Subgroups — C₆₇×#34012 #30713263

Order 17152.

3158 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	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2285	C₂	1	6	yes	maximal, complemented
3	C₆₇×#2283	C₂	1	12	yes	maximal, complemented
4	C₆₇×#2280	C₂	1	6	yes	maximal
5	C₄×#117×C₆₇	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
7	C₂×#117×C₆₇	C₂²	1	12	yes	—
8	C₂²×C₄²×C₆₇	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₆₇	C₂²	1	6	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
12	#384×C₆₇	C₂²	1	4	yes	complemented
13	#383×C₆₇	C₂²	1	8	yes	complemented
14	#382×C₆₇	C₂²	1	12	yes	complemented
15	#381×C₆₇	C₂²	1	12	yes	—
16	#380×C₆₇	C₂²	1	24	yes	—
17	#379×C₆₇	C₂²	1	4	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	24	yes	complemented
19	C₄×C₂²⋊C₄×C₆₇	C₂²	1	24	yes	—
20	#376×C₆₇	C₂²	1	12	yes	complemented
21	C₄³×C₆₇	C₂²	1	4	yes	complemented
22	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	2	24	no	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	24	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
29	#117×C₆₇	—	2	48	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
31	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	48	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	24	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#95×C₆₇	C₂³	1	48	yes	—
43	C₂⁴×C₆₇	—	2	12	no	—
44	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₆₇	—	2	12	no	—
47	C₂²×C₄×C₆₇	—	2	96	no	—
48	C₂²×C₄×C₆₇	—	2	24	no	—
49	C₂²×C₄×C₆₇	—	2	24	no	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₆₇	#41	1	24	yes	—
55	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
56	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
59	C₄⋊C₄×C₆₇	—	2	48	no	—
60	C₂²⋊C₄×C₆₇	—	2	48	no	—
61	C₄²×C₆₇	—	2	48	no	—
62	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
64	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
67	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
68	C₂³×C₆₇	—	2	32	no	—
69	C₂³×C₆₇	—	2	24	no	—
70	C₂³×C₆₇	—	2	12	no	—
71	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
72	C₂³×C₆₇	C₂×#41	1	12	yes	—
73	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
74	C₂×C₄×C₆₇	—	2	24	no	—
75	C₂×C₄×C₆₇	—	2	12	no	—
76	C₂×C₄×C₆₇	—	2	48	no	—
77	C₂×C₄×C₆₇	—	2	48	no	—
78	C₂×C₄×C₆₇	—	2	48	no	—
79	C₂×C₄×C₆₇	#117	1	48	yes	complemented
80	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
82	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
83	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
84	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
85	C₂²×C₆₇	—	2	32	no	—
86	C₂²×C₆₇	—	2	12	no	—
87	C₂²×C₆₇	—	2	24	no	—
88	C₂²×C₆₇	C₂×#117	1	12	yes	—
89	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
90	C₂²×C₆₇	C₂²×#41	1	6	yes	—
91	C₄×C₆₇	—	2	12	no	—
92	C₄×C₆₇	—	4	16	no	—
93	C₄×C₆₇	C₂×#117	1	12	yes	—
94	C₄×C₆₇	C₂×#117	1	12	yes	—
95	#34012	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₆₇	—	2	12	no	—
97	C₂×C₆₇	C₂²×#117	1	3	yes	—
98	C₂×C₆₇	#3444	1	4	yes	—
99	C₂³×C₄²	C₂×C₆₇	1	1	yes	characteristic
100	#2285	C₂×C₆₇	1	6	yes	complemented
101	#2283	C₂×C₆₇	1	12	yes	complemented
102	#2280	C₂×C₆₇	1	6	yes	—
103	C₄×#117	C₂×C₆₇	1	6	yes	complemented
104	C₆₇	#34012	1	1	yes	characteristic, direct product factor, Sylow
105	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
106	C₂×#117	C₂²×C₆₇	1	12	yes	—
107	C₂²×C₄²	C₄×C₆₇	1	16	yes	complemented
108	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
109	C₂²×C₄²	C₂²×C₆₇	1	6	yes	—
110	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
111	#384	C₂²×C₆₇	1	4	yes	complemented
112	#383	C₂²×C₆₇	1	8	yes	complemented
113	#382	C₂²×C₆₇	1	12	yes	complemented
114	#381	C₂²×C₆₇	1	12	yes	—
115	#380	C₂²×C₆₇	1	24	yes	—
116	#379	C₂²×C₆₇	1	4	yes	—
117	C₄×C₄⋊C₄	C₂²×C₆₇	1	24	yes	complemented
118	C₄×C₂²⋊C₄	C₂²×C₆₇	1	24	yes	—
119	#376	C₂²×C₆₇	1	12	yes	complemented
120	C₄³	C₂²×C₆₇	1	4	yes	complemented
121	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
