Subgroups — C₆₇×#33957 #30713208

Order 17152.

444 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3383	C₂	1	2	yes	maximal
2	C₆₇×#2255	C₂	1	2	yes	maximal
3	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2229	C₂	1	2	yes	maximal
5	C₆₇×#2229	C₂	1	2	yes	maximal
6	C₆₇×#2227	C₂	1	2	yes	maximal
7	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#2176	C₂	1	2	yes	maximal
9	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#557×C₆₇	—	2	2	no	—
11	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
12	#500×C₆₇	C₂²	1	2	yes	—
13	#499×C₆₇	C₂²	1	2	yes	—
14	#499×C₆₇	C₂²	1	2	yes	—
15	#479×C₆₇	—	2	2	no	—
16	#479×C₆₇	—	2	2	no	—
17	#475×C₆₇	—	2	2	no	—
18	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
22	#367×C₆₇	C₂²	1	2	yes	—
23	#367×C₆₇	C₂²	1	2	yes	—
24	#366×C₆₇	C₂²	1	1	yes	characteristic
25	#366×C₆₇	C₂²	1	1	yes	characteristic
26	#366×C₆₇	C₂²	1	1	yes	characteristic
27	#366×C₆₇	C₂²	1	2	yes	—
28	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	#358×C₆₇	C₂²	1	2	yes	—
31	#358×C₆₇	C₂²	1	2	yes	—
32	#358×C₆₇	C₂²	1	2	yes	—
33	#346×C₆₇	C₂²	1	2	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₆₇	—	2	2	no	—
37	C₄⋊Q₈×C₆₇	—	4	2	no	—
38	#125×C₆₇	—	2	2	no	—
39	#125×C₆₇	—	2	2	no	—
40	#125×C₆₇	—	4	2	no	—
41	#125×C₆₇	—	4	2	no	—
42	C₄×Q₈×C₆₇	—	2	2	no	—
43	C₄×Q₈×C₆₇	—	2	2	no	—
44	C₄×Q₈×C₆₇	—	2	2	no	—
45	C₄×Q₈×C₆₇	—	4	2	no	—
46	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
47	Q₃₂×C₆₇	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
51	#107×C₆₇	—	2	2	no	—
52	#107×C₆₇	C₂³	1	2	yes	—
53	#107×C₆₇	C₂³	1	2	yes	—
54	#107×C₆₇	C₂³	1	2	yes	—
55	#106×C₆₇	—	2	2	no	—
56	#106×C₆₇	—	2	2	no	—
57	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
58	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
59	#103×C₆₇	—	2	2	no	—
60	#103×C₆₇	—	2	2	no	—
61	#103×C₆₇	—	2	2	no	—
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂×Q₈×C₆₇	—	2	2	no	—
65	C₂×Q₈×C₆₇	—	4	2	no	—
66	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
67	Q₁₆×C₆₇	—	2	2	no	—
68	Q₁₆×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	4	2	no	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	4	2	no	—
81	C₄⋊C₄×C₆₇	—	4	2	no	—
82	C₄⋊C₄×C₆₇	—	4	2	no	—
83	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
84	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
85	C₄²×C₆₇	—	4	2	no	—
86	C₄²×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×D₄	1	2	yes	—
90	C₁₆×C₆₇	—	2	1	no	—
91	C₁₆×C₆₇	—	2	2	no	—
92	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
93	Q₈×C₆₇	—	2	4	no	—
94	Q₈×C₆₇	—	4	1	no	—
95	Q₈×C₆₇	—	4	2	no	—
96	Q₈×C₆₇	—	8	4	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	2	1	no	—
114	C₄×C₆₇	—	2	2	no	—
115	C₄×C₆₇	—	4	4	no	—
116	C₄×C₆₇	—	8	2	no	—
117	C₄×C₆₇	—	8	2	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
120	C₄×C₆₇	#578	1	1	yes	characteristic
121	#33957	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₆₇	Q₈×D₈	1	1	yes	characteristic
123	C₂×C₆₇	#3414	1	1	yes	characteristic
124	C₂×C₆₇	#3417	1	1	yes	characteristic
125	#3383	C₂×C₆₇	1	2	yes	—
126	#2255	C₂×C₆₇	1	2	yes	—
127	#2253	C₂×C₆₇	1	1	yes	characteristic
128	#2229	C₂×C₆₇	1	2	yes	—
129	#2229	C₂×C₆₇	1	2	yes	—
130	#2227	C₂×C₆₇	1	2	yes	—
131	#2183	C₂×C₆₇	1	1	yes	characteristic
132	#2176	C₂×C₆₇	1	2	yes	—
133	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic
134	C₆₇	#33957	1	1	yes	characteristic, direct product factor, Sylow
135	#557	—	2	2	no	—
136	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
137	#500	C₂²×C₆₇	1	2	yes	—
138	#499	C₂²×C₆₇	1	2	yes	—
139	#499	C₂²×C₆₇	1	2	yes	—
140	#479	—	2	2	no	—
141	#479	—	2	2	no	—
142	#475	—	2	2	no	—
143	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
144	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
145	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
146	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
147	#367	C₂²×C₆₇	1	2	yes	—
148	#367	C₂²×C₆₇	1	2	yes	—
149	#366	C₂²×C₆₇	1	1	yes	characteristic
150	#366	C₂²×C₆₇	1	1	yes	characteristic
151	#366	C₂²×C₆₇	1	2	yes	—
152	#366	C₂²×C₆₇	1	1	yes	characteristic
153	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
154	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
155	#358	C₂²×C₆₇	1	2	yes	—
156	#358	C₂²×C₆₇	1	2	yes	—
157	#358	C₂²×C₆₇	1	2	yes	—
158	#346	C₂²×C₆₇	1	2	yes	—
159	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	4	2	no	—
163	#125	—	2	2	no	—
164	#125	—	2	2	no	—
165	#125	—	4	2	no	—
166	#125	—	4	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	—	4	2	no	—
171	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
172	Q₃₂	Q₈×C₆₇	1	4	yes	complemented
173	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
174	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
175	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
176	#107	—	2	2	no	—
177	#107	C₂³×C₆₇	1	2	yes	—
178	#107	C₂³×C₆₇	1	2	yes	—
179	#107	C₂³×C₆₇	1	2	yes	—
180	#106	—	2	2	no	—
181	#106	—	2	2	no	—
182	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
183	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	#103	—	2	2	no	—
187	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₆₇	1	2	yes	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
192	Q₁₆	—	2	2	no	—
193	Q₁₆	C₂×Q₈×C₆₇	1	4	yes	—
194	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
196	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
209	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
210	C₄²	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₆₇	1	2	yes	—
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×Q₈×C₆₇	1	2	yes	—
218	Q₈	—	2	4	no	—
219	Q₈	—	4	1	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	8	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
227	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
234	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
235	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
245	C₄	#578×C₆₇	1	1	yes	characteristic
246	C₂	Q₈×D₈×C₆₇	1	1	yes	characteristic
247	C₂	C₆₇×#3414	1	1	yes	characteristic
248	C₂	C₆₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
47. Q₃₂×C₆₇	Q₈	semidirect
121. #33957	C₆₇	direct (N × H)
134. C₆₇	#33957	direct (N × H)
172. Q₃₂	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3383	2
C₆₇×#2255	2
C₆₇×#2253	2
C₆₇×#2229	2
C₆₇×#2227	2
C₆₇×#2183	2
C₆₇×#2176	2
C₄×Q₃₂×C₆₇	2
#33957	67

Subgroups — C67×#33957 #30713208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33957 #30713208