Subgroups — C₂×C₆₇×#2177 #30712701

Order 17152.

798 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2177	C₂	1	8	yes	maximal, direct product factor
4	C₂×#367×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#357×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₆₇	—	2	2	no	—
9	C₂×D₁₆×C₆₇	C₄	1	8	yes	complemented
10	C₂×D₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×D₈×C₆₇	—	2	4	no	—
14	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#102×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#367×C₆₇	C₂²	1	4	yes	complemented
19	#357×C₆₇	C₂²	1	8	yes	complemented
20	#346×C₆₇	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂³×C₄×C₆₇	—	4	2	no	—
23	C₂×D₈×C₆₇	—	2	2	no	—
24	C₂×D₈×C₆₇	—	2	4	no	—
25	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
26	C₂×D₈×C₆₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₆₇	—	2	8	no	—
30	C₄×D₄×C₆₇	—	4	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	D₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
37	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
39	#107×C₆₇	—	2	2	no	—
40	#107×C₆₇	C₂³	1	4	yes	—
41	#102×C₆₇	—	2	8	no	—
42	C₄×C₈×C₆₇	D₄	1	2	yes	—
43	C₄×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂⁴×C₆₇	—	4	2	no	—
45	C₂×D₄×C₆₇	—	2	8	no	—
46	C₂×D₄×C₆₇	—	2	4	no	—
47	C₂×D₄×C₆₇	—	4	2	no	—
48	C₂×D₄×C₆₇	—	4	4	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	—	4	8	no	—
51	C₂²×C₄×C₆₇	—	8	8	no	—
52	C₂²×C₄×C₆₇	D₈	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	D₈×C₆₇	—	2	8	no	—
55	D₈×C₆₇	—	2	4	no	—
56	D₈×C₆₇	C₂²×C₄	1	8	yes	complemented
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₂²⋊C₄×C₆₇	—	4	8	no	—
66	C₄²×C₆₇	D₈	1	2	yes	—
67	C₄²×C₆₇	C₂×D₄	1	2	yes	—
68	C₁₆×C₆₇	—	2	2	no	—
69	C₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
70	C₂³×C₆₇	—	4	4	no	—
71	C₂³×C₆₇	—	4	8	no	—
72	C₂³×C₆₇	—	8	4	no	—
73	C₂³×C₆₇	—	8	4	no	—
74	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
75	D₄×C₆₇	—	2	8	no	—
76	D₄×C₆₇	—	4	4	no	—
77	D₄×C₆₇	—	4	8	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	8	16	no	—
81	C₂×C₄×C₆₇	—	8	4	no	—
82	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#135	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	C₄×D₄	1	2	yes	—
89	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
91	C₂²×C₆₇	—	4	8	no	—
92	C₂²×C₆₇	—	8	4	no	—
93	C₂²×C₆₇	—	8	4	no	—
94	C₂²×C₆₇	—	8	16	no	—
95	C₂²×C₆₇	C₄×D₈	1	2	yes	—
96	C₂²×C₆₇	#509	1	4	yes	—
97	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₇	—	2	4	no	—
99	C₄×C₆₇	—	8	4	no	—
100	C₄×C₆₇	C₄×D₈	1	2	yes	—
101	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
103	C₂×#2177	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	—	8	8	no	—
105	C₂×C₆₇	#2177	1	4	yes	direct product factor
106	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
107	C₂×C₆₇	C₂×#509	1	2	yes	—
108	C₂²×D₁₆	C₂×C₆₇	1	1	yes	characteristic
109	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	complemented
110	#2177	C₂×C₆₇	1	8	yes	direct product factor
111	C₂×#367	C₂×C₆₇	1	1	yes	characteristic, complemented
112	C₂×#357	C₂×C₆₇	1	2	yes	complemented
113	C₂×#346	C₂×C₆₇	1	1	yes	characteristic, complemented
114	C₆₇	C₂×#2177	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
116	C₂×C₄×D₄	—	2	2	no	—
117	C₂×D₁₆	C₄×C₆₇	1	8	yes	complemented
118	C₂×D₁₆	C₂²×C₆₇	1	4	yes	—
119	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
120	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
121	C₄×D₈	—	2	4	no	—
122	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
123	C₂×#107	C₂²×C₆₇	1	2	yes	—
124	C₂×#102	—	2	2	no	—
125	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	#367	C₂²×C₆₇	1	4	yes	complemented
127	#357	C₂²×C₆₇	1	8	yes	complemented
128	#346	C₂²×C₆₇	1	4	yes	complemented
129	C₂²×D₄	—	2	2	no	—
130	C₂³×C₄	—	4	2	no	—
131	C₂×D₈	—	2	2	no	—
