Subgroups — C₆₇×#33441 #30712692

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2955	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2171	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2170	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2117	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#575×C₆₇	C₄	1	1	yes	characteristic
9	C₂×#131×C₆₇	C₂²	1	2	yes	—
10	#504×C₆₇	—	2	2	no	—
11	#504×C₆₇	—	2	2	no	—
12	#504×C₆₇	C₂²	1	4	yes	complemented
13	#504×C₆₇	C₂²	1	4	yes	complemented
14	C₂×M₃₂×C₆₇	—	2	2	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₇	—	2	2	no	—
18	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	#444×C₆₇	C₄	1	4	yes	—
20	#443×C₆₇	C₂²	1	4	yes	—
21	#430×C₆₇	C₄	1	1	yes	characteristic
22	#421×C₆₇	C₄	1	2	yes	—
23	#405×C₆₇	C₂²	1	1	yes	characteristic
24	#364×C₆₇	C₂²	1	4	yes	complemented
25	#350×C₆₇	C₂²	1	8	yes	complemented
26	#346×C₆₇	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
29	#137×C₆₇	C₈	1	4	yes	—
30	AGL(1,ℤ/8)×C₆₇	C₈	1	4	yes	—
31	#135×C₆₇	C₂×C₄	1	4	yes	—
32	#131×C₆₇	—	2	2	no	—
33	#131×C₆₇	—	2	2	no	—
34	#131×C₆₇	C₂×C₄	1	4	yes	—
35	#131×C₆₇	C₂³	1	4	yes	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₆₇	—	2	2	no	—
43	M₃₂×C₆₇	—	2	2	no	—
44	M₃₂×C₆₇	—	2	4	no	—
45	M₃₂×C₆₇	—	2	4	no	—
46	M₃₂×C₆₇	C₂³	1	4	yes	—
47	C₂×C₁₆×C₆₇	—	2	2	no	—
48	C₂×C₁₆×C₆₇	—	2	4	no	—
49	C₂×C₁₆×C₆₇	—	2	4	no	—
50	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
51	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
52	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	C₂×C₄	1	4	yes	—
56	#104×C₆₇	C₂×C₄	1	8	yes	—
57	#97×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	2	yes	—
59	#41×C₆₇	—	2	2	no	—
60	#41×C₆₇	—	2	2	no	—
61	#41×C₆₇	C₂×C₈	1	4	yes	—
62	#41×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆×C₆₇	C₂×C₈	1	4	yes	—
69	SD₁₆×C₆₇	C₂×C₈	1	8	yes	—
70	D₈×C₆₇	C₂×C₈	1	4	yes	—
71	M₁₆×C₆₇	—	2	2	no	—
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	2	4	no	—
74	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
75	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₆₇	#41	1	2	yes	—
83	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
87	C₁₆×C₆₇	—	2	4	no	—
88	C₁₆×C₆₇	—	2	2	no	—
89	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
90	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	#131	1	1	yes	characteristic
94	Q₈×C₆₇	—	2	2	no	—
95	Q₈×C₆₇	C₂²×C₈	1	4	yes	—
96	D₄×C₆₇	—	2	4	no	—
97	D₄×C₆₇	—	2	4	no	—
98	D₄×C₆₇	—	2	2	no	—
99	D₄×C₆₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₇	—	2	4	no	—
109	C₈×C₆₇	C₄×D₄	1	2	yes	—
110	C₈×C₆₇	C₂²×C₈	1	4	yes	—
111	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	4	1	no	—
115	C₂²×C₆₇	—	4	4	no	—
116	C₂²×C₆₇	C₈×D₄	1	2	yes	—
117	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
118	C₄×C₆₇	—	2	4	no	—
119	C₄×C₆₇	—	4	2	no	—
120	C₄×C₆₇	C₈×D₄	1	2	yes	—
121	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
123	#33441	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	—	4	4	no	—
127	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
128	C₂×#504	C₂×C₆₇	1	2	yes	complemented
129	#2955	C₂×C₆₇	1	1	yes	characteristic
130	#2171	C₂×C₆₇	1	4	yes	complemented
131	#2170	C₂×C₆₇	1	4	yes	complemented
132	#2153	C₂×C₆₇	1	1	yes	characteristic, complemented
133	#2117	C₂×C₆₇	1	2	yes	complemented
134	#2108	C₂×C₆₇	1	1	yes	characteristic, complemented
135	C₆₇	#33441	1	1	yes	characteristic, direct product factor, Sylow
136	#575	C₄×C₆₇	1	1	yes	characteristic
137	C₂×#131	C₂²×C₆₇	1	2	yes	—
138	#504	—	2	2	no	—
139	#504	—	2	2	no	—
140	#504	C₂²×C₆₇	1	4	yes	complemented
141	#504	C₂²×C₆₇	1	4	yes	complemented
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
144	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
145	C₂²×C₁₆	—	2	2	no	—
