Subgroups — C₆₇×#32872 #30712123

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2103	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2098	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2082	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2070	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2049	C₂	1	2	yes	maximal
6	C₆₇×#2047	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2020	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1902	C₂	1	2	yes	maximal
9	C₆₇×#1858	C₂	1	1	yes	maximal, characteristic
10	C₂×#137×C₆₇	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#568×C₆₇	C₂²	1	1	yes	characteristic
13	#430×C₆₇	C₂²	1	2	yes	—
14	#430×C₆₇	C₂²	1	1	yes	characteristic
15	#430×C₆₇	C₂²	1	2	yes	—
16	#424×C₆₇	C₂²	1	2	yes	—
17	#421×C₆₇	C₂²	1	2	yes	—
18	#419×C₆₇	C₂²	1	2	yes	—
19	#419×C₆₇	C₂²	1	2	yes	—
20	#413×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#101×C₆₇	C₂²	1	2	yes	—
22	C₂×#101×C₆₇	C₂²	1	2	yes	—
23	C₂×#101×C₆₇	C₂²	1	2	yes	—
24	#408×C₆₇	C₂²	1	1	yes	characteristic
25	#344×C₆₇	C₂²	1	2	yes	—
26	#344×C₆₇	C₂²	1	2	yes	—
27	#341×C₆₇	C₂²	1	1	yes	characteristic
28	#341×C₆₇	C₂²	1	2	yes	—
29	#338×C₆₇	C₂²	1	2	yes	—
30	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
33	#137×C₆₇	—	4	2	no	—
34	#137×C₆₇	—	4	2	no	—
35	C₂×Q₁₆×C₆₇	—	2	2	no	—
36	C₂×Q₁₆×C₆₇	—	4	2	no	—
37	C₂×Q₁₆×C₆₇	—	4	1	no	—
38	C₂×SD₁₆×C₆₇	—	2	2	no	—
39	#131×C₆₇	—	4	2	no	—
40	C₂×M₁₆×C₆₇	—	4	1	no	—
41	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	#117×C₆₇	C₂³	1	2	yes	—
48	#108×C₆₇	—	4	2	no	—
49	#108×C₆₇	—	4	2	no	—
50	#108×C₆₇	—	4	1	no	—
51	C₄⋊C₈×C₆₇	—	4	2	no	—
52	#104×C₆₇	—	4	2	no	—
53	#103×C₆₇	—	4	2	no	—
54	#103×C₆₇	—	4	2	no	—
55	#101×C₆₇	—	2	1	no	—
56	#101×C₆₇	—	4	2	no	—
57	#101×C₆₇	—	4	2	no	—
58	#101×C₆₇	—	4	2	no	—
59	#100×C₆₇	—	2	1	no	—
60	C₂²⋊C₈×C₆₇	—	4	1	no	—
61	#41×C₆₇	—	2	2	no	—
62	#41×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	—	2	2	no	—
65	C₂×Q₈×C₆₇	—	2	2	no	—
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	—	2	2	no	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
70	C₂×D₄×C₆₇	—	2	1	no	—
71	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
75	Q₁₆×C₆₇	—	4	4	no	—
76	Q₁₆×C₆₇	—	4	4	no	—
77	SD₁₆×C₆₇	—	4	4	no	—
78	M₁₆×C₆₇	—	2	2	no	—
79	M₁₆×C₆₇	—	2	2	no	—
80	M₁₆×C₆₇	—	2	2	no	—
81	M₁₆×C₆₇	—	2	2	no	—
82	M₁₆×C₆₇	—	2	2	no	—
83	M₁₆×C₆₇	—	2	2	no	—
84	M₁₆×C₆₇	—	2	2	no	—
85	M₁₆×C₆₇	—	2	2	no	—
86	M₁₆×C₆₇	—	2	2	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	4	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	2	no	—
102	Q₈×C₆₇	—	4	2	no	—
103	Q₈×C₆₇	—	4	2	no	—
104	Q₈×C₆₇	—	4	2	no	—
105	Q₈×C₆₇	—	4	2	no	—
106	D₄×C₆₇	—	2	2	no	—
107	D₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	4	1	no	—
110	C₂×C₄×C₆₇	—	4	1	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
117	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
119	C₈×C₆₇	—	4	2	no	—
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	—	4	1	no	—
122	C₈×C₆₇	—	4	2	no	—
123	C₈×C₆₇	—	4	2	no	—
124	C₈×C₆₇	—	4	1	no	—
125	C₂²×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	8	1	no	—
128	C₂²×C₆₇	#539	1	1	yes	characteristic
129	C₂²×C₆₇	#562	1	2	yes	—
130	C₄×C₆₇	—	4	1	no	—
131	C₄×C₆₇	—	8	2	no	—
132	C₄×C₆₇	—	8	2	no	—
133	C₄×C₆₇	#538	1	2	yes	—
134	C₄×C₆₇	#560	1	1	yes	characteristic
135	C₄×C₆₇	#560	1	1	yes	characteristic
136	#32872	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	8	1	no	—
