Subgroups — C₆₇×#33442 #30712693

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3407	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	C₂×#350×C₆₇	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	4	no	—
11	#504×C₆₇	C₂²	1	8	yes	complemented
12	C₂×M₃₂×C₆₇	—	2	4	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#444×C₆₇	C₄	1	4	yes	—
17	#443×C₆₇	C₂²	1	4	yes	—
18	#430×C₆₇	C₄	1	1	yes	characteristic
19	#421×C₆₇	C₄	1	2	yes	—
20	#405×C₆₇	C₂²	1	1	yes	characteristic
21	#364×C₆₇	C₂²	1	4	yes	complemented
22	#350×C₆₇	C₂²	1	4	yes	complemented
23	#350×C₆₇	C₂²	1	4	yes	complemented
24	#346×C₆₇	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
27	#137×C₆₇	C₈	1	4	yes	—
28	AGL(1,ℤ/8)×C₆₇	C₈	1	4	yes	—
29	#135×C₆₇	C₂×C₄	1	4	yes	—
30	#131×C₆₇	—	2	2	no	—
31	#131×C₆₇	—	2	2	no	—
32	#131×C₆₇	C₂×C₄	1	4	yes	—
33	#131×C₆₇	C₂³	1	4	yes	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₆₇	—	2	2	no	—
41	M₃₂×C₆₇	—	2	8	no	—
42	M₃₂×C₆₇	D₄	1	4	yes	—
43	M₃₂×C₆₇	C₂³	1	4	yes	—
44	C₂×C₁₆×C₆₇	—	2	2	no	—
45	C₂×C₁₆×C₆₇	—	2	2	no	—
46	C₂×C₁₆×C₆₇	—	2	8	no	—
47	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
50	#108×C₆₇	C₂×C₄	1	4	yes	—
51	#104×C₆₇	C₂×C₄	1	8	yes	—
52	#97×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	#41×C₆₇	—	2	2	no	—
55	#41×C₆₇	—	2	2	no	—
56	#41×C₆₇	C₂×C₈	1	4	yes	—
57	#41×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	Q₁₆×C₆₇	C₂×C₈	1	4	yes	—
64	SD₁₆×C₆₇	C₂×C₈	1	8	yes	—
65	D₈×C₆₇	C₂×C₈	1	4	yes	—
66	M₁₆×C₆₇	—	2	2	no	—
67	M₁₆×C₆₇	—	2	4	no	—
68	M₁₆×C₆₇	—	2	4	no	—
69	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
70	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₇	#41	1	2	yes	—
78	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₆₇	—	2	1	no	—
80	C₂²⋊C₄×C₆₇	—	2	1	no	—
81	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
82	C₁₆×C₆₇	—	2	4	no	—
83	C₁₆×C₆₇	—	2	1	no	—
84	C₁₆×C₆₇	—	2	1	no	—
85	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	#131	1	1	yes	characteristic
88	Q₈×C₆₇	—	2	2	no	—
89	Q₈×C₆₇	C₂²×C₈	1	4	yes	—
90	D₄×C₆₇	—	2	4	no	—
91	D₄×C₆₇	—	2	4	no	—
92	D₄×C₆₇	—	2	2	no	—
93	D₄×C₆₇	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
102	C₈×C₆₇	—	2	4	no	—
103	C₈×C₆₇	C₄×D₄	1	2	yes	—
104	C₈×C₆₇	C₂²×C₈	1	4	yes	—
105	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₆₇	—	2	4	no	—
108	C₂²×C₆₇	—	4	1	no	—
109	C₂²×C₆₇	—	8	2	no	—
110	C₂²×C₆₇	C₈×D₄	1	2	yes	—
111	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
112	C₄×C₆₇	—	2	4	no	—
113	C₄×C₆₇	—	4	2	no	—
114	C₄×C₆₇	C₈×D₄	1	2	yes	—
115	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
117	#33442	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₆₇	—	2	1	no	—
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	—	4	4	no	—
121	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
122	#3407	C₂×C₆₇	1	2	yes	complemented
123	#2955	C₂×C₆₇	1	1	yes	characteristic
124	#2171	C₂×C₆₇	1	4	yes	complemented
125	#2170	C₂×C₆₇	1	4	yes	complemented
126	#2153	C₂×C₆₇	1	1	yes	characteristic, complemented
127	C₂×#350	C₂×C₆₇	1	2	yes	complemented
128	#2108	C₂×C₆₇	1	1	yes	characteristic, complemented
129	C₆₇	#33442	1	1	yes	characteristic, direct product factor, Sylow
130	#575	C₄×C₆₇	1	1	yes	characteristic
131	C₂×#131	C₂²×C₆₇	1	2	yes	—
132	#504	—	2	4	no	—
133	#504	C₂²×C₆₇	1	8	yes	complemented
134	C₂×M₃₂	—	2	4	no	—
135	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
136	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
138	#444	C₄×C₆₇	1	4	yes	—
139	#443	C₂²×C₆₇	1	4	yes	—
140	#430	C₄×C₆₇	1	1	yes	characteristic
141	#421	C₄×C₆₇	1	2	yes	—
142	#405	C₂²×C₆₇	1	1	yes	characteristic
