Subgroups — C₆₇×#59816 #30739067

Order 17152.

4828 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	C₂×D₄²×C₆₇	C₂	1	6	yes	maximal, complemented
2	C₆₇×#2679	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2537	C₂	1	6	yes	maximal, complemented
4	C₆₇×#2408	C₂	1	12	yes	maximal, complemented
5	C₄×#127×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂³×D₄×C₆₇	C₂²	1	12	yes	complemented
7	D₄²×C₆₇	—	2	48	no	—
8	C₂×#127×C₆₇	C₂²	1	3	yes	—
9	C₂×#121×C₆₇	C₂²	1	24	yes	complemented
10	C₂×#120×C₆₇	C₂²	1	24	yes	complemented
11	C₂×C₄×D₄×C₆₇	C₂²	1	12	yes	complemented
12	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	complemented
13	#394×C₆₇	C₂²	1	24	yes	complemented
14	#392×C₆₇	C₂²	1	8	yes	complemented
15	#390×C₆₇	C₂²	1	12	yes	complemented
16	#390×C₆₇	C₂²	1	6	yes	complemented
17	#384×C₆₇	C₂²	1	3	yes	complemented
18	#380×C₆₇	C₂²	1	12	yes	complemented
19	#379×C₆₇	C₂²	1	8	yes	complemented
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	C₂⁵×C₆₇	C₂³	1	8	yes	—
22	C₂²×D₄×C₆₇	—	2	96	no	—
23	C₂²×D₄×C₆₇	—	4	24	no	—
24	C₂²×D₄×C₆₇	C₂³	1	24	yes	—
25	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
26	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
28	#127×C₆₇	D₄	1	24	yes	complemented
29	#121×C₆₇	—	2	96	no	—
30	#120×C₆₇	—	2	96	no	—
31	C₄×D₄×C₆₇	—	2	48	no	—
32	C₄×D₄×C₆₇	—	4	12	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
35	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂³	1	12	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
41	#95×C₆₇	C₂³	1	8	yes	—
42	C₂⁴×C₆₇	—	2	48	no	—
43	C₂⁴×C₆₇	—	4	24	no	—
44	C₂⁴×C₆₇	—	4	8	no	—
45	C₂⁴×C₆₇	C₂⁴	1	12	yes	—
46	C₂×D₄×C₆₇	—	2	96	no	—
47	C₂×D₄×C₆₇	—	2	24	no	—
48	C₂×D₄×C₆₇	—	2	24	no	—
49	C₂×D₄×C₆₇	—	4	6	no	—
50	C₂×D₄×C₆₇	—	4	24	no	—
51	C₂×D₄×C₆₇	—	4	6	no	—
52	C₂×D₄×C₆₇	—	4	192	no	—
53	C₂×D₄×C₆₇	C₂×D₄	1	48	yes	complemented
54	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	complemented
55	C₂²×C₄×C₆₇	—	2	48	no	—
56	C₂²×C₄×C₆₇	—	2	24	no	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
61	C₄⋊C₄×C₆₇	—	2	24	no	—
62	C₄⋊C₄×C₆₇	—	4	6	no	—
63	C₂²⋊C₄×C₆₇	—	2	48	no	—
64	C₂²⋊C₄×C₆₇	—	2	48	no	—
65	C₂²⋊C₄×C₆₇	—	4	12	no	—
66	C₄²×C₆₇	—	2	6	no	—
67	C₄²×C₆₇	—	4	1	no	—
68	C₄²×C₆₇	C₂×D₄	1	12	yes	complemented
69	C₂³×C₆₇	—	2	24	no	—
70	C₂³×C₆₇	—	2	12	no	—
71	C₂³×C₆₇	—	2	24	no	—
72	C₂³×C₆₇	—	2	12	no	—
73	C₂³×C₆₇	—	4	96	no	—
74	C₂³×C₆₇	—	8	48	no	—
75	C₂³×C₆₇	—	8	16	no	—
76	C₂³×C₆₇	C₂²×D₄	1	24	yes	—
77	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
78	D₄×C₆₇	—	2	96	no	—
79	D₄×C₆₇	—	4	24	no	—
80	D₄×C₆₇	—	4	96	no	—
81	C₂×C₄×C₆₇	—	2	3	no	—
82	C₂×C₄×C₆₇	—	2	3	no	—
83	C₂×C₄×C₆₇	—	2	24	no	—
84	C₂×C₄×C₆₇	—	2	3	no	—
85	C₂×C₄×C₆₇	—	2	12	no	—
86	C₂×C₄×C₆₇	—	4	48	no	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
89	C₂×C₄×C₆₇	H(5,𝔽₂)	1	6	yes	—
90	C₂²×C₆₇	—	2	48	no	—
91	C₂²×C₆₇	—	2	24	no	—
