
144	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	8	4	no	—
155	C₄⋊C₄	—	8	4	no	—
156	C₄⋊C₄	—	8	4	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	1	no	—
161	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
176	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
177	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
178	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	—	4	2	no	—
181	C₂²	#536×C₆₇	1	2	yes	—
182	C₂²	#537×C₆₇	1	1	yes	characteristic
183	C₂²	#576×C₆₇	1	2	yes	—
184	C₂²	#578×C₆₇	1	2	yes	—
185	C₄	—	4	4	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	8	4	no	—
189	C₄	—	8	4	no	—
190	C₄	#563×C₆₇	1	1	yes	characteristic
191	C₄	#563×C₆₇	1	1	yes	characteristic
192	C₄	#564×C₆₇	1	2	yes	—
193	C₂	C₆₇×#2822	1	1	yes	characteristic
194	C₂	C₆₇×#3243	1	4	yes	—
195	C₂	C₆₇×#3259	1	1	yes	characteristic
196	C₂	C₆₇×#3261	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
94. #32856	C₆₇	direct (N × H)
105. C₆₇	#32856	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×#2647	2
C₆₇×#2101	2
C₆₇×#2087	2
C₆₇×#2086	2
C₆₇×#2083	2
C₆₇×#1832	2
#32856	67

Subgroups — C67×#32856 #30712107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32856 #30712107