Subgroups — C₆₇×#33451 #30712702

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#135×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2177	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2176	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2175	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₂²	1	2	yes	—
10	C₄×#41×C₆₇	—	2	2	no	—
11	#508×C₆₇	C₄	1	8	yes	complemented
12	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
18	C₄×SD₁₆×C₆₇	—	2	4	no	—
19	C₄×D₈×C₆₇	C₂²	1	4	yes	—
20	#427×C₆₇	C₂²	1	2	yes	—
21	#416×C₆₇	—	2	2	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#367×C₆₇	C₂²	1	2	yes	—
24	#366×C₆₇	C₂²	1	2	yes	—
25	#358×C₆₇	C₂²	1	4	yes	—
26	#357×C₆₇	C₂²	1	4	yes	—
27	#346×C₆₇	C₂²	1	2	yes	—
28	#346×C₆₇	C₂²	1	2	yes	—
29	C₂×#41×C₆₇	—	2	2	no	—
30	#135×C₆₇	—	2	4	no	—
31	#135×C₆₇	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₆₇	—	2	2	no	—
34	C₂×D₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₆₇	—	2	4	no	—
38	C₄×D₄×C₆₇	—	2	4	no	—
39	C₄×D₄×C₆₇	—	4	4	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	4	2	no	—
42	C₂×C₄²×C₆₇	—	4	2	no	—
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	Q₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
45	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
46	D₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₆₇	—	2	2	no	—
48	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
52	#107×C₆₇	C₂³	1	4	yes	—
53	#106×C₆₇	—	2	2	no	—
54	#103×C₆₇	—	2	4	no	—
55	#102×C₆₇	—	2	4	no	—
56	C₄×C₈×C₆₇	D₄	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	#41×C₆₇	—	2	8	no	—
59	#41×C₆₇	—	4	4	no	—
60	C₂×Q₈×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	—	4	4	no	—
65	C₂²×C₄×C₆₇	D₈	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₆₇	—	2	2	no	—
68	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₆₇	—	2	8	no	—
70	D₈×C₆₇	—	2	2	no	—
71	D₈×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	#41	1	2	yes	—
77	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	4	4	no	—
80	C₂²⋊C₄×C₆₇	—	4	2	no	—
81	C₂²⋊C₄×C₆₇	—	4	4	no	—
82	C₄²×C₆₇	—	4	4	no	—
83	C₄²×C₆₇	—	4	2	no	—
84	C₄²×C₆₇	D₈	1	2	yes	—
85	C₄²×C₆₇	C₂×D₄	1	2	yes	—
86	C₁₆×C₆₇	—	2	2	no	—
87	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
88	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂³×C₆₇	—	4	2	no	—
90	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₆₇	—	2	4	no	—
92	Q₈×C₆₇	—	4	2	no	—
93	D₄×C₆₇	—	2	4	no	—
94	D₄×C₆₇	—	4	8	no	—
95	D₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	8	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	8	8	no	—
101	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#135	1	2	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	C₄×D₄	1	2	yes	—
108	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	4	no	—
112	C₂²×C₆₇	—	8	4	no	—
113	C₂²×C₆₇	C₄×D₈	1	2	yes	—
114	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₆₇	—	2	4	no	—
116	C₄×C₆₇	—	4	4	no	—
117	C₄×C₆₇	—	8	4	no	—
118	C₄×C₆₇	C₄×D₈	1	2	yes	—
119	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
121	#33451	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₆₇	—	2	2	no	—
123	C₂×C₆₇	—	8	4	no	—
124	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
125	C₂×C₆₇	#3414	1	2	yes	—
126	C₂×#508	C₂×C₆₇	1	1	yes	characteristic
127	C₄×#135	C₂×C₆₇	1	2	yes	complemented
128	#2177	C₂×C₆₇	1	2	yes	complemented
129	#2176	C₂×C₆₇	1	2	yes	complemented
130	#2175	C₂×C₆₇	1	4	yes	complemented
131	#2158	C₂×C₆₇	1	1	yes	characteristic, complemented
132	#2138	C₂×C₆₇	1	2	yes	complemented
133	C₂×#346	C₂×C₆₇	1	1	yes	characteristic, complemented
134	C₆₇	#33451	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#135	C₂²×C₆₇	1	2	yes	—
136	C₄×#41	—	2	2	no	—
137	#508	C₄×C₆₇	1	8	yes	complemented
138	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
