Subgroups — C₂×C₆₇×#2264 #30713231

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2264	C₂	1	8	yes	maximal, direct product factor
4	C₂×Q₆₄×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×SD₆₄×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₆₇	—	2	1	no	—
8	C₂²×Q₁₆×C₆₇	—	2	1	no	—
9	#508×C₆₇	—	2	2	no	—
10	#508×C₆₇	C₂²	1	4	yes	—
11	C₂×Q₃₂×C₆₇	—	2	1	no	—
12	C₂×Q₃₂×C₆₇	—	2	2	no	—
13	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	complemented
16	C₂×SD₃₂×C₆₇	—	2	1	no	—
17	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	Q₆₄×C₆₇	C₂²	1	8	yes	complemented
20	SD₆₄×C₆₇	C₂²	1	8	yes	complemented
21	M₆₄×C₆₇	C₂²	1	4	yes	complemented
22	C₂×C₃₂×C₆₇	C₂²	1	2	yes	—
23	C₂×#41×C₆₇	—	4	1	no	—
24	C₂²×Q₈×C₆₇	—	4	1	no	—
25	#135×C₆₇	—	2	4	no	—
26	#135×C₆₇	—	4	2	no	—
27	C₂×Q₁₆×C₆₇	—	2	1	no	—
28	C₂×Q₁₆×C₆₇	—	2	2	no	—
29	C₂×Q₁₆×C₆₇	—	2	4	no	—
30	C₂×Q₁₆×C₆₇	—	4	1	no	—
31	C₂×Q₁₆×C₆₇	—	4	2	no	—
32	C₂×SD₁₆×C₆₇	—	4	1	no	—
33	C₂×D₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	Q₃₂×C₆₇	—	2	2	no	—
36	Q₃₂×C₆₇	—	2	4	no	—
37	Q₃₂×C₆₇	—	2	1	no	—
38	Q₃₂×C₆₇	C₂³	1	4	yes	—
39	Q₃₂×C₆₇	C₂³	1	2	yes	—
40	SD₃₂×C₆₇	—	2	4	no	—
41	D₁₆×C₆₇	—	2	1	no	—
42	D₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
45	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
47	C₃₂×C₆₇	C₂³	1	4	yes	—
48	#41×C₆₇	—	4	4	no	—
49	#41×C₆₇	—	8	2	no	—
50	C₂×Q₈×C₆₇	—	4	1	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	8	2	no	—
54	C₂×Q₈×C₆₇	—	8	1	no	—
55	C₂×D₄×C₆₇	—	4	1	no	—
56	C₂×D₄×C₆₇	—	8	1	no	—
57	C₂²×C₄×C₆₇	—	8	1	no	—
58	C₂²×C₄×C₆₇	—	8	1	no	—
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	Q₁₆×C₆₇	—	2	2	no	—
61	Q₁₆×C₆₇	—	2	4	no	—
62	Q₁₆×C₆₇	—	4	2	no	—
63	Q₁₆×C₆₇	—	4	4	no	—
64	Q₁₆×C₆₇	—	4	1	no	—
65	SD₁₆×C₆₇	—	4	4	no	—
66	D₈×C₆₇	—	2	2	no	—
67	D₈×C₆₇	—	4	1	no	—
68	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
69	C₂×C₈×C₆₇	D₈	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
72	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
74	C₁₆×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₂³×C₆₇	—	8	1	no	—
76	C₂³×C₆₇	D₁₆	1	1	yes	characteristic
77	Q₈×C₆₇	—	4	2	no	—
78	Q₈×C₆₇	—	4	4	no	—
79	Q₈×C₆₇	—	8	4	no	—
80	Q₈×C₆₇	—	8	2	no	—
81	Q₈×C₆₇	—	8	1	no	—
82	D₄×C₆₇	—	4	2	no	—
83	D₄×C₆₇	—	8	4	no	—
84	D₄×C₆₇	—	8	1	no	—
85	C₂×C₄×C₆₇	—	8	4	no	—
86	C₂×C₄×C₆₇	—	8	1	no	—
87	C₂×C₄×C₆₇	—	8	2	no	—
88	C₂×C₄×C₆₇	—	8	4	no	—
89	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
90	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
94	C₈×C₆₇	C₂×D₈	1	2	yes	—
95	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	8	2	no	—
98	C₂²×C₆₇	—	16	2	no	—
99	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
100	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
101	C₄×C₆₇	—	8	2	no	—
102	C₄×C₆₇	—	8	4	no	—
103	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
104	C₄×C₆₇	C₂×D₁₆	1	1	yes	characteristic
105	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
106	C₂×#2264	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₆₇	—	2	2	no	—
108	C₂×C₆₇	—	16	2	no	—
109	C₂×C₆₇	#2264	1	2	yes	direct product factor
110	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
111	C₂×#508	C₂×C₆₇	1	1	yes	characteristic
112	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
113	#2264	C₂×C₆₇	1	8	yes	direct product factor
114	C₂×Q₆₄	C₂×C₆₇	1	2	yes	complemented
115	C₂×SD₆₄	C₂×C₆₇	1	2	yes	complemented
116	C₂×M₆₄	C₂×C₆₇	1	1	yes	characteristic, complemented
117	C₆₇	C₂×#2264	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×#135	—	2	1	no	—
119	C₂²×Q₁₆	—	2	1	no	—
120	#508	—	2	2	no	—
121	#508	C₂²×C₆₇	1	4	yes	—
122	C₂×Q₃₂	—	2	1	no	—
123	C₂×Q₃₂	—	2	2	no	—
124	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
