Subgroups — C₆₇×#22282 #30701533

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2932	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1963	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1876	C₂	1	8	yes	maximal, complemented
5	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₆₇	C₂²	1	4	yes	—
8	C₂×#131×C₆₇	C₂²	1	4	yes	—
9	#532×C₆₇	C₂²	1	2	yes	—
10	#495×C₆₇	C₄	1	4	yes	complemented
11	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
12	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#492×C₆₇	C₄	1	2	yes	complemented
14	#435×C₆₇	—	2	8	no	—
15	#423×C₆₇	C₂²	1	2	yes	—
16	C₂×#104×C₆₇	C₂²	1	8	yes	—
17	#408×C₆₇	C₂²	1	4	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#376×C₆₇	C₂²	1	1	yes	characteristic
21	#341×C₆₇	C₂²	1	4	yes	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	C₂×#41×C₆₇	C₂³	1	4	yes	—
24	#135×C₆₇	—	4	8	no	—
25	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
26	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
27	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
28	#131×C₆₇	—	4	8	no	—
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₇	—	2	4	no	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	#127×C₆₇	C₂×C₄	1	2	yes	—
36	#124×C₆₇	C₂×C₄	1	4	yes	—
37	#121×C₆₇	—	2	4	no	—
38	C₄×D₄×C₆₇	—	2	4	no	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	#104×C₆₇	—	4	8	no	—
43	C₂²⋊C₈×C₆₇	—	2	8	no	—
44	#97×C₆₇	D₄	1	4	yes	—
45	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	#95×C₆₇	—	2	2	no	—
49	#41×C₆₇	—	2	8	no	—
50	#41×C₆₇	—	2	8	no	—
51	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	2	4	no	—
54	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	#41	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₆₇	—	4	4	no	—
60	SD₁₆×C₆₇	—	4	8	no	—
61	D₈×C₆₇	—	4	4	no	—
62	M₁₆×C₆₇	—	2	8	no	—
63	M₁₆×C₆₇	—	4	8	no	—
64	C₂×C₈×C₆₇	—	2	4	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	4	8	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
71	C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₂²⋊C₄×C₆₇	—	2	8	no	—
73	C₄²×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	2	yes	—
77	C₂³×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
79	Q₈×C₆₇	—	2	8	no	—
80	D₄×C₆₇	—	2	8	no	—
81	D₄×C₆₇	—	4	8	no	—
82	D₄×C₆₇	—	8	4	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	4	4	no	—
87	C₂×C₄×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
94	C₈×C₆₇	—	2	2	no	—
95	C₈×C₆₇	—	2	2	no	—
96	C₈×C₆₇	—	2	4	no	—
97	C₈×C₆₇	—	4	4	no	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	4	4	no	—
100	C₂²×C₆₇	—	4	8	no	—
101	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²×C₆₇	#540	1	2	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	—	4	4	no	—
106	C₄×C₆₇	—	8	2	no	—
107	C₄×C₆₇	#518	1	2	yes	—
108	C₄×C₆₇	#535	1	2	yes	—
109	#22282	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₆₇	—	2	2	no	—
111	C₂×C₆₇	—	8	4	no	—
112	C₂×C₆₇	#2328	1	1	yes	characteristic
113	C₂×C₆₇	#2957	1	2	yes	—
114	#3147	C₂×C₆₇	1	1	yes	characteristic
115	#2932	C₂×C₆₇	1	2	yes	complemented
116	#1963	C₂×C₆₇	1	2	yes	complemented
117	#1876	C₂×C₆₇	1	8	yes	complemented
118	#1837	C₂×C₆₇	1	1	yes	characteristic, complemented
119	#1746	C₂×C₆₇	1	1	yes	characteristic, complemented
120	C₆₇	#22282	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#135	C₂²×C₆₇	1	4	yes	—
122	C₂×#131	C₂²×C₆₇	1	4	yes	—
123	#532	C₂²×C₆₇	1	2	yes	—
124	#495	C₄×C₆₇	1	4	yes	complemented
125	#494	C₄×C₆₇	1	1	yes	characteristic, complemented
126	#493	C₄×C₆₇	1	1	yes	characteristic, complemented
127	#492	C₄×C₆₇	1	2	yes	complemented
128	#435	—	2	8	no	—
129	#423	C₂²×C₆₇	1	2	yes	—
130	C₂×#104	C₂²×C₆₇	1	8	yes	—
131	#408	C₂²×C₆₇	1	4	yes	—
132	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
134	#376	C₂²×C₆₇	1	1	yes	characteristic
135	#341	C₂²×C₆₇	1	4	yes	—
136	#336	C₂²×C₆₇	1	4	yes	—
137	C₂×#41	C₂³×C₆₇	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
140	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	—
141	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
142	#131	—	4	8	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
147	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
149	#127	C₂×C₄×C₆₇	1	2	yes	—
150	#124	C₂×C₄×C₆₇	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
154	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	#104	—	4	8	no	—
157	C₂²⋊C₈	—	2	8	no	—
158	#97	D₄×C₆₇	1	4	yes	—
159	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
161	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
162	#95	—	2	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	#41×C₆₇	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
184	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	2	4	no	—
188	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
189	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₆₇	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
204	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
205	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
207	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	4	no	—
211	C₈	—	4	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
216	C₂²	#540×C₆₇	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	#518×C₆₇	1	2	yes	—
222	C₄	#535×C₆₇	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	C₆₇×#2328	1	1	yes	characteristic
226	C₂	C₆₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2932	C₂	semidirect
3. C₆₇×#1963	C₂	semidirect
4. C₆₇×#1876	C₂	semidirect
5. C₆₇×#1837	C₂	semidirect
6. C₆₇×#1746	C₂	semidirect
10. #495×C₆₇	C₄	semidirect
11. #494×C₆₇	C₄	semidirect
12. #493×C₆₇	C₄	semidirect
13. #492×C₆₇	C₄	semidirect
109. #22282	C₆₇	direct (N × H)
115. #2932	C₂×C₆₇	semidirect
116. #1963	C₂×C₆₇	semidirect
117. #1876	C₂×C₆₇	semidirect
118. #1837	C₂×C₆₇	semidirect
119. #1746	C₂×C₆₇	semidirect
120. C₆₇	#22282	direct (N × H)
124. #495	C₄×C₆₇	semidirect
125. #494	C₄×C₆₇	semidirect
126. #493	C₄×C₆₇	semidirect
127. #492	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3147	2
C₆₇×#2932	2
C₆₇×#1963	2
C₆₇×#1876	2
C₆₇×#1837	2
C₆₇×#1746	2
#22282	67

Subgroups — C67×#22282 #30701533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22282 #30701533