Subgroups — C₆₇×#32947 #30712198

Order 17152.

1306 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3057	C₂	1	3	yes	maximal, complemented
3	C₆₇×#3028	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3027	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1910	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1909	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1812	C₂	1	3	yes	maximal, complemented
8	#585×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
10	#568×C₆₇	—	2	6	no	—
11	#568×C₆₇	C₂²	1	3	yes	—
12	C₂×#131×C₆₇	—	2	6	no	—
13	C₂×#131×C₆₇	C₂²	1	3	yes	—
14	#536×C₆₇	C₂²	1	3	yes	—
15	#535×C₆₇	C₂²	1	3	yes	—
16	#468×C₆₇	—	2	6	no	—
17	#465×C₆₇	—	2	6	no	—
18	#458×C₆₇	—	2	4	no	—
19	B(4,𝔽₂)×C₆₇	—	2	4	no	—
20	#427×C₆₇	C₂²	1	3	yes	complemented
21	#419×C₆₇	C₂²	1	3	yes	complemented
22	#418×C₆₇	C₂²	1	3	yes	complemented
23	#410×C₆₇	C₂²	1	3	yes	—
24	#410×C₆₇	C₂²	1	3	yes	—
25	H(3,ℤ/4)×C₆₇	C₂²	1	6	yes	complemented
26	2₋¹⁺⁴×C₆₇	—	2	3	no	—
27	H(5,𝔽₂)×C₆₇	—	2	3	no	—
28	H(5,𝔽₂)×C₆₇	D₄	1	4	yes	—
29	C₂×#41×C₆₇	—	2	6	no	—
30	C₂×#41×C₆₇	C₂³	1	3	yes	—
31	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×SD₁₆×C₆₇	—	2	3	no	—
34	C₂×SD₁₆×C₆₇	—	2	3	no	—
35	#131×C₆₇	—	2	4	no	—
36	#131×C₆₇	—	2	12	no	—
37	#131×C₆₇	—	2	6	no	—
38	#131×C₆₇	—	2	12	no	—
39	#131×C₆₇	—	2	6	no	—
40	C₂×M₁₆×C₆₇	—	2	12	no	—
41	C₂×M₁₆×C₆₇	—	2	6	no	—
42	C₂×M₁₆×C₆₇	D₄	1	6	yes	complemented
43	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	D₄	1	6	yes	complemented
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	—	2	3	no	—
48	#127×C₆₇	—	2	3	no	—
49	#124×C₆₇	—	2	3	no	—
50	#124×C₆₇	—	2	3	no	—
51	#123×C₆₇	—	2	6	no	—
52	C₂²⋊Q₈×C₆₇	—	2	6	no	—
53	#121×C₆₇	—	2	6	no	—
54	#120×C₆₇	—	2	6	no	—
55	#117×C₆₇	C₂³	1	3	yes	—
56	#117×C₆₇	C₂³	1	3	yes	—
57	#107×C₆₇	—	2	3	no	—
58	#106×C₆₇	—	2	3	no	—
59	#103×C₆₇	—	2	6	no	—
60	#102×C₆₇	—	2	6	no	—
61	#99×C₆₇	—	2	6	no	—
62	#99×C₆₇	—	2	6	no	—
63	C₂⁴×C₆₇	—	2	1	no	—
64	#41×C₆₇	—	2	4	no	—
65	#41×C₆₇	—	2	12	no	—
66	#41×C₆₇	—	2	12	no	—
67	#41×C₆₇	—	2	6	no	—
68	#41×C₆₇	—	2	6	no	—
69	C₂×Q₈×C₆₇	—	2	6	no	—
70	C₂×Q₈×C₆₇	—	2	3	no	—
71	C₂×Q₈×C₆₇	—	2	3	no	—
72	C₂×Q₈×C₆₇	#41	1	3	yes	—
73	C₂×D₄×C₆₇	—	2	6	no	—
74	C₂×D₄×C₆₇	—	2	12	no	—
75	C₂×D₄×C₆₇	—	2	3	no	—
76	C₂×D₄×C₆₇	—	2	3	no	—
77	C₂×D₄×C₆₇	—	4	6	no	—
78	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
