Subgroups — C₆₇×#33166 #30712417

Order 17152.

1238 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#463×C₆₇	C₂	1	3	yes	maximal, complemented
2	C₂×#459×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2657	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1983	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1982	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₆₇	C₂²	1	3	yes	—
10	C₂×#127×C₆₇	C₂²	1	3	yes	—
11	C₂×#124×C₆₇	C₂²	1	3	yes	—
12	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
13	#463×C₆₇	—	2	12	no	—
14	#459×C₆₇	—	2	12	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
17	C₂×#103×C₆₇	C₂²	1	3	yes	—
18	C₂×#102×C₆₇	C₂²	1	6	yes	—
19	C₂×#102×C₆₇	C₂²	1	3	yes	—
20	#392×C₆₇	—	2	1	no	—
21	#392×C₆₇	—	2	3	no	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
23	C₄×C₂²⋊C₄×C₆₇	—	2	3	no	—
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#95×C₆₇	—	2	1	no	—
26	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×D₄×C₆₇	—	2	3	no	—
28	C₂²×D₄×C₆₇	—	2	3	no	—
29	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
30	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	—	2	3	no	—
32	C₂×SD₁₆×C₆₇	—	2	6	no	—
33	C₂×D₈×C₆₇	—	2	12	no	—
34	C₂×D₈×C₆₇	—	2	6	no	—
35	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	#127×C₆₇	—	2	6	no	—
38	#127×C₆₇	—	2	6	no	—
39	#124×C₆₇	—	2	6	no	—
40	#124×C₆₇	—	2	6	no	—
41	C₄×D₄×C₆₇	—	2	6	no	—
42	C₄×D₄×C₆₇	D₄	1	12	yes	complemented
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
47	C₂×C₄²×C₆₇	—	2	3	no	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₇	—	2	6	no	—
50	C₄⋊C₈×C₆₇	—	2	6	no	—
51	#103×C₆₇	—	2	6	no	—
52	#102×C₆₇	—	2	12	no	—
53	#102×C₆₇	—	2	6	no	—
54	#95×C₆₇	—	2	3	no	—
55	#95×C₆₇	D₄	1	1	yes	characteristic
56	#95×C₆₇	D₄	1	3	yes	—
57	C₂⁴×C₆₇	—	2	1	no	—
58	C₂⁴×C₆₇	—	2	3	no	—
59	C₂×Q₈×C₆₇	—	2	3	no	—
60	C₂×Q₈×C₆₇	—	2	3	no	—
61	C₂×D₄×C₆₇	—	2	3	no	—
62	C₂×D₄×C₆₇	—	2	6	no	—
63	C₂×D₄×C₆₇	—	2	3	no	—
64	C₂×D₄×C₆₇	—	2	3	no	—
65	C₂×D₄×C₆₇	—	2	6	no	—
66	C₂×D₄×C₆₇	—	4	6	no	—
67	C₂×D₄×C₆₇	—	4	6	no	—
68	C₂×D₄×C₆₇	—	8	3	no	—
69	C₂×D₄×C₆₇	—	8	6	no	—
70	C₂×D₄×C₆₇	—	8	3	no	—
71	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₆₇	—	2	3	no	—
73	C₂²×C₄×C₆₇	—	2	12	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	8	6	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	SD₁₆×C₆₇	—	4	4	no	—
80	D₈×C₆₇	—	4	12	no	—
81	C₂×C₈×C₆₇	—	2	6	no	—
82	C₂×C₈×C₆₇	—	2	3	no	—
83	C₂×C₈×C₆₇	—	2	3	no	—
84	C₄⋊C₄×C₆₇	—	2	3	no	—
85	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
86	C₂²⋊C₄×C₆₇	—	2	12	no	—
87	C₂²⋊C₄×C₆₇	—	4	6	no	—
88	C₂²⋊C₄×C₆₇	—	4	6	no	—
89	C₄²×C₆₇	—	8	3	no	—
90	C₄²×C₆₇	C₂×D₄	1	6	yes	—
91	C₄²×C₆₇	#41	1	6	yes	—
92	C₂³×C₆₇	—	2	6	no	—
93	C₂³×C₆₇	—	4	3	no	—
94	C₂³×C₆₇	—	4	1	no	—
95	C₂³×C₆₇	—	4	3	no	—
96	C₂³×C₆₇	—	4	6	no	—
97	C₂³×C₆₇	—	4	3	no	—
98	C₂³×C₆₇	—	4	6	no	—
99	C₂³×C₆₇	—	4	3	no	—
100	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	4	1	no	—
102	Q₈×C₆₇	—	4	3	no	—
103	D₄×C₆₇	—	2	6	no	—
104	D₄×C₆₇	—	4	3	no	—
105	D₄×C₆₇	—	4	3	no	—
106	D₄×C₆₇	—	4	6	no	—
107	D₄×C₆₇	—	8	12	no	—
108	D₄×C₆₇	—	8	12	no	—
109	D₄×C₆₇	#127	1	4	yes	—