139	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
140	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
141	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
143	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
144	C₄×SD₁₆	—	2	4	no	—
145	C₄×D₈	C₂²×C₆₇	1	4	yes	—
146	#427	C₂²×C₆₇	1	2	yes	—
147	#416	—	2	2	no	—
148	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
149	#367	C₂²×C₆₇	1	2	yes	—
150	#366	C₂²×C₆₇	1	2	yes	—
151	#358	C₂²×C₆₇	1	4	yes	—
152	#357	C₂²×C₆₇	1	4	yes	—
153	#346	C₂²×C₆₇	1	2	yes	—
154	#346	C₂²×C₆₇	1	2	yes	—
155	C₂×#41	—	2	2	no	—
156	#135	—	2	4	no	—
157	#135	C₂×C₄×C₆₇	1	8	yes	complemented
158	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂³×C₆₇	1	2	yes	—
161	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
163	C₄×Q₈	—	2	4	no	—
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	—	4	4	no	—
166	#117	—	2	2	no	—
167	#117	—	4	2	no	—
168	C₂×C₄²	—	4	2	no	—
169	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
170	Q₃₂	C₂×C₄×C₆₇	1	4	yes	complemented
171	SD₃₂	C₂×C₄×C₆₇	1	8	yes	complemented
172	D₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
175	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
176	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
177	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
178	#107	C₂³×C₆₇	1	4	yes	—
179	#106	—	2	2	no	—
180	#103	—	2	4	no	—
181	#102	—	2	4	no	—
182	C₄×C₈	D₄×C₆₇	1	2	yes	—
183	C₄×C₈	C₂³×C₆₇	1	2	yes	—
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	D₈×C₆₇	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×C₄×C₆₇	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×C₄×C₆₇	1	4	yes	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
202	C₂×C₈	#41×C₆₇	1	2	yes	—
203	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	4	no	—
210	C₄²	D₈×C₆₇	1	2	yes	—
211	C₄²	C₂×D₄×C₆₇	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂²×C₄×C₆₇	1	2	yes	—
214	C₁₆	C₂²×C₄×C₆₇	1	2	yes	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	8	no	—
227	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
229	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
230	C₂×C₄	#135×C₆₇	1	2	yes	—
231	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₄×C₆₇	1	2	yes	—
234	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
235	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₄×D₈×C₆₇	1	2	yes	—
240	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	4	4	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄×D₈×C₆₇	1	2	yes	—
245	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
246	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
250	C₂	C₆₇×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#508×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#135×C₆₇	C₂	semidirect
3. C₆₇×#2177	C₂	semidirect
4. C₆₇×#2176	C₂	semidirect
5. C₆₇×#2175	C₂	semidirect
6. C₆₇×#2158	C₂	semidirect
7. C₆₇×#2138	C₂	semidirect
8. C₂×#346×C₆₇	C₂	semidirect
11. #508×C₆₇	C₄	semidirect
31. #135×C₆₇	C₂×C₄	semidirect
44. Q₃₂×C₆₇	C₂×C₄	semidirect
45. SD₃₂×C₆₇	C₂×C₄	semidirect
46. D₁₆×C₆₇	C₂×C₄	semidirect
48. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
121. #33451	C₆₇	direct (N × H)
127. C₄×#135	C₂×C₆₇	semidirect
128. #2177	C₂×C₆₇	semidirect
129. #2176	C₂×C₆₇	semidirect
130. #2175	C₂×C₆₇	semidirect
131. #2158	C₂×C₆₇	semidirect
132. #2138	C₂×C₆₇	semidirect
133. C₂×#346	C₂×C₆₇	semidirect
134. C₆₇	#33451	direct (N × H)
137. #508	C₄×C₆₇	semidirect
157. #135	C₂×C₄×C₆₇	semidirect
170. Q₃₂	C₂×C₄×C₆₇	semidirect
171. SD₃₂	C₂×C₄×C₆₇	semidirect
172. D₁₆	C₂×C₄×C₆₇	semidirect
174. C₂×C₁₆	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₄×#135×C₆₇	2
C₆₇×#2177	2
C₆₇×#2176	2
C₆₇×#2175	2
C₆₇×#2158	2
C₆₇×#2138	2
C₂×#346×C₆₇	2
#33451	67

Subgroups — C67×#33451 #30712702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33451 #30712702