79	C₂×D₄×C₆₇	#41	1	3	yes	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₆₇	—	4	1	no	—
85	SD₁₆×C₆₇	—	4	3	no	—
86	M₁₆×C₆₇	—	2	12	no	—
87	M₁₆×C₆₇	—	2	6	no	—
88	M₁₆×C₆₇	—	2	12	no	—
89	C₂×C₈×C₆₇	—	2	12	no	—
90	C₂×C₈×C₆₇	—	2	6	no	—
91	C₂×C₈×C₆₇	—	2	6	no	—
92	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
95	C₄⋊C₄×C₆₇	—	2	3	no	—
96	C₄⋊C₄×C₆₇	—	2	3	no	—
97	C₄⋊C₄×C₆₇	—	4	6	no	—
98	C₂²⋊C₄×C₆₇	—	2	3	no	—
99	C₂²⋊C₄×C₆₇	—	2	3	no	—
100	C₂²⋊C₄×C₆₇	—	4	6	no	—
101	C₂²⋊C₄×C₆₇	—	4	6	no	—
102	C₄²×C₆₇	—	2	3	no	—
103	C₄²×C₆₇	—	2	3	no	—
104	C₂³×C₆₇	—	2	4	no	—
105	C₂³×C₆₇	—	4	3	no	—
106	C₂³×C₆₇	—	16	1	no	—
107	C₂³×C₆₇	#121	1	6	yes	—
108	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	4	no	—
110	Q₈×C₆₇	—	2	6	no	—
111	Q₈×C₆₇	—	4	1	no	—
112	Q₈×C₆₇	—	4	3	no	—
113	D₄×C₆₇	—	2	12	no	—
114	D₄×C₆₇	—	2	12	no	—
115	D₄×C₆₇	—	2	6	no	—
116	D₄×C₆₇	—	4	1	no	—
117	D₄×C₆₇	—	4	3	no	—
118	D₄×C₆₇	—	16	6	no	—
119	C₂×C₄×C₆₇	—	2	12	no	—
120	C₂×C₄×C₆₇	—	2	6	no	—
121	C₂×C₄×C₆₇	—	2	6	no	—
122	C₂×C₄×C₆₇	—	4	3	no	—
123	C₂×C₄×C₆₇	—	4	3	no	—
124	C₂×C₄×C₆₇	—	4	3	no	—
125	C₂×C₄×C₆₇	#121	1	6	yes	—
126	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
127	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
128	C₈×C₆₇	—	2	6	no	—
129	C₈×C₆₇	#120	1	1	yes	characteristic
130	C₈×C₆₇	#120	1	3	yes	—
131	C₂²×C₆₇	—	2	6	no	—
132	C₂²×C₆₇	—	4	1	no	—
133	C₂²×C₆₇	—	4	6	no	—
134	C₂²×C₆₇	—	8	6	no	—
135	C₂²×C₆₇	—	8	4	no	—
136	C₂²×C₆₇	—	8	1	no	—
137	C₂²×C₆₇	C₂×#121	1	3	yes	—
138	C₄×C₆₇	—	2	6	no	—
139	C₄×C₆₇	—	8	3	no	—
140	C₄×C₆₇	—	8	3	no	—
141	C₄×C₆₇	—	8	1	no	—
142	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
143	C₄×C₆₇	#525	1	3	yes	—
144	#32947	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	2	3	no	—
146	C₂×C₆₇	—	4	6	no	—
147	C₂×C₆₇	—	16	1	no	—
148	C₂×C₆₇	#2847	1	1	yes	characteristic
149	#3573	C₂×C₆₇	1	1	yes	characteristic, complemented
150	#3057	C₂×C₆₇	1	3	yes	complemented
151	#3028	C₂×C₆₇	1	1	yes	characteristic, complemented
152	#3027	C₂×C₆₇	1	1	yes	characteristic
153	#1910	C₂×C₆₇	1	3	yes	complemented
154	#1909	C₂×C₆₇	1	3	yes	complemented
155	#1812	C₂×C₆₇	1	3	yes	complemented
156	C₆₇	#32947	1	1	yes	characteristic, direct product factor, Sylow
157	#585	C₂²×C₆₇	1	1	yes	characteristic
158	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
159	#568	—	2	6	no	—
160	#568	C₂²×C₆₇	1	3	yes	—
161	C₂×#131	—	2	6	no	—
162	C₂×#131	C₂²×C₆₇	1	3	yes	—
