Subgroups — C₆₇×#33340 #30712591

Order 17152.

1114 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3573	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2117	C₂	1	6	yes	maximal, complemented
5	C₂×#350×C₆₇	C₂	1	6	yes	maximal, complemented
6	#585×C₆₇	C₄	1	1	yes	characteristic
7	#568×C₆₇	—	2	4	no	—
8	#568×C₆₇	C₄	1	6	yes	—
9	#568×C₆₇	C₄	1	1	yes	characteristic
10	C₂×#131×C₆₇	—	2	4	no	—
11	C₂×#131×C₆₇	C₂²	1	6	yes	—
12	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
13	#504×C₆₇	—	2	4	no	—
14	#504×C₆₇	—	2	4	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	6	yes	complemented
16	C₂×M₃₂×C₆₇	C₂²	1	3	yes	complemented
17	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	complemented
18	#350×C₆₇	C₂²	1	4	yes	complemented
19	#350×C₆₇	C₂²	1	12	yes	complemented
20	2₋¹⁺⁴×C₆₇	—	2	1	no	—
21	2₋¹⁺⁴×C₆₇	C₈	1	4	yes	—
22	H(5,𝔽₂)×C₆₇	—	2	3	no	—
23	H(5,𝔽₂)×C₆₇	C₈	1	4	yes	—
24	C₂×#41×C₆₇	—	2	4	no	—
25	C₂×#41×C₆₇	C₂×C₄	1	6	yes	—
26	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
27	#131×C₆₇	—	2	4	no	—
28	#131×C₆₇	—	2	12	no	—
29	#131×C₆₇	—	2	6	no	—
30	#131×C₆₇	—	2	6	no	—
31	#131×C₆₇	C₂×C₄	1	12	yes	—
32	#131×C₆₇	D₄	1	4	yes	—
33	#131×C₆₇	D₄	1	4	yes	—
34	#131×C₆₇	C₂³	1	4	yes	—
35	C₂×M₁₆×C₆₇	—	2	12	no	—
36	C₂×M₁₆×C₆₇	—	2	6	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	6	yes	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
39	C₂²×C₈×C₆₇	—	2	6	no	—
40	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
41	M₃₂×C₆₇	—	2	6	no	—
42	M₃₂×C₆₇	—	2	6	no	—
43	M₃₂×C₆₇	C₂³	1	4	yes	—
44	C₂×C₁₆×C₆₇	—	2	6	no	—
45	C₂×C₁₆×C₆₇	—	2	6	no	—
46	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
48	#41×C₆₇	—	2	4	no	—
49	#41×C₆₇	—	2	12	no	—
50	#41×C₆₇	—	2	6	no	—
51	#41×C₆₇	—	2	6	no	—
52	#41×C₆₇	C₂²⋊C₄	1	4	yes	—
53	#41×C₆₇	C₂×C₈	1	12	yes	—
54	#41×C₆₇	C₂×C₈	1	4	yes	—
55	#41×C₆₇	C₂²×C₄	1	4	yes	—
56	C₂×Q₈×C₆₇	—	2	4	no	—
57	C₂×Q₈×C₆₇	C₂×C₈	1	6	yes	—
58	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
59	C₂×D₄×C₆₇	—	2	6	no	—
60	C₂×D₄×C₆₇	—	2	12	no	—
61	C₂×D₄×C₆₇	C₂×C₈	1	6	yes	—
62	C₂×D₄×C₆₇	M₁₆	1	3	yes	—
63	C₂²×C₄×C₆₇	—	2	6	no	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
65	M₁₆×C₆₇	—	2	6	no	—
66	M₁₆×C₆₇	—	2	12	no	—
67	M₁₆×C₆₇	—	2	4	no	—
68	M₁₆×C₆₇	C₂²⋊C₄	1	12	yes	—
69	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₇	—	2	6	no	—
71	C₂×C₈×C₆₇	—	2	12	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
76	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₁₆×C₆₇	—	2	1	no	—
78	C₁₆×C₆₇	—	2	3	no	—
79	C₁₆×C₆₇	—	2	4	no	—
80	C₂³×C₆₇	—	2	6	no	—
81	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
82	Q₈×C₆₇	—	2	6	no	—
83	Q₈×C₆₇	C₂²⋊C₈	1	4	yes	—
84	Q₈×C₆₇	C₂²×C₈	1	4	yes	—
85	D₄×C₆₇	—	2	4	no	—
86	D₄×C₆₇	—	2	12	no	—
87	D₄×C₆₇	—	2	6	no	—
88	D₄×C₆₇	C₂²⋊C₈	1	12	yes	—
89	D₄×C₆₇	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₆₇	—	2	6	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	2	12	no	—
93	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
95	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
96	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₆₇	—	2	4	no	—
98	C₈×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
99	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₆₇	—	2	4	no	—
102	C₂²×C₆₇	—	4	3	no	—
103	C₂²×C₆₇	—	8	6	no	—
104	C₂²×C₆₇	C₂×C₂²⋊C₈	1	6	yes	—
105	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	C₂×C₂²⋊C₈	1	6	yes	—
108	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
110	#33340	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	6	no	—
113	C₂×C₆₇	—	4	4	no	—
114	C₂×C₆₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
115	#3573	C₂×C₆₇	1	1	yes	characteristic
116	#3407	C₂×C₆₇	1	1	yes	characteristic, complemented
117	C₂×#504	C₂×C₆₇	1	1	yes	characteristic, complemented
118	#2117	C₂×C₆₇	1	6	yes	complemented
119	C₂×#350	C₂×C₆₇	1	6	yes	complemented
120	C₆₇	#33340	1	1	yes	characteristic, direct product factor, Sylow
121	#585	C₄×C₆₇	1	1	yes	characteristic
122	#568	—	2	4	no	—
123	#568	C₄×C₆₇	1	6	yes	—
124	#568	C₄×C₆₇	1	1	yes	characteristic