146	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
147	#444	C₄×C₆₇	1	4	yes	—
148	#443	C₂²×C₆₇	1	4	yes	—
149	#430	C₄×C₆₇	1	1	yes	characteristic
150	#421	C₄×C₆₇	1	2	yes	—
151	#405	C₂²×C₆₇	1	1	yes	characteristic
152	#364	C₂²×C₆₇	1	4	yes	complemented
153	#350	C₂²×C₆₇	1	8	yes	complemented
154	#346	C₂²×C₆₇	1	2	yes	complemented
155	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
156	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
157	#137	C₈×C₆₇	1	4	yes	—
158	AGL(1,ℤ/8)	C₈×C₆₇	1	4	yes	—
159	#135	C₂×C₄×C₆₇	1	4	yes	—
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×C₄×C₆₇	1	4	yes	—
163	#131	C₂³×C₆₇	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
166	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
169	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	—	2	4	no	—
174	M₃₂	C₂³×C₆₇	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
179	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
180	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
181	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
182	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
183	#108	C₂×C₄×C₆₇	1	4	yes	—
184	#104	C₂×C₄×C₆₇	1	8	yes	—
185	#97	C₂×C₄×C₆₇	1	2	yes	—
186	C₄×C₈	C₂³×C₆₇	1	2	yes	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×C₈×C₆₇	1	4	yes	—
190	#41	C₂²×C₄×C₆₇	1	4	yes	—
191	C₂×Q₈	C₂×C₈×C₆₇	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
196	Q₁₆	C₂×C₈×C₆₇	1	4	yes	—
197	SD₁₆	C₂×C₈×C₆₇	1	8	yes	—
198	D₈	C₂×C₈×C₆₇	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂×C₈×C₆₇	1	4	yes	—
203	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
210	C₂×C₈	#41×C₆₇	1	2	yes	—
211	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×D₄×C₆₇	1	1	yes	characteristic
218	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
219	C₁₆	C₂×D₄×C₆₇	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	#131×C₆₇	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×C₈×C₆₇	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×C₈×C₆₇	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
233	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
234	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	C₄×D₄×C₆₇	1	2	yes	—
238	C₈	C₂²×C₈×C₆₇	1	4	yes	—
239	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
240	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	C₈×D₄×C₆₇	1	2	yes	—
245	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	C₈×D₄×C₆₇	1	2	yes	—
249	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
250	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×C₈×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₆₇	C₂×D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	#575×C₆₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₆₇ ◁ C₂²×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#504×C₆₇	C₂	semidirect
3. C₆₇×#2171	C₂	semidirect
4. C₆₇×#2170	C₂	semidirect
5. C₆₇×#2153	C₂	semidirect
6. C₆₇×#2117	C₂	semidirect
7. C₆₇×#2108	C₂	semidirect
12. #504×C₆₇	C₂²	semidirect
13. #504×C₆₇	C₂²	semidirect
24. #364×C₆₇	C₂²	semidirect
25. #350×C₆₇	C₂²	semidirect
26. #346×C₆₇	C₂²	semidirect
27. C₄×C₁₆×C₆₇	C₂²	semidirect
123. #33441	C₆₇	direct (N × H)
128. C₂×#504	C₂×C₆₇	semidirect
130. #2171	C₂×C₆₇	semidirect
131. #2170	C₂×C₆₇	semidirect
132. #2153	C₂×C₆₇	semidirect
133. #2117	C₂×C₆₇	semidirect
134. #2108	C₂×C₆₇	semidirect
135. C₆₇	#33441	direct (N × H)
140. #504	C₂²×C₆₇	semidirect
141. #504	C₂²×C₆₇	semidirect
152. #364	C₂²×C₆₇	semidirect
153. #350	C₂²×C₆₇	semidirect
154. #346	C₂²×C₆₇	semidirect
155. 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×#504×C₆₇	2
C₆₇×#2955	2
C₆₇×#2171	2
C₆₇×#2170	2
C₆₇×#2153	2
C₆₇×#2117	2
C₆₇×#2108	2
#33441	67

Subgroups — C67×#33441 #30712692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33441 #30712692