140	C₂×C₆₇	#2826	1	1	yes	characteristic
141	#2103	C₂×C₆₇	1	2	yes	complemented
142	#2098	C₂×C₆₇	1	1	yes	characteristic
143	#2082	C₂×C₆₇	1	2	yes	complemented
144	#2070	C₂×C₆₇	1	2	yes	complemented
145	#2049	C₂×C₆₇	1	2	yes	—
146	#2047	C₂×C₆₇	1	2	yes	complemented
147	#2020	C₂×C₆₇	1	1	yes	characteristic
148	#1902	C₂×C₆₇	1	2	yes	—
149	#1858	C₂×C₆₇	1	1	yes	characteristic
150	C₆₇	#32872	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#137	C₂²×C₆₇	1	2	yes	—
152	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
153	#568	C₂²×C₆₇	1	1	yes	characteristic
154	#430	C₂²×C₆₇	1	2	yes	—
155	#430	C₂²×C₆₇	1	1	yes	characteristic
156	#430	C₂²×C₆₇	1	2	yes	—
157	#424	C₂²×C₆₇	1	2	yes	—
158	#421	C₂²×C₆₇	1	2	yes	—
159	#419	C₂²×C₆₇	1	2	yes	—
160	#419	C₂²×C₆₇	1	2	yes	—
161	#413	C₂²×C₆₇	1	1	yes	characteristic
162	C₂×#101	C₂²×C₆₇	1	2	yes	—
163	C₂×#101	C₂²×C₆₇	1	2	yes	—
164	C₂×#101	C₂²×C₆₇	1	2	yes	—
165	#408	C₂²×C₆₇	1	1	yes	characteristic
166	#344	C₂²×C₆₇	1	2	yes	—
167	#344	C₂²×C₆₇	1	2	yes	—
168	#341	C₂²×C₆₇	1	1	yes	characteristic
169	#341	C₂²×C₆₇	1	2	yes	—
170	#338	C₂²×C₆₇	1	2	yes	—
171	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
173	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
174	#137	—	4	2	no	—
175	#137	—	4	2	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	—	4	1	no	—
178	C₂×Q₁₆	—	4	2	no	—
179	C₂×SD₁₆	—	2	2	no	—
180	#131	—	4	2	no	—
181	C₂×M₁₆	—	4	1	no	—
182	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
183	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
184	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
185	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
186	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
188	#117	C₂³×C₆₇	1	2	yes	—
189	#108	—	4	1	no	—
190	#108	—	4	2	no	—
191	#108	—	4	2	no	—
192	C₄⋊C₈	—	4	2	no	—
193	#104	—	4	2	no	—
194	#103	—	4	2	no	—
195	#103	—	4	2	no	—
196	#101	—	2	1	no	—
197	#101	—	4	2	no	—
198	#101	—	4	2	no	—
199	#101	—	4	2	no	—
200	#100	—	2	1	no	—
201	C₂²⋊C₈	—	4	1	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	#41×C₆₇	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	#41×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
216	Q₁₆	—	4	4	no	—
217	Q₁₆	—	4	4	no	—
218	SD₁₆	—	4	4	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
257	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
258	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
259	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	#539×C₆₇	1	1	yes	characteristic
270	C₂²	#562×C₆₇	1	2	yes	—
271	C₄	—	4	1	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	2	no	—
274	C₄	#538×C₆₇	1	2	yes	—
275	C₄	#560×C₆₇	1	1	yes	characteristic
276	C₄	#560×C₆₇	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₆₇×#2826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	C₂³
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×#2103	C₂	semidirect
3. C₆₇×#2082	C₂	semidirect
4. C₆₇×#2070	C₂	semidirect
6. C₆₇×#2047	C₂	semidirect
136. #32872	C₆₇	direct (N × H)
141. #2103	C₂×C₆₇	semidirect
143. #2082	C₂×C₆₇	semidirect
144. #2070	C₂×C₆₇	semidirect
146. #2047	C₂×C₆₇	semidirect
150. C₆₇	#32872	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2103	2
C₆₇×#2098	2
C₆₇×#2082	2
C₆₇×#2070	2
C₆₇×#2049	2
C₆₇×#2047	2
C₆₇×#2020	2
C₆₇×#1902	2
C₆₇×#1858	2
#32872	67

Subgroups — C67×#32872 #30712123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32872 #30712123