Subgroups — C₆₇×#33419 #30712670

Order 17152.

590 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×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2164	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2150	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2112	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
10	#405×C₆₇	C₄	1	4	yes	—
11	#405×C₆₇	C₂²	1	4	yes	—
12	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₆₇	C₂²	1	8	yes	complemented
17	C₂²⋊C₁₆×C₆₇	C₂²	1	8	yes	—
18	#346×C₆₇	C₂²	1	4	yes	complemented
19	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₆₇	—	2	4	no	—
22	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₇	—	2	4	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	#117×C₆₇	C₈	1	4	yes	—
31	#117×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₆₇	—	2	8	no	—
37	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
38	#97×C₆₇	C₂×C₄	1	4	yes	—
39	#97×C₆₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₄×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	8	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₆₇	#41	1	4	yes	—
64	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
66	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
67	C₄²×C₆₇	C₂×C₈	1	2	yes	—
68	C₄²×C₆₇	C₂×C₈	1	2	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
71	C₁₆×C₆₇	—	2	8	no	—
72	C₂³×C₆₇	—	2	2	no	—
73	C₂³×C₆₇	—	4	2	no	—
74	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
75	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	—	4	2	no	—
82	C₂×C₄×C₆₇	#117	1	2	yes	—
83	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	—	2	2	no	—
91	C₈×C₆₇	#117	1	2	yes	—
92	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
93	C₈×C₆₇	C₂×#41	1	2	yes	—
94	C₂²×C₆₇	—	2	2	no	—
95	C₂²×C₆₇	—	2	4	no	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	4	4	no	—
98	C₂²×C₆₇	#431	1	2	yes	—
99	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
100	C₄×C₆₇	—	2	4	no	—
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	2	2	no	—
103	C₄×C₆₇	#431	1	2	yes	—
104	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
105	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
106	#33419	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₆₇	—	2	2	no	—
108	C₂×C₆₇	—	4	2	no	—
109	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
110	C₂×C₆₇	#3407	1	2	yes	—
111	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
112	#2164	C₂×C₆₇	1	8	yes	complemented
113	#2150	C₂×C₆₇	1	2	yes	complemented
114	#2112	C₂×C₆₇	1	2	yes	—
115	C₄×M₃₂	C₂×C₆₇	1	2	yes	complemented
116	C₆₇	#33419	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×M₁₆	C₄×C₆₇	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
119	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
121	#405	C₄×C₆₇	1	4	yes	—
122	#405	C₂²×C₆₇	1	4	yes	—
123	C₂×#97	C₄×C₆₇	1	1	yes	characteristic
124	C₂×#97	C₄×C₆₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
127	C₄⋊C₁₆	C₂²×C₆₇	1	8	yes	complemented
128	C₂²⋊C₁₆	C₂²×C₆₇	1	8	yes	—
129	#346	C₂²×C₆₇	1	4	yes	complemented
130	C₄×C₁₆	C₂²×C₆₇	1	4	yes	complemented
131	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
140	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
141	#117	C₈×C₆₇	1	4	yes	—
142	#117	C₂×C₄×C₆₇	1	4	yes	—
143	C₂×C₄⋊C₄	C₈×C₆₇	1	2	yes	—
144	C₂×C₂²⋊C₄	C₈×C₆₇	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
147	M₃₂	—	2	8	no	—
148	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
149	#97	C₂×C₄×C₆₇	1	4	yes	—
150	#97	C₂×C₄×C₆₇	1	4	yes	—
151	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
152	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
153	C₄×C₈	C₂³×C₆₇	1	2	yes	—
154	C₄×C₈	C₂³×C₆₇	1	2	yes	—
155	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
160	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
161	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
162	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	M₁₆×C₆₇	1	4	yes	—
170	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
171	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
172	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
174	C₂×C₈	#41×C₆₇	1	4	yes	—
175	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
177	C₂²⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
178	C₄²	C₂×C₈×C₆₇	1	2	yes	—
179	C₄²	C₂×C₈×C₆₇	1	2	yes	—
180	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
181	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
182	C₁₆	—	2	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
186	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#117×C₆₇	1	2	yes	—
194	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
195	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
196	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
197	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
199	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	#117×C₆₇	1	2	yes	—
203	C₈	C₂×M₁₆×C₆₇	1	4	yes	—
204	C₈	C₂×#41×C₆₇	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#431×C₆₇	1	2	yes	—
210	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	#431×C₆₇	1	2	yes	—
215	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
216	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#431×C₆₇	1	1	yes	characteristic
220	C₂	C₆₇×#3407	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2164	C₂	semidirect
3. C₆₇×#2150	C₂	semidirect
5. C₄×M₃₂×C₆₇	C₂	semidirect
16. C₄⋊C₁₆×C₆₇	C₂²	semidirect
18. #346×C₆₇	C₂²	semidirect
19. C₄×C₁₆×C₆₇	C₂²	semidirect
106. #33419	C₆₇	direct (N × H)
112. #2164	C₂×C₆₇	semidirect
113. #2150	C₂×C₆₇	semidirect
115. C₄×M₃₂	C₂×C₆₇	semidirect
116. C₆₇	#33419	direct (N × H)
127. C₄⋊C₁₆	C₂²×C₆₇	semidirect
129. #346	C₂²×C₆₇	semidirect
130. C₄×C₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405×C₆₇	2
C₆₇×#2164	2
C₆₇×#2150	2
C₆₇×#2112	2
C₄×M₃₂×C₆₇	2
#33419	67

Subgroups — C67×#33419 #30712670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33419 #30712670