Subgroups — C₆₇×#32870 #30712121

Order 17152.

502 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	C₆₇×#2103	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2098	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2081	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2069	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2048	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2045	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2018	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1903	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1858	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
11	C₂²×D₈×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	#418×C₆₇	C₂²	1	2	yes	—
19	#418×C₆₇	C₂²	1	2	yes	—
20	#413×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#100×C₆₇	C₂²	1	2	yes	—
22	C₂×#100×C₆₇	C₂²	1	2	yes	—
23	C₂×#100×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×D₄×C₆₇	C₂³	1	2	yes	—
33	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
34	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
35	C₂×SD₁₆×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	—	2	2	no	—
37	C₂×D₈×C₆₇	—	4	2	no	—
38	C₂×D₈×C₆₇	—	4	1	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	#102×C₆₇	—	4	2	no	—
54	#102×C₆₇	—	4	2	no	—
55	#101×C₆₇	—	2	1	no	—
56	#100×C₆₇	—	2	1	no	—
57	#100×C₆₇	—	4	2	no	—
58	#100×C₆₇	—	4	2	no	—
59	#100×C₆₇	—	4	2	no	—
60	C₂²⋊C₈×C₆₇	—	4	1	no	—
61	C₂⁴×C₆₇	—	2	2	no	—
62	#41×C₆₇	—	2	2	no	—
63	#41×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
65	C₂×D₄×C₆₇	—	2	1	no	—
66	C₂×D₄×C₆₇	—	2	2	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	—	2	2	no	—
70	C₂×D₄×C₆₇	—	2	2	no	—
71	C₂×D₄×C₆₇	—	2	2	no	—
72	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
75	SD₁₆×C₆₇	—	4	4	no	—
76	D₈×C₆₇	—	4	4	no	—
77	D₈×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	—	4	2	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	—	16	2	no	—
104	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
105	Q₈×C₆₇	—	2	2	no	—
106	D₄×C₆₇	—	2	2	no	—
107	D₄×C₆₇	—	4	2	no	—
108	D₄×C₆₇	—	4	2	no	—
109	D₄×C₆₇	—	4	2	no	—
110	D₄×C₆₇	—	4	2	no	—
111	D₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	—	4	2	no	—
122	C₈×C₆₇	—	4	1	no	—
123	C₈×C₆₇	—	4	2	no	—
124	C₈×C₆₇	—	4	2	no	—
125	C₈×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	8	1	no	—
129	C₂²×C₆₇	—	8	2	no	—
130	C₂²×C₆₇	—	16	2	no	—
131	C₂²×C₆₇	—	16	2	no	—
132	C₂²×C₆₇	—	16	2	no	—
133	C₂²×C₆₇	#539	1	1	yes	characteristic
134	C₂²×C₆₇	#562	1	2	yes	—
135	C₄×C₆₇	—	4	1	no	—
136	C₄×C₆₇	—	8	2	no	—
137	C₄×C₆₇	#538	1	2	yes	—
138	C₄×C₆₇	#560	1	1	yes	characteristic
139	C₄×C₆₇	#560	1	1	yes	characteristic
140	#32870	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₆₇	—	2	2	no	—
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	8	1	no	—
144	C₂×C₆₇	—	16	2	no	—
145	C₂×C₆₇	#2826	1	1	yes	characteristic
146	#2103	C₂×C₆₇	1	2	yes	complemented
147	#2098	C₂×C₆₇	1	1	yes	characteristic, complemented
148	#2081	C₂×C₆₇	1	2	yes	complemented
149	#2069	C₂×C₆₇	1	2	yes	complemented
150	#2048	C₂×C₆₇	1	2	yes	complemented
151	#2045	C₂×C₆₇	1	2	yes	complemented
152	#2018	C₂×C₆₇	1	1	yes	characteristic
153	#1903	C₂×C₆₇	1	2	yes	complemented
154	#1858	C₂×C₆₇	1	1	yes	characteristic, complemented
155	C₆₇	#32870	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