163	#536	C₂²×C₆₇	1	3	yes	—
164	#535	C₂²×C₆₇	1	3	yes	—
165	#468	—	2	6	no	—
166	#465	—	2	6	no	—
167	#458	—	2	4	no	—
168	B(4,𝔽₂)	—	2	4	no	—
169	#427	C₂²×C₆₇	1	3	yes	complemented
170	#419	C₂²×C₆₇	1	3	yes	complemented
171	#418	C₂²×C₆₇	1	3	yes	complemented
172	#410	C₂²×C₆₇	1	3	yes	—
173	#410	C₂²×C₆₇	1	3	yes	—
174	H(3,ℤ/4)	C₂²×C₆₇	1	6	yes	complemented
175	2₋¹⁺⁴	—	2	3	no	—
176	H(5,𝔽₂)	—	2	3	no	—
177	H(5,𝔽₂)	D₄×C₆₇	1	4	yes	—
178	C₂×#41	—	2	6	no	—
179	C₂×#41	C₂³×C₆₇	1	3	yes	—
180	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
181	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
182	C₂×SD₁₆	—	2	3	no	—
183	C₂×SD₁₆	—	2	3	no	—
184	#131	—	2	4	no	—
185	#131	—	2	12	no	—
186	#131	—	2	6	no	—
187	#131	—	2	12	no	—
188	#131	—	2	6	no	—
189	C₂×M₁₆	—	2	12	no	—
190	C₂×M₁₆	—	2	6	no	—
191	C₂×M₁₆	D₄×C₆₇	1	6	yes	complemented
192	C₂×M₁₆	C₂³×C₆₇	1	3	yes	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	D₄×C₆₇	1	6	yes	complemented
195	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	3	no	—
197	#127	—	2	3	no	—
198	#124	—	2	3	no	—
199	#124	—	2	3	no	—
200	#123	—	2	6	no	—
201	C₂²⋊Q₈	—	2	6	no	—
202	#121	—	2	6	no	—
203	#120	—	2	6	no	—
204	#117	C₂³×C₆₇	1	3	yes	—
205	#117	C₂³×C₆₇	1	3	yes	—
206	#107	—	2	3	no	—
207	#106	—	2	3	no	—
208	#103	—	2	6	no	—
209	#102	—	2	6	no	—
210	#99	—	2	6	no	—
211	#99	—	2	6	no	—
212	C₂⁴	—	2	1	no	—
213	#41	—	2	4	no	—
214	#41	—	2	12	no	—
215	#41	—	2	12	no	—
216	#41	—	2	6	no	—
217	#41	—	2	6	no	—
218	C₂×Q₈	—	2	6	no	—
219	C₂×Q₈	—	2	3	no	—
220	C₂×Q₈	—	2	3	no	—
221	C₂×Q₈	#41×C₆₇	1	3	yes	—
222	C₂×D₄	—	2	6	no	—
223	C₂×D₄	—	2	12	no	—
224	C₂×D₄	—	2	3	no	—
225	C₂×D₄	—	2	3	no	—
226	C₂×D₄	—	4	6	no	—
227	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
228	C₂×D₄	#41×C₆₇	1	3	yes	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
232	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
233	SD₁₆	—	4	1	no	—
234	SD₁₆	—	4	3	no	—
235	M₁₆	—	2	12	no	—
236	M₁₆	—	2	6	no	—
237	M₁₆	—	2	12	no	—
238	C₂×C₈	—	2	12	no	—
239	C₂×C₈	—	2	6	no	—
240	C₂×C₈	—	2	6	no	—
241	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
242	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
243	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
244	C₄⋊C₄	—	2	3	no	—
245	C₄⋊C₄	—	2	3	no	—
246	C₄⋊C₄	—	4	6	no	—
247	C₂²⋊C₄	—	2	3	no	—
248	C₂²⋊C₄	—	2	3	no	—
249	C₂²⋊C₄	—	4	6	no	—
250	C₂²⋊C₄	—	4	6	no	—
251	C₄²	—	2	3	no	—