92	C₂²×C₆₇	—	4	12	no	—
93	C₂²×C₆₇	—	4	24	no	—
94	C₂²×C₆₇	—	4	12	no	—
95	C₂²×C₆₇	—	8	64	no	—
96	C₂²×C₆₇	C₂³×D₄	1	3	yes	—
97	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	3	yes	—
98	C₂²×C₆₇	#585	1	1	yes	characteristic
99	C₄×C₆₇	—	2	6	no	—
100	C₄×C₆₇	—	4	12	no	—
101	C₄×C₆₇	—	4	1	no	—
102	C₄×C₆₇	D₄²	1	12	yes	complemented
103	#59816	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	—	4	24	no	—
105	C₂×C₆₇	C₂×D₄²	1	3	yes	—
106	C₂×C₆₇	#3496	1	3	yes	—
107	C₂×C₆₇	#3549	1	1	yes	characteristic
108	C₂×D₄²	C₂×C₆₇	1	6	yes	complemented
109	#2679	C₂×C₆₇	1	4	yes	complemented
110	#2537	C₂×C₆₇	1	6	yes	complemented
111	#2408	C₂×C₆₇	1	12	yes	complemented
112	C₄×#127	C₂×C₆₇	1	3	yes	complemented
113	C₆₇	#59816	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×D₄	C₂²×C₆₇	1	12	yes	complemented
115	D₄²	—	2	48	no	—
116	C₂×#127	C₂²×C₆₇	1	3	yes	—
117	C₂×#121	C₂²×C₆₇	1	24	yes	complemented
118	C₂×#120	C₂²×C₆₇	1	24	yes	complemented
119	C₂×C₄×D₄	C₂²×C₆₇	1	12	yes	complemented
120	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	complemented
121	#394	C₂²×C₆₇	1	24	yes	complemented
122	#392	C₂²×C₆₇	1	8	yes	complemented
123	#390	C₂²×C₆₇	1	12	yes	complemented
124	#390	C₂²×C₆₇	1	6	yes	complemented
125	#384	C₂²×C₆₇	1	3	yes	complemented
126	#380	C₂²×C₆₇	1	12	yes	complemented
127	#379	C₂²×C₆₇	1	8	yes	complemented
128	C₄³	C₂²×C₆₇	1	1	yes	characteristic, complemented
129	C₂⁵	C₂³×C₆₇	1	8	yes	—
130	C₂²×D₄	—	2	96	no	—
131	C₂²×D₄	—	4	24	no	—
132	C₂²×D₄	C₂³×C₆₇	1	24	yes	—
133	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
134	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
135	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
136	#127	D₄×C₆₇	1	24	yes	complemented
137	#121	—	2	96	no	—
138	#120	—	2	96	no	—
139	C₄×D₄	—	2	48	no	—
140	C₄×D₄	—	4	12	no	—
141	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
142	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
143	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
144	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
145	C₂×C₂²⋊C₄	C₂³×C₆₇	1	12	yes	—
146	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
147	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
149	#95	C₂³×C₆₇	1	8	yes	—
150	C₂⁴	—	2	48	no	—
151	C₂⁴	—	4	24	no	—
152	C₂⁴	—	4	8	no	—
153	C₂⁴	C₂⁴×C₆₇	1	12	yes	—
154	C₂×D₄	—	2	96	no	—
155	C₂×D₄	—	2	24	no	—
156	C₂×D₄	—	2	24	no	—
157	C₂×D₄	—	4	6	no	—
158	C₂×D₄	—	4	24	no	—
159	C₂×D₄	—	4	6	no	—
160	C₂×D₄	—	4	192	no	—
161	C₂×D₄	C₂×D₄×C₆₇	1	48	yes	complemented
162	C₂×D₄	C₂×D₄×C₆₇	1	24	yes	complemented
163	C₂²×C₄	—	2	48	no	—
164	C₂²×C₄	—	2	24	no	—
165	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
166	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
167	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
168	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
169	C₄⋊C₄	—	2	24	no	—