143	#364	C₂²×C₆₇	1	4	yes	complemented
144	#350	C₂²×C₆₇	1	4	yes	complemented
145	#350	C₂²×C₆₇	1	4	yes	complemented
146	#346	C₂²×C₆₇	1	2	yes	complemented
147	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
148	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
149	#137	C₈×C₆₇	1	4	yes	—
150	AGL(1,ℤ/8)	C₈×C₆₇	1	4	yes	—
151	#135	C₂×C₄×C₆₇	1	4	yes	—
152	#131	—	2	2	no	—
153	#131	—	2	2	no	—
154	#131	C₂×C₄×C₆₇	1	4	yes	—
155	#131	C₂³×C₆₇	1	4	yes	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
158	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
161	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	8	no	—
164	M₃₂	D₄×C₆₇	1	4	yes	—
165	M₃₂	C₂³×C₆₇	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	8	no	—
169	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
170	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
171	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
172	#108	C₂×C₄×C₆₇	1	4	yes	—
173	#104	C₂×C₄×C₆₇	1	8	yes	—
174	#97	C₂×C₄×C₆₇	1	2	yes	—
175	C₄×C₈	C₂³×C₆₇	1	2	yes	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	C₂×C₈×C₆₇	1	4	yes	—
179	#41	C₂²×C₄×C₆₇	1	4	yes	—
180	C₂×Q₈	C₂×C₈×C₆₇	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
185	Q₁₆	C₂×C₈×C₆₇	1	4	yes	—
186	SD₁₆	C₂×C₈×C₆₇	1	8	yes	—
187	D₈	C₂×C₈×C₆₇	1	4	yes	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	C₂×C₈×C₆₇	1	4	yes	—
192	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
197	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
199	C₂×C₈	#41×C₆₇	1	2	yes	—
200	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	2	1	no	—
207	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	#131×C₆₇	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×C₈×C₆₇	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×C₈×C₆₇	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
221	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
222	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
223	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₄×D₄×C₆₇	1	2	yes	—
226	C₈	C₂²×C₈×C₆₇	1	4	yes	—
227	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
228	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	C₈×D₄×C₆₇	1	2	yes	—
233	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₈×D₄×C₆₇	1	2	yes	—
237	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
238	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	C₂	C₂×C₈×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	C₂²×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 ◁ C₈×C₆₇ ◁ C₂×M₃₂×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3407	C₂	semidirect
3. C₆₇×#2171	C₂	semidirect
4. C₆₇×#2170	C₂	semidirect
5. C₆₇×#2153	C₂	semidirect
6. C₂×#350×C₆₇	C₂	semidirect
7. C₆₇×#2108	C₂	semidirect
11. #504×C₆₇	C₂²	semidirect
21. #364×C₆₇	C₂²	semidirect
22. #350×C₆₇	C₂²	semidirect
23. #350×C₆₇	C₂²	semidirect
24. #346×C₆₇	C₂²	semidirect
25. C₄×C₁₆×C₆₇	C₂²	semidirect
117. #33442	C₆₇	direct (N × H)
122. #3407	C₂×C₆₇	semidirect
124. #2171	C₂×C₆₇	semidirect
125. #2170	C₂×C₆₇	semidirect
126. #2153	C₂×C₆₇	semidirect
127. C₂×#350	C₂×C₆₇	semidirect
128. #2108	C₂×C₆₇	semidirect
129. C₆₇	#33442	direct (N × H)
133. #504	C₂²×C₆₇	semidirect
143. #364	C₂²×C₆₇	semidirect
144. #350	C₂²×C₆₇	semidirect
145. #350	C₂²×C₆₇	semidirect
146. #346	C₂²×C₆₇	semidirect
147. 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
C₆₇×#3407	2
C₆₇×#2955	2
C₆₇×#2171	2
C₆₇×#2170	2
C₆₇×#2153	2
C₂×#350×C₆₇	2
C₆₇×#2108	2
#33442	67

Subgroups — C67×#33442 #30712693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33442 #30712693