122	C₂³×C₄	—	2	24	no	—
123	C₂³×C₄	C₂×C₄×C₆₇	1	24	yes	—
124	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
125	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
126	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
127	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
128	#117	—	2	48	no	—
129	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
130	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
131	C₂×C₄²	C₂×C₄×C₆₇	1	48	yes	—
132	C₂×C₄²	C₂×C₄×C₆₇	1	24	yes	—
133	C₂×C₄²	C₂×C₄×C₆₇	1	24	yes	—
134	C₂×C₄²	C₂³×C₆₇	1	24	yes	—
135	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
136	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
137	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
138	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
139	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
141	#95	C₂³×C₆₇	1	48	yes	—
142	C₂⁴	—	2	12	no	—
143	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
144	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	2	96	no	—
147	C₂²×C₄	—	2	24	no	—
148	C₂²×C₄	—	2	24	no	—
149	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
150	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
151	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
152	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
153	C₂²×C₄	#41×C₆₇	1	24	yes	—
154	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
155	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
156	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
157	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
158	C₄⋊C₄	—	2	48	no	—
159	C₂²⋊C₄	—	2	48	no	—
160	C₄²	—	2	48	no	—
161	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
162	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
163	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
164	C₄²	C₂²×C₄×C₆₇	1	24	yes	—
165	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
166	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
167	C₂³	—	2	32	no	—
168	C₂³	—	2	24	no	—
169	C₂³	—	2	12	no	—
170	C₂³	C₂³×C₄×C₆₇	1	6	yes	—
171	C₂³	C₂×#41×C₆₇	1	12	yes	—
172	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	2	12	no	—
175	C₂×C₄	—	2	48	no	—
176	C₂×C₄	—	2	48	no	—
177	C₂×C₄	—	2	48	no	—
178	C₂×C₄	#117×C₆₇	1	48	yes	complemented
179	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
180	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
181	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
182	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
183	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
184	C₂²	—	2	32	no	—
185	C₂²	—	2	12	no	—
186	C₂²	—	2	24	no	—
187	C₂²	C₂×#117×C₆₇	1	12	yes	—
188	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
189	C₂²	C₂²×#41×C₆₇	1	6	yes	—
190	C₄	—	2	12	no	—
191	C₄	—	4	16	no	—
192	C₄	C₂×#117×C₆₇	1	12	yes	—
193	C₄	C₂×#117×C₆₇	1	12	yes	—
194	C₂	—	2	12	no	—
195	C₂	C₂²×#117×C₆₇	1	3	yes	—
196	C₂	C₆₇×#3444	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2285	C₂	semidirect
3. C₆₇×#2283	C₂	semidirect
5. C₄×#117×C₆₇	C₂	semidirect
8. C₂²×C₄²×C₆₇	C₄	semidirect
12. #384×C₆₇	C₂²	semidirect
13. #383×C₆₇	C₂²	semidirect
14. #382×C₆₇	C₂²	semidirect
18. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
20. #376×C₆₇	C₂²	semidirect
21. C₄³×C₆₇	C₂²	semidirect
79. C₂×C₄×C₆₇	#117	semidirect
95. #34012	C₆₇	direct (N × H)
100. #2285	C₂×C₆₇	semidirect
101. #2283	C₂×C₆₇	semidirect
103. C₄×#117	C₂×C₆₇	semidirect
104. C₆₇	#34012	direct (N × H)
107. C₂²×C₄²	C₄×C₆₇	semidirect
111. #384	C₂²×C₆₇	semidirect
112. #383	C₂²×C₆₇	semidirect
113. #382	C₂²×C₆₇	semidirect
117. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
119. #376	C₂²×C₆₇	semidirect
120. C₄³	C₂²×C₆₇	semidirect
178. C₂×C₄	#117×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₆₇	2
C₆₇×#2285	2
C₆₇×#2283	2
C₆₇×#2280	2
C₄×#117×C₆₇	2
#34012	67

Subgroups — C67×#34012 #30713263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34012 #30713263