252	C₄²	—	2	3	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	4	3	no	—
255	C₂³	—	16	1	no	—
256	C₂³	#121×C₆₇	1	6	yes	—
257	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
258	Q₈	—	2	4	no	—
259	Q₈	—	2	6	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	4	3	no	—
262	D₄	—	2	12	no	—
263	D₄	—	2	12	no	—
264	D₄	—	2	6	no	—
265	D₄	—	4	1	no	—
266	D₄	—	4	3	no	—
267	D₄	—	16	6	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	6	no	—
270	C₂×C₄	—	2	6	no	—
271	C₂×C₄	—	4	3	no	—
272	C₂×C₄	—	4	3	no	—
273	C₂×C₄	—	4	3	no	—
274	C₂×C₄	#121×C₆₇	1	6	yes	—
275	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
276	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
277	C₈	—	2	6	no	—
278	C₈	#120×C₆₇	1	1	yes	characteristic
279	C₈	#120×C₆₇	1	3	yes	—
280	C₂²	—	2	6	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	6	no	—
283	C₂²	—	8	6	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	8	1	no	—
286	C₂²	C₂×#121×C₆₇	1	3	yes	—
287	C₄	—	2	6	no	—
288	C₄	—	8	3	no	—
289	C₄	—	8	3	no	—
290	C₄	—	8	1	no	—
291	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
292	C₄	#525×C₆₇	1	3	yes	—
293	C₂	—	2	3	no	—
294	C₂	—	4	6	no	—
295	C₂	—	16	1	no	—
296	C₂	C₆₇×#2847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3027	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3573	C₂	semidirect
2. C₆₇×#3057	C₂	semidirect
3. C₆₇×#3028	C₂	semidirect
5. C₆₇×#1910	C₂	semidirect
6. C₆₇×#1909	C₂	semidirect
7. C₆₇×#1812	C₂	semidirect
9. C₂²×SD₁₆×C₆₇	C₂²	semidirect
20. #427×C₆₇	C₂²	semidirect
21. #419×C₆₇	C₂²	semidirect
22. #418×C₆₇	C₂²	semidirect
25. H(3,ℤ/4)×C₆₇	C₂²	semidirect
42. C₂×M₁₆×C₆₇	D₄	semidirect
45. C₂²×C₈×C₆₇	D₄	semidirect
144. #32947	C₆₇	direct (N × H)
149. #3573	C₂×C₆₇	semidirect
150. #3057	C₂×C₆₇	semidirect
151. #3028	C₂×C₆₇	semidirect
153. #1910	C₂×C₆₇	semidirect
154. #1909	C₂×C₆₇	semidirect
155. #1812	C₂×C₆₇	semidirect
156. C₆₇	#32947	direct (N × H)
158. C₂²×SD₁₆	C₂²×C₆₇	semidirect
169. #427	C₂²×C₆₇	semidirect
170. #419	C₂²×C₆₇	semidirect
171. #418	C₂²×C₆₇	semidirect
174. H(3,ℤ/4)	C₂²×C₆₇	semidirect
191. C₂×M₁₆	D₄×C₆₇	semidirect
194. C₂²×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3573	2
C₆₇×#3057	2
C₆₇×#3028	2
C₆₇×#3027	2
C₆₇×#1910	2
C₆₇×#1909	2
C₆₇×#1812	2
#32947	67

Subgroups — C67×#32947 #30712198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32947 #30712198