125	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
126	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	complemented
127	C₂×SD₃₂	—	2	1	no	—
128	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
130	Q₆₄	C₂²×C₆₇	1	8	yes	complemented
131	SD₆₄	C₂²×C₆₇	1	8	yes	complemented
132	M₆₄	C₂²×C₆₇	1	4	yes	complemented
133	C₂×C₃₂	C₂²×C₆₇	1	2	yes	—
134	C₂×#41	—	4	1	no	—
135	C₂²×Q₈	—	4	1	no	—
136	#135	—	2	4	no	—
137	#135	—	4	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	—	4	1	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×SD₁₆	—	4	1	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
146	Q₃₂	—	2	2	no	—
147	Q₃₂	—	2	4	no	—
148	Q₃₂	—	2	1	no	—
149	Q₃₂	C₂³×C₆₇	1	4	yes	—
150	Q₃₂	C₂³×C₆₇	1	2	yes	—
151	SD₃₂	—	2	4	no	—
152	D₁₆	—	2	1	no	—
153	D₁₆	C₂³×C₆₇	1	2	yes	—
154	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
155	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
156	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
157	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
158	C₃₂	C₂³×C₆₇	1	4	yes	—
159	#41	—	4	4	no	—
160	#41	—	8	2	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	8	1	no	—
165	C₂×Q₈	—	8	2	no	—
166	C₂×D₄	—	4	1	no	—
167	C₂×D₄	—	8	1	no	—
168	C₂²×C₄	—	8	1	no	—
169	C₂²×C₄	—	8	1	no	—
170	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
171	Q₁₆	—	2	2	no	—
172	Q₁₆	—	2	4	no	—
173	Q₁₆	—	4	2	no	—
174	Q₁₆	—	4	4	no	—
175	Q₁₆	—	4	1	no	—
176	SD₁₆	—	4	4	no	—
177	D₈	—	2	2	no	—
178	D₈	—	4	1	no	—
179	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
180	C₂×C₈	D₈×C₆₇	1	2	yes	—
181	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
183	C₁₆	C₂×D₄×C₆₇	1	1	yes	characteristic
184	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
185	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
186	C₂³	—	8	1	no	—
187	C₂³	D₁₆×C₆₇	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	4	no	—
191	Q₈	—	8	2	no	—
192	Q₈	—	8	1	no	—
193	D₄	—	4	2	no	—
194	D₄	—	8	4	no	—
195	D₄	—	8	1	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	D₁₆×C₆₇	1	1	yes	characteristic
201	C₂×C₄	D₁₆×C₆₇	1	2	yes	—
202	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
204	C₈	C₂×D₈×C₆₇	1	1	yes	characteristic
205	C₈	C₂×D₈×C₆₇	1	2	yes	—
206	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	16	2	no	—
210	C₂²	C₂×D₁₆×C₆₇	1	2	yes	—
211	C₂²	C₂×D₁₆×C₆₇	1	1	yes	characteristic
212	C₄	—	8	2	no	—
213	C₄	—	8	4	no	—
214	C₄	C₂×D₁₆×C₆₇	1	2	yes	—
215	C₄	C₂×D₁₆×C₆₇	1	1	yes	characteristic
216	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	16	2	no	—
219	C₂	C₆₇×#2264	1	2	yes	direct product factor
220	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₆₇
prime radical	C₂×#508×C₆₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ C₂²×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₃₂×C₆₇	C₂	semidirect
3. C₆₇×#2264	C₂	direct (N × H)
4. C₂×Q₆₄×C₆₇	C₂	semidirect
5. C₂×SD₆₄×C₆₇	C₂	semidirect
6. C₂×M₆₄×C₆₇	C₂	semidirect
15. C₂×Q₃₂×C₆₇	C₂²	semidirect
19. Q₆₄×C₆₇	C₂²	semidirect
20. SD₆₄×C₆₇	C₂²	semidirect
21. M₆₄×C₆₇	C₂²	semidirect
106. C₂×#2264	C₆₇	direct (N × H)
109. C₂×C₆₇	#2264	direct (N × H)
112. C₂²×Q₃₂	C₂×C₆₇	semidirect
113. #2264	C₂×C₆₇	direct (N × H)
114. C₂×Q₆₄	C₂×C₆₇	semidirect
115. C₂×SD₆₄	C₂×C₆₇	semidirect
116. C₂×M₆₄	C₂×C₆₇	semidirect
117. C₆₇	C₂×#2264	direct (N × H)
126. C₂×Q₃₂	C₂²×C₆₇	semidirect
130. Q₆₄	C₂²×C₆₇	semidirect
131. SD₆₄	C₂²×C₆₇	semidirect
132. M₆₄	C₂²×C₆₇	semidirect
219. C₂	C₆₇×#2264	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₂²×Q₃₂×C₆₇	2
C₆₇×#2264	2
C₂×Q₆₄×C₆₇	2
C₂×SD₆₄×C₆₇	2
C₂×M₆₄×C₆₇	2
C₂×#2264	67

Subgroups — C2×C67×#2264 #30713231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2264 #30713231