170	C₄⋊C₄	—	4	6	no	—
171	C₂²⋊C₄	—	2	48	no	—
172	C₂²⋊C₄	—	2	48	no	—
173	C₂²⋊C₄	—	4	12	no	—
174	C₄²	—	2	6	no	—
175	C₄²	—	4	1	no	—
176	C₄²	C₂×D₄×C₆₇	1	12	yes	complemented
177	C₂³	—	2	24	no	—
178	C₂³	—	2	12	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	4	96	no	—
182	C₂³	—	8	48	no	—
183	C₂³	—	8	16	no	—
184	C₂³	C₂²×D₄×C₆₇	1	24	yes	—
185	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
186	D₄	—	2	96	no	—
187	D₄	—	4	24	no	—
188	D₄	—	4	96	no	—
189	C₂×C₄	—	2	3	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	4	48	no	—
195	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
196	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
197	C₂×C₄	H(5,𝔽₂)×C₆₇	1	6	yes	—
198	C₂²	—	2	48	no	—
199	C₂²	—	2	24	no	—
200	C₂²	—	4	12	no	—
201	C₂²	—	4	12	no	—
202	C₂²	—	4	24	no	—
203	C₂²	—	8	64	no	—
204	C₂²	C₂³×D₄×C₆₇	1	3	yes	—
205	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	3	yes	—
206	C₂²	#585×C₆₇	1	1	yes	characteristic
207	C₄	—	2	6	no	—
208	C₄	—	4	1	no	—
209	C₄	—	4	12	no	—
210	C₄	D₄²×C₆₇	1	12	yes	complemented
211	C₂	—	4	24	no	—
212	C₂	C₂×D₄²×C₆₇	1	3	yes	—
213	C₂	C₆₇×#3496	1	3	yes	—
214	C₂	C₆₇×#3549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄²×C₆₇	C₂	semidirect
2. C₆₇×#2679	C₂	semidirect
3. C₆₇×#2537	C₂	semidirect
4. C₆₇×#2408	C₂	semidirect
5. C₄×#127×C₆₇	C₂	semidirect
6. C₂³×D₄×C₆₇	C₂²	semidirect
9. C₂×#121×C₆₇	C₂²	semidirect
10. C₂×#120×C₆₇	C₂²	semidirect
11. C₂×C₄×D₄×C₆₇	C₂²	semidirect
12. C₂×C₄×D₄×C₆₇	C₂²	semidirect
13. #394×C₆₇	C₂²	semidirect
14. #392×C₆₇	C₂²	semidirect
15. #390×C₆₇	C₂²	semidirect
16. #390×C₆₇	C₂²	semidirect
17. #384×C₆₇	C₂²	semidirect
18. #380×C₆₇	C₂²	semidirect
19. #379×C₆₇	C₂²	semidirect
20. C₄³×C₆₇	C₂²	semidirect
28. #127×C₆₇	D₄	semidirect
53. C₂×D₄×C₆₇	C₂×D₄	semidirect
54. C₂×D₄×C₆₇	C₂×D₄	semidirect
68. C₄²×C₆₇	C₂×D₄	semidirect
102. C₄×C₆₇	D₄²	semidirect
103. #59816	C₆₇	direct (N × H)
108. C₂×D₄²	C₂×C₆₇	semidirect
109. #2679	C₂×C₆₇	semidirect
110. #2537	C₂×C₆₇	semidirect
111. #2408	C₂×C₆₇	semidirect
112. C₄×#127	C₂×C₆₇	semidirect
113. C₆₇	#59816	direct (N × H)
114. C₂³×D₄	C₂²×C₆₇	semidirect
117. C₂×#121	C₂²×C₆₇	semidirect
118. C₂×#120	C₂²×C₆₇	semidirect
119. C₂×C₄×D₄	C₂²×C₆₇	semidirect
120. C₂×C₄×D₄	C₂²×C₆₇	semidirect
121. #394	C₂²×C₆₇	semidirect
122. #392	C₂²×C₆₇	semidirect
123. #390	C₂²×C₆₇	semidirect
124. #390	C₂²×C₆₇	semidirect
125. #384	C₂²×C₆₇	semidirect
126. #380	C₂²×C₆₇	semidirect
127. #379	C₂²×C₆₇	semidirect
128. C₄³	C₂²×C₆₇	semidirect
136. #127	D₄×C₆₇	semidirect
161. C₂×D₄	C₂×D₄×C₆₇	semidirect
162. C₂×D₄	C₂×D₄×C₆₇	semidirect
176. C₄²	C₂×D₄×C₆₇	semidirect
210. C₄	D₄²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄²×C₆₇	2
C₆₇×#2679	2
C₆₇×#2537	2
C₆₇×#2408	2
C₄×#127×C₆₇	2
#59816	67

Subgroups — C67×#59816 #30739067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59816 #30739067