Subgroups — C₆₇×#20752 #30700003

Order 17152.

1158 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1831	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1728	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
6	C₂²×C₄²×C₆₇	C₄	1	4	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	#423×C₆₇	—	2	8	no	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
11	C₄³×C₆₇	C₄	1	2	yes	—
12	C₄³×C₆₇	C₄	1	2	yes	—
13	C₄³×C₆₇	C₂²	1	4	yes	—
14	#336×C₆₇	C₂²	1	16	yes	—
15	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₆₇	—	2	8	no	—
19	C₂×M₁₆×C₆₇	—	2	16	no	—
20	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
21	C₂×C₄²×C₆₇	—	2	8	no	—
22	C₂×C₄²×C₆₇	—	2	16	no	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
24	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₇	D₄	1	4	yes	—
30	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
33	C₄⋊C₈×C₆₇	—	2	16	no	—
34	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	—	2	8	no	—
36	C₂²×C₄×C₆₇	—	2	16	no	—
37	C₂²×C₄×C₆₇	—	2	8	no	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₆₇	#41	1	8	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₆₇	—	4	16	no	—
48	C₂×C₈×C₆₇	—	2	8	no	—
49	C₂×C₈×C₆₇	—	2	16	no	—
50	C₄²×C₆₇	—	2	8	no	—
51	C₄²×C₆₇	—	2	16	no	—
52	C₄²×C₆₇	—	2	8	no	—
53	C₄²×C₆₇	—	2	8	no	—
54	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
55	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
56	C₄²×C₆₇	M₁₆	1	16	yes	complemented
57	C₄²×C₆₇	C₂×D₄	1	4	yes	—
58	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
59	C₂³×C₆₇	—	2	4	no	—
60	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
61	C₂³×C₆₇	#117	1	4	yes	—
62	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	32	no	—
64	C₂×C₄×C₆₇	—	2	8	no	—
65	C₂×C₄×C₆₇	—	2	4	no	—
66	C₂×C₄×C₆₇	—	2	8	no	—
67	C₂×C₄×C₆₇	—	2	16	no	—
68	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₆₇	#117	1	4	yes	—
72	C₂×C₄×C₆₇	#124	1	8	yes	—
73	C₂×C₄×C₆₇	#125	1	8	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	16	yes	—
75	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
79	C₈×C₆₇	—	4	8	no	—
80	C₂²×C₆₇	—	2	4	no	—
81	C₂²×C₆₇	—	2	8	no	—
82	C₂²×C₆₇	#382	1	4	yes	—
83	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×#117	1	2	yes	—
85	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
86	C₄×C₆₇	—	2	4	no	—
87	C₄×C₆₇	—	2	16	no	—
88	C₄×C₆₇	#382	1	4	yes	—
89	C₄×C₆₇	#423	1	8	yes	—
90	C₄×C₆₇	C₂×#124	1	2	yes	—
91	C₄×C₆₇	C₂×#125	1	2	yes	—
92	#20752	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₆₇	—	2	4	no	—
94	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
95	C₂×C₆₇	C₂×#423	1	2	yes	—
96	C₂×C₆₇	#2920	1	4	yes	—
97	C₂×#423	C₂×C₆₇	1	2	yes	—
98	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic
99	#1831	C₂×C₆₇	1	8	yes	complemented
100	#1728	C₂×C₆₇	1	4	yes	—
101	C₆₇	#20752	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
103	C₂²×C₄²	C₄×C₆₇	1	4	yes	—
104	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
105	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
106	#423	—	2	8	no	—
107	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
108	C₄³	C₄×C₆₇	1	2	yes	—
109	C₄³	C₄×C₆₇	1	2	yes	—
110	C₄³	C₂²×C₆₇	1	4	yes	—
111	#336	C₂²×C₆₇	1	16	yes	—
112	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
113	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
114	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
115	C₂×M₁₆	—	2	8	no	—
116	C₂×M₁₆	—	2	16	no	—
117	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
118	C₂×C₄²	—	2	8	no	—
119	C₂×C₄²	—	2	16	no	—
120	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
121	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
122	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
126	C₂×C₄²	D₄×C₆₇	1	4	yes	—
127	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
128	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
129	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
130	C₄⋊C₈	—	2	16	no	—
131	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	—	2	16	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
136	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
137	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
138	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
140	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
141	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
142	C₂²×C₄	#41×C₆₇	1	8	yes	—
143	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
144	M₁₆	—	4	16	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	16	no	—
147	C₄²	—	2	8	no	—
148	C₄²	—	2	16	no	—
149	C₄²	—	2	8	no	—
150	C₄²	—	2	8	no	—
151	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
152	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
153	C₄²	M₁₆×C₆₇	1	16	yes	complemented
154	C₄²	C₂×D₄×C₆₇	1	4	yes	—
155	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
158	C₂³	#117×C₆₇	1	4	yes	—
159	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
160	C₂×C₄	—	2	32	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
167	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
168	C₂×C₄	#117×C₆₇	1	4	yes	—
169	C₂×C₄	#124×C₆₇	1	8	yes	—
170	C₂×C₄	#125×C₆₇	1	8	yes	—
171	C₂×C₄	C₂×M₁₆×C₆₇	1	16	yes	—
172	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
173	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
175	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
176	C₈	—	4	8	no	—
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	#382×C₆₇	1	4	yes	—
180	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
181	C₂²	C₂×#117×C₆₇	1	2	yes	—
182	C₂²	C₂²×M₁₆×C₆₇	1	4	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	16	no	—
185	C₄	#382×C₆₇	1	4	yes	—
186	C₄	#423×C₆₇	1	8	yes	—
187	C₄	C₂×#124×C₆₇	1	2	yes	—
188	C₄	C₂×#125×C₆₇	1	2	yes	—
189	C₂	—	2	4	no	—
190	C₂	C₂×#382×C₆₇	1	1	yes	characteristic
191	C₂	C₂×#423×C₆₇	1	2	yes	—
192	C₂	C₆₇×#2920	1	4	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
3. C₆₇×#1831	C₂	semidirect
56. C₄²×C₆₇	M₁₆	semidirect
92. #20752	C₆₇	direct (N × H)
99. #1831	C₂×C₆₇	semidirect
101. C₆₇	#20752	direct (N × H)
153. C₄²	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₆₇	2
C₂×C₄³×C₆₇	2
C₆₇×#1831	2
C₆₇×#1728	2
#20752	67

Subgroups — C67×#20752 #30700003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20752 #30700003