125	C₂×#131	—	2	4	no	—
126	C₂×#131	C₂²×C₆₇	1	6	yes	—
127	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
128	#504	—	2	4	no	—
129	#504	—	2	4	no	—
130	C₂×M₃₂	C₂²×C₆₇	1	6	yes	complemented
131	C₂×M₃₂	C₂²×C₆₇	1	3	yes	complemented
132	C₂²×C₁₆	C₂²×C₆₇	1	3	yes	complemented
133	#350	C₂²×C₆₇	1	4	yes	complemented
134	#350	C₂²×C₆₇	1	12	yes	complemented
135	2₋¹⁺⁴	—	2	1	no	—
136	2₋¹⁺⁴	C₈×C₆₇	1	4	yes	—
137	H(5,𝔽₂)	—	2	3	no	—
138	H(5,𝔽₂)	C₈×C₆₇	1	4	yes	—
139	C₂×#41	—	2	4	no	—
140	C₂×#41	C₂×C₄×C₆₇	1	6	yes	—
141	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
142	#131	—	2	4	no	—
143	#131	—	2	12	no	—
144	#131	—	2	6	no	—
145	#131	—	2	6	no	—
146	#131	C₂×C₄×C₆₇	1	12	yes	—
147	#131	D₄×C₆₇	1	4	yes	—
148	#131	D₄×C₆₇	1	4	yes	—
149	#131	C₂³×C₆₇	1	4	yes	—
150	C₂×M₁₆	—	2	12	no	—
151	C₂×M₁₆	—	2	6	no	—
152	C₂×M₁₆	C₂×C₄×C₆₇	1	6	yes	—
153	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
154	C₂²×C₈	—	2	6	no	—
155	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
156	M₃₂	—	2	6	no	—
157	M₃₂	—	2	6	no	—
158	M₃₂	C₂³×C₆₇	1	4	yes	—
159	C₂×C₁₆	—	2	6	no	—
160	C₂×C₁₆	—	2	6	no	—
161	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
163	#41	—	2	4	no	—
164	#41	—	2	12	no	—
165	#41	—	2	6	no	—
166	#41	—	2	6	no	—
167	#41	C₂²⋊C₄×C₆₇	1	4	yes	—
168	#41	C₂×C₈×C₆₇	1	12	yes	—
169	#41	C₂×C₈×C₆₇	1	4	yes	—
170	#41	C₂²×C₄×C₆₇	1	4	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂×C₈×C₆₇	1	6	yes	—
173	C₂×Q₈	M₁₆×C₆₇	1	1	yes	characteristic
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	2	12	no	—
176	C₂×D₄	C₂×C₈×C₆₇	1	6	yes	—
177	C₂×D₄	M₁₆×C₆₇	1	3	yes	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
180	M₁₆	—	2	6	no	—
181	M₁₆	—	2	12	no	—
182	M₁₆	—	2	4	no	—
183	M₁₆	C₂²⋊C₄×C₆₇	1	12	yes	—
184	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
185	C₂×C₈	—	2	6	no	—
186	C₂×C₈	—	2	12	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
189	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
190	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
191	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
192	C₁₆	—	2	1	no	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	2	3	no	—
195	C₂³	—	2	6	no	—
196	C₂³	C₂×M₁₆×C₆₇	1	3	yes	—
197	Q₈	—	2	6	no	—
198	Q₈	C₂²⋊C₈×C₆₇	1	4	yes	—
199	Q₈	C₂²×C₈×C₆₇	1	4	yes	—
200	D₄	—	2	4	no	—
201	D₄	—	2	12	no	—
202	D₄	—	2	6	no	—
203	D₄	C₂²⋊C₈×C₆₇	1	12	yes	—
204	D₄	C₂²×C₈×C₆₇	1	4	yes	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	12	no	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
209	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
210	C₂×C₄	C₂×M₁₆×C₆₇	1	3	yes	—
211	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
214	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
215	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
216	C₂²	—	2	4	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	8	6	no	—
219	C₂²	C₂×C₂²⋊C₈×C₆₇	1	6	yes	—
220	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	C₂×C₂²⋊C₈×C₆₇	1	6	yes	—
223	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
224	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂²×C₂²⋊C₈×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂²×C₂²⋊C₈
prime radical	#585×C₆₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3407	C₂	semidirect
3. C₂×#504×C₆₇	C₂	semidirect
4. C₆₇×#2117	C₂	semidirect
5. C₂×#350×C₆₇	C₂	semidirect
15. C₂×M₃₂×C₆₇	C₂²	semidirect
16. C₂×M₃₂×C₆₇	C₂²	semidirect
17. C₂²×C₁₆×C₆₇	C₂²	semidirect
18. #350×C₆₇	C₂²	semidirect
19. #350×C₆₇	C₂²	semidirect
110. #33340	C₆₇	direct (N × H)
116. #3407	C₂×C₆₇	semidirect
117. C₂×#504	C₂×C₆₇	semidirect
118. #2117	C₂×C₆₇	semidirect
119. C₂×#350	C₂×C₆₇	semidirect
120. C₆₇	#33340	direct (N × H)
130. C₂×M₃₂	C₂²×C₆₇	semidirect
131. C₂×M₃₂	C₂²×C₆₇	semidirect
132. C₂²×C₁₆	C₂²×C₆₇	semidirect
133. #350	C₂²×C₆₇	semidirect
134. #350	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3573	2
C₆₇×#3407	2
C₂×#504×C₆₇	2
C₆₇×#2117	2
C₂×#350×C₆₇	2
#33340	67

Subgroups — C67×#33340 #30712591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33340 #30712591