110	C₂×C₄×C₆₇	—	2	3	no	—
111	C₂×C₄×C₆₇	—	2	6	no	—
112	C₂×C₄×C₆₇	—	2	3	no	—
113	C₂×C₄×C₆₇	—	2	6	no	—
114	C₂×C₄×C₆₇	—	4	6	no	—
115	C₂×C₄×C₆₇	—	4	3	no	—
116	C₂×C₄×C₆₇	—	8	12	no	—
117	C₂×C₄×C₆₇	#121	1	6	yes	—
118	C₂×C₄×C₆₇	#121	1	6	yes	—
119	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
120	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
121	C₈×C₆₇	—	4	3	no	—
122	C₈×C₆₇	—	4	1	no	—
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	4	6	no	—
125	C₂²×C₆₇	—	4	6	no	—
126	C₂²×C₆₇	—	8	3	no	—
127	C₂²×C₆₇	—	8	6	no	—
128	C₂²×C₆₇	—	8	3	no	—
129	C₂²×C₆₇	—	8	6	no	—
130	C₂²×C₆₇	—	8	3	no	—
131	C₂²×C₆₇	C₂×#121	1	3	yes	—
132	C₂²×C₆₇	#576	1	3	yes	—
133	C₂²×C₆₇	#577	1	1	yes	characteristic
134	C₄×C₆₇	—	2	6	no	—
135	C₄×C₆₇	—	2	6	no	—
136	C₄×C₆₇	—	4	6	no	—
137	C₄×C₆₇	—	8	1	no	—
138	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
139	C₄×C₆₇	#532	1	3	yes	—
140	#33166	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₆₇	—	4	4	no	—
142	C₂×C₆₇	—	16	3	no	—
143	C₂×C₆₇	#2856	1	1	yes	characteristic
144	C₂×C₆₇	#3041	1	3	yes	—
145	C₂×C₆₇	#3045	1	3	yes	—
146	C₂×#463	C₂×C₆₇	1	3	yes	complemented
147	C₂×#459	C₂×C₆₇	1	3	yes	complemented
148	#2657	C₂×C₆₇	1	1	yes	characteristic, complemented
149	#2275	C₂×C₆₇	1	1	yes	characteristic, complemented
150	#1983	C₂×C₆₇	1	3	yes	complemented
151	#1982	C₂×C₆₇	1	3	yes	complemented
152	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
153	C₆₇	#33166	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
155	C₂²×D₈	C₂²×C₆₇	1	3	yes	—
156	C₂×#127	C₂²×C₆₇	1	3	yes	—
157	C₂×#124	C₂²×C₆₇	1	3	yes	—
158	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
159	#463	—	2	12	no	—
160	#459	—	2	12	no	—
161	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
162	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
163	C₂×#103	C₂²×C₆₇	1	3	yes	—
164	C₂×#102	C₂²×C₆₇	1	3	yes	—
165	C₂×#102	C₂²×C₆₇	1	6	yes	—
166	#392	—	2	1	no	—
167	#392	—	2	3	no	—
168	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
169	C₄×C₂²⋊C₄	—	2	3	no	—
170	#376	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×#95	—	2	1	no	—
172	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
173	C₂²×D₄	—	2	3	no	—
174	C₂²×D₄	—	2	3	no	—
175	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
176	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
177	C₂³×C₄	—	2	3	no	—
178	C₂×SD₁₆	—	2	6	no	—
179	C₂×D₈	—	2	12	no	—
180	C₂×D₈	—	2	6	no	—
181	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
182	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
183	#127	—	2	6	no	—
184	#127	—	2	6	no	—
185	#124	—	2	6	no	—
186	#124	—	2	6	no	—
187	C₄×D₄	—	2	6	no	—
188	C₄×D₄	D₄×C₆₇	1	12	yes	complemented
189	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
190	C₂×C₂²⋊C₄	—	2	3	no	—
191	C₂×C₂²⋊C₄	—	2	3	no	—
192	C₂×C₂²⋊C₄	—	2	3	no	—
193	C₂×C₄²	—	2	3	no	—
194	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
195	C₄⋊C₈	—	2	6	no	—
196	C₄⋊C₈	—	2	6	no	—
197	#103	—	2	6	no	—
198	#102	—	2	12	no	—
199	#102	—	2	6	no	—
200	#95	—	2	3	no	—
201	#95	D₄×C₆₇	1	1	yes	characteristic
202	#95	D₄×C₆₇	1	3	yes	—
203	C₂⁴	—	2	1	no	—
204	C₂⁴	—	2	3	no	—
