Subgroups — C₂×C₆₇×#2216 #30712849

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#467×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2216	C₂	1	8	yes	maximal, direct product factor
5	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#358×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
10	C₂×Q₃₂×C₆₇	—	2	4	no	—
11	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	complemented
12	C₂²×C₁₆×C₆₇	—	2	4	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#467×C₆₇	—	2	4	no	—
17	#467×C₆₇	C₂²	1	8	yes	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	—	2	2	no	—
20	#366×C₆₇	C₂²	1	4	yes	complemented
21	#358×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂×Q₁₆×C₆₇	—	2	4	no	—
25	C₂×Q₁₆×C₆₇	—	2	2	no	—
26	C₂×Q₁₆×C₆₇	—	2	4	no	—
27	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²⋊Q₈×C₆₇	—	2	8	no	—
35	C₂²⋊Q₈×C₆₇	—	4	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
38	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
39	Q₃₂×C₆₇	—	2	8	no	—
40	C₂×C₁₆×C₆₇	—	2	2	no	—
41	C₂×C₁₆×C₆₇	—	2	8	no	—
42	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
43	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
45	#107×C₆₇	—	2	2	no	—
46	#107×C₆₇	C₂³	1	4	yes	—
47	#103×C₆₇	—	2	8	no	—
48	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂×Q₈×C₆₇	—	4	4	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	D₈	1	2	yes	—
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₆₇	—	2	4	no	—
62	Q₁₆×C₆₇	—	2	4	no	—
63	Q₁₆×C₆₇	—	4	4	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₄⋊C₄×C₆₇	—	4	8	no	—
76	C₂²⋊C₄×C₆₇	—	4	8	no	—
77	C₁₆×C₆₇	—	2	2	no	—
78	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	Q₃₂	1	4	yes	complemented
82	C₂³×C₆₇	C₂×D₈	1	2	yes	—
83	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₆₇	—	4	4	no	—
85	Q₈×C₆₇	—	4	4	no	—
86	Q₈×C₆₇	—	8	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	4	4	no	—
92	C₂×C₄×C₆₇	—	4	4	no	—
93	C₂×C₄×C₆₇	—	8	4	no	—
94	C₂×C₄×C₆₇	—	8	4	no	—
95	C₂×C₄×C₆₇	#121	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#135	1	2	yes	—
99	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	#121	1	2	yes	—
102	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	—	2	8	no	—
106	C₂²×C₆₇	—	2	2	no	—
107	C₂²×C₆₇	#466	1	2	yes	—
108	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
109	C₂²×C₆₇	C₂×Q₃₂	1	4	yes	complemented
110	C₂²×C₆₇	#508	1	2	yes	—
111	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	8	4	no	—
114	C₄×C₆₇	—	8	4	no	—
115	C₄×C₆₇	#466	1	2	yes	—
116	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
117	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
118	C₂×#2216	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₆₇	—	2	4	no	—
120	C₂×C₆₇	#2216	1	4	yes	direct product factor
121	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
122	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
123	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
124	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
125	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic
126	C₂×#467	C₂×C₆₇	1	2	yes	—
127	#2216	C₂×C₆₇	1	8	yes	direct product factor
128	C₂×#366	C₂×C₆₇	1	1	yes	characteristic, complemented
129	C₂×#358	C₂×C₆₇	1	2	yes	complemented
130	C₆₇	C₂×#2216	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
132	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
133	C₂×C₂²⋊Q₈	—	2	2	no	—
134	C₂×Q₃₂	—	2	4	no	—
135	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	complemented
136	C₂²×C₁₆	—	2	4	no	—
137	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
138	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	—
139	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
140	#467	—	2	4	no	—
141	#467	C₂²×C₆₇	1	8	yes	—
142	C₂×#107	C₂²×C₆₇	1	2	yes	—
143	C₂×#103	—	2	2	no	—
144	#366	C₂²×C₆₇	1	4	yes	complemented
145	#358	C₂²×C₆₇	1	8	yes	complemented
146	C₂²×Q₈	—	2	2	no	—
147	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	—	2	4	no	—
151	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
155	C₂²×C₈	D₄×C₆₇	1	2	yes	—
156	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
157	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
158	C₂²⋊Q₈	—	2	8	no	—
159	C₂²⋊Q₈	—	4	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	4	2	no	—
162	C₂×C₂²⋊C₄	—	4	2	no	—
163	Q₃₂	—	2	8	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	8	no	—
166	C₂×C₁₆	D₄×C₆₇	1	4	yes	—
167	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
168	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
169	#107	—	2	2	no	—
170	#107	C₂³×C₆₇	1	4	yes	—
171	#103	—	2	8	no	—
172	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
182	C₂²×C₄	D₈×C₆₇	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	4	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
195	C₂×C₈	#41×C₆₇	1	2	yes	—
196	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	Q₃₂×C₆₇	1	4	yes	complemented
206	C₂³	C₂×D₈×C₆₇	1	2	yes	—
207	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	4	4	no	—
210	Q₈	—	8	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#121×C₆₇	1	2	yes	—
220	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
221	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
222	C₂×C₄	#135×C₆₇	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	#121×C₆₇	1	2	yes	—
226	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
227	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	8	no	—
231	C₂²	#466×C₆₇	1	2	yes	—
232	C₂²	C₂×Q₃₂×C₆₇	1	2	yes	—
233	C₂²	C₂×Q₃₂×C₆₇	1	4	yes	complemented
234	C₂²	#508×C₆₇	1	2	yes	—
235	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	#466×C₆₇	1	2	yes	—
240	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
241	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	C₆₇×#2216	1	4	yes	direct product factor
244	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
245	C₂	C₂²×Q₃₂×C₆₇	1	1	yes	characteristic
246	C₂	C₂×#508×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₆₇	C₂	semidirect
4. C₆₇×#2216	C₂	direct (N × H)
5. C₂×#366×C₆₇	C₂	semidirect
6. C₂×#358×C₆₇	C₂	semidirect
11. C₂×Q₃₂×C₆₇	C₂²	semidirect
20. #366×C₆₇	C₂²	semidirect
21. #358×C₆₇	C₂²	semidirect
81. C₂³×C₆₇	Q₃₂	semidirect
109. C₂²×C₆₇	C₂×Q₃₂	semidirect
118. C₂×#2216	C₆₇	direct (N × H)
120. C₂×C₆₇	#2216	direct (N × H)
124. C₂²×Q₃₂	C₂×C₆₇	semidirect
127. #2216	C₂×C₆₇	direct (N × H)
128. C₂×#366	C₂×C₆₇	semidirect
129. C₂×#358	C₂×C₆₇	semidirect
130. C₆₇	C₂×#2216	direct (N × H)
135. C₂×Q₃₂	C₂²×C₆₇	semidirect
144. #366	C₂²×C₆₇	semidirect
145. #358	C₂²×C₆₇	semidirect
205. C₂³	Q₃₂×C₆₇	semidirect
233. C₂²	C₂×Q₃₂×C₆₇	semidirect
243. C₂	C₆₇×#2216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₆₇	2
C₂³×C₁₆×C₆₇	2
C₂×#467×C₆₇	2
C₆₇×#2216	2
C₂×#366×C₆₇	2
C₂×#358×C₆₇	2
C₂×#2216	67

Subgroups — C2×C67×#2216 #30712849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2216 #30712849