157	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
158	#568	C₂²×C₆₇	1	1	yes	characteristic
159	#430	C₂²×C₆₇	1	2	yes	—
160	#430	C₂²×C₆₇	1	1	yes	characteristic
161	#430	C₂²×C₆₇	1	2	yes	—
162	#424	C₂²×C₆₇	1	2	yes	—
163	#421	C₂²×C₆₇	1	2	yes	—
164	#418	C₂²×C₆₇	1	2	yes	—
165	#418	C₂²×C₆₇	1	2	yes	—
166	#413	C₂²×C₆₇	1	1	yes	characteristic
167	C₂×#100	C₂²×C₆₇	1	2	yes	—
168	C₂×#100	C₂²×C₆₇	1	2	yes	—
169	C₂×#100	C₂²×C₆₇	1	2	yes	—
170	#408	C₂²×C₆₇	1	1	yes	characteristic
171	#344	C₂²×C₆₇	1	2	yes	—
172	#344	C₂²×C₆₇	1	2	yes	—
173	#341	C₂²×C₆₇	1	1	yes	characteristic
174	#341	C₂²×C₆₇	1	2	yes	—
175	#338	C₂²×C₆₇	1	2	yes	—
176	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
178	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
179	AGL(1,ℤ/8)	—	4	2	no	—
180	AGL(1,ℤ/8)	—	4	2	no	—
181	C₂×SD₁₆	—	2	2	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	—	4	1	no	—
184	C₂×D₈	—	4	2	no	—
185	#131	—	4	2	no	—
186	C₂×M₁₆	—	4	1	no	—
187	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
188	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
189	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
190	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
191	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
192	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
193	#117	C₂³×C₆₇	1	2	yes	—
194	#108	—	4	1	no	—
195	#108	—	4	2	no	—
196	#108	—	4	2	no	—
197	C₄⋊C₈	—	4	2	no	—
198	#104	—	4	2	no	—
199	#102	—	4	2	no	—
200	#102	—	4	2	no	—
201	#101	—	2	1	no	—
202	#100	—	2	1	no	—
203	#100	—	4	2	no	—
204	#100	—	4	2	no	—
205	#100	—	4	2	no	—
206	C₂²⋊C₈	—	4	1	no	—
207	C₂⁴	—	2	2	no	—
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	C₂×Q₈	#41×C₆₇	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	#41×C₆₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
221	SD₁₆	—	4	4	no	—
222	D₈	—	4	4	no	—
223	D₈	—	4	4	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	16	2	no	—
250	C₂³	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
263	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
264	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
265	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	2	no	—
278	C₂²	—	16	2	no	—
279	C₂²	#539×C₆₇	1	1	yes	characteristic
280	C₂²	#562×C₆₇	1	2	yes	—
281	C₄	—	4	1	no	—
282	C₄	—	8	2	no	—
283	C₄	#538×C₆₇	1	2	yes	—
284	C₄	#560×C₆₇	1	1	yes	characteristic
285	C₄	#560×C₆₇	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	1	no	—
289	C₂	—	16	2	no	—
290	C₂	C₆₇×#2826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2018	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×#2103	C₂	semidirect
2. C₆₇×#2098	C₂	semidirect
3. C₆₇×#2081	C₂	semidirect
4. C₆₇×#2069	C₂	semidirect
5. C₆₇×#2048	C₂	semidirect
6. C₆₇×#2045	C₂	semidirect
8. C₆₇×#1903	C₂	semidirect
9. C₆₇×#1858	C₂	semidirect
140. #32870	C₆₇	direct (N × H)
146. #2103	C₂×C₆₇	semidirect
147. #2098	C₂×C₆₇	semidirect
148. #2081	C₂×C₆₇	semidirect
149. #2069	C₂×C₆₇	semidirect
150. #2048	C₂×C₆₇	semidirect
151. #2045	C₂×C₆₇	semidirect
153. #1903	C₂×C₆₇	semidirect
154. #1858	C₂×C₆₇	semidirect
155. C₆₇	#32870	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×#2081	2
C₆₇×#2069	2
C₆₇×#2048	2
C₆₇×#2045	2
C₆₇×#2018	2
C₆₇×#1903	2
C₆₇×#1858	2
#32870	67

Subgroups — C67×#32870 #30712121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32870 #30712121