205	C₂×Q₈	—	2	3	no	—
206	C₂×Q₈	—	2	3	no	—
207	C₂×D₄	—	2	3	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	2	3	no	—
210	C₂×D₄	—	2	3	no	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	4	6	no	—
213	C₂×D₄	—	4	6	no	—
214	C₂×D₄	—	8	3	no	—
215	C₂×D₄	—	8	6	no	—
216	C₂×D₄	—	8	3	no	—
217	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
218	C₂²×C₄	—	2	3	no	—
219	C₂²×C₄	—	2	12	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	8	6	no	—
222	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
223	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
224	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
225	SD₁₆	—	4	4	no	—
226	D₈	—	4	12	no	—
227	C₂×C₈	—	2	6	no	—
228	C₂×C₈	—	2	3	no	—
229	C₂×C₈	—	2	3	no	—
230	C₄⋊C₄	—	2	3	no	—
231	C₄⋊C₄	C₂×D₄×C₆₇	1	6	yes	—
232	C₂²⋊C₄	—	2	12	no	—
233	C₂²⋊C₄	—	4	6	no	—
234	C₂²⋊C₄	—	4	6	no	—
235	C₄²	—	8	3	no	—
236	C₄²	C₂×D₄×C₆₇	1	6	yes	—
237	C₄²	#41×C₆₇	1	6	yes	—
238	C₂³	—	2	6	no	—
239	C₂³	—	4	3	no	—
240	C₂³	—	4	1	no	—
241	C₂³	—	4	3	no	—
242	C₂³	—	4	6	no	—
243	C₂³	—	4	3	no	—
244	C₂³	—	4	6	no	—
245	C₂³	—	4	3	no	—
246	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
247	Q₈	—	4	1	no	—
248	Q₈	—	4	3	no	—
249	D₄	—	2	6	no	—
250	D₄	—	4	3	no	—
251	D₄	—	4	3	no	—
252	D₄	—	4	6	no	—
253	D₄	—	8	12	no	—
254	D₄	—	8	12	no	—
255	D₄	#127×C₆₇	1	4	yes	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	6	no	—
258	C₂×C₄	—	2	3	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	4	6	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	8	12	no	—
263	C₂×C₄	#121×C₆₇	1	6	yes	—
264	C₂×C₄	#121×C₆₇	1	6	yes	—
265	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
266	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
267	C₈	—	4	3	no	—
268	C₈	—	4	1	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	6	no	—
271	C₂²	—	4	6	no	—
272	C₂²	—	8	6	no	—
273	C₂²	—	8	3	no	—
274	C₂²	—	8	6	no	—
275	C₂²	—	8	3	no	—
276	C₂²	—	8	3	no	—
277	C₂²	C₂×#121×C₆₇	1	3	yes	—
278	C₂²	#576×C₆₇	1	3	yes	—
279	C₂²	#577×C₆₇	1	1	yes	characteristic
280	C₄	—	2	6	no	—
281	C₄	—	2	6	no	—
282	C₄	—	4	6	no	—
283	C₄	—	8	1	no	—
284	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
285	C₄	#532×C₆₇	1	3	yes	—
286	C₂	—	4	4	no	—
287	C₂	—	16	3	no	—
288	C₂	C₆₇×#2856	1	1	yes	characteristic
289	C₂	C₆₇×#3041	1	3	yes	—
290	C₂	C₆₇×#3045	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#463×C₆₇	C₂	semidirect
2. C₂×#459×C₆₇	C₂	semidirect
3. C₆₇×#2657	C₂	semidirect
4. C₆₇×#2275	C₂	semidirect
5. C₆₇×#1983	C₂	semidirect
6. C₆₇×#1982	C₂	semidirect
7. C₆₇×#1843	C₂	semidirect
42. C₄×D₄×C₆₇	D₄	semidirect
140. #33166	C₆₇	direct (N × H)
146. C₂×#463	C₂×C₆₇	semidirect
147. C₂×#459	C₂×C₆₇	semidirect
148. #2657	C₂×C₆₇	semidirect
149. #2275	C₂×C₆₇	semidirect
150. #1983	C₂×C₆₇	semidirect
151. #1982	C₂×C₆₇	semidirect
152. #1843	C₂×C₆₇	semidirect
153. C₆₇	#33166	direct (N × H)
188. C₄×D₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#463×C₆₇	2
C₂×#459×C₆₇	2
C₆₇×#2657	2
C₆₇×#2275	2
C₆₇×#1983	2
C₆₇×#1982	2
C₆₇×#1843	2
#33166	67

Subgroups — C67×#33166 #30712417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33166 #30712417