132	C₂×D₈	—	2	4	no	—
133	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
134	C₂×D₈	C₂³×C₆₇	1	4	yes	—
135	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
137	C₄×D₄	—	2	8	no	—
138	C₄×D₄	—	4	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₂²⋊C₄	—	4	2	no	—
141	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
142	D₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
145	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
146	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
147	#107	—	2	2	no	—
148	#107	C₂³×C₆₇	1	4	yes	—
149	#102	—	2	8	no	—
150	C₄×C₈	D₄×C₆₇	1	2	yes	—
151	C₄×C₈	C₂³×C₆₇	1	2	yes	—
152	C₂⁴	—	4	2	no	—
153	C₂×D₄	—	2	8	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	8	8	no	—
160	C₂²×C₄	D₈×C₆₇	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
162	D₈	—	2	8	no	—
163	D₈	—	2	4	no	—
164	D₈	C₂²×C₄×C₆₇	1	8	yes	complemented
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
167	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
168	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
169	C₂×C₈	#41×C₆₇	1	2	yes	—
170	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	8	no	—
174	C₄²	D₈×C₆₇	1	2	yes	—
175	C₄²	C₂×D₄×C₆₇	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²×C₄×C₆₇	1	4	yes	complemented
178	C₂³	—	4	4	no	—
179	C₂³	—	4	8	no	—
180	C₂³	—	8	4	no	—
181	C₂³	—	8	4	no	—
182	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
183	D₄	—	2	8	no	—
184	D₄	—	4	4	no	—
185	D₄	—	4	8	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	8	16	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
192	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
193	C₂×C₄	#135×C₆₇	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	C₄×D₄×C₆₇	1	2	yes	—
197	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
198	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
199	C₂²	—	4	8	no	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	4	no	—
202	C₂²	—	8	16	no	—
203	C₂²	C₄×D₈×C₆₇	1	2	yes	—
204	C₂²	#509×C₆₇	1	4	yes	—
205	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	8	4	no	—
208	C₄	C₄×D₈×C₆₇	1	2	yes	—
209	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
210	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
211	C₂	—	8	8	no	—
212	C₂	C₆₇×#2177	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
214	C₂	C₂×#509×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×D₁₆×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₆₇	C₂	semidirect
3. C₆₇×#2177	C₂	direct (N × H)
4. C₂×#367×C₆₇	C₂	semidirect
5. C₂×#357×C₆₇	C₂	semidirect
6. C₂×#346×C₆₇	C₂	semidirect
9. C₂×D₁₆×C₆₇	C₄	semidirect
14. C₄×D₈×C₆₇	C₂²	semidirect
18. #367×C₆₇	C₂²	semidirect
19. #357×C₆₇	C₂²	semidirect
20. #346×C₆₇	C₂²	semidirect
25. C₂×D₈×C₆₇	C₂×C₄	semidirect
34. D₁₆×C₆₇	C₂×C₄	semidirect
36. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
56. D₈×C₆₇	C₂²×C₄	semidirect
69. C₁₆×C₆₇	C₂²×C₄	semidirect
103. C₂×#2177	C₆₇	direct (N × H)
105. C₂×C₆₇	#2177	direct (N × H)
109. C₂×C₄×D₈	C₂×C₆₇	semidirect
110. #2177	C₂×C₆₇	direct (N × H)
111. C₂×#367	C₂×C₆₇	semidirect
112. C₂×#357	C₂×C₆₇	semidirect
113. C₂×#346	C₂×C₆₇	semidirect
114. C₆₇	C₂×#2177	direct (N × H)
117. C₂×D₁₆	C₄×C₆₇	semidirect
122. C₄×D₈	C₂²×C₆₇	semidirect
126. #367	C₂²×C₆₇	semidirect
127. #357	C₂²×C₆₇	semidirect
128. #346	C₂²×C₆₇	semidirect
133. C₂×D₈	C₂×C₄×C₆₇	semidirect
142. D₁₆	C₂×C₄×C₆₇	semidirect
144. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
164. D₈	C₂²×C₄×C₆₇	semidirect
177. C₁₆	C₂²×C₄×C₆₇	semidirect
212. C₂	C₆₇×#2177	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×D₁₆×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2177	2
C₂×#367×C₆₇	2
C₂×#357×C₆₇	2
C₂×#346×C₆₇	2
C₂×#2177	67

Subgroups — C2×C67×#2177 #30712701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2177 #30712701