Subgroups — C₆₇×#9734 #30688985

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1982	C₂	1	2	yes	maximal, complemented
3	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₆₇	—	2	1	no	—
7	C₂²×D₈×C₆₇	—	2	1	no	—
8	C₂×#127×C₆₇	C₂²	1	2	yes	—
9	#493×C₆₇	—	2	2	no	—
10	#493×C₆₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#102×C₆₇	—	2	4	no	—
14	C₂×C₄×C₈×C₆₇	—	2	1	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	#336×C₆₇	—	2	1	no	—
18	#335×C₆₇	—	2	2	no	—
19	#331×C₆₇	—	2	4	no	—
20	C₂²×D₄×C₆₇	—	2	2	no	—
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂²×D₄×C₆₇	—	2	2	no	—
23	C₂×D₈×C₆₇	—	2	4	no	—
24	C₂×D₈×C₆₇	—	2	4	no	—
25	C₂×D₈×C₆₇	—	4	2	no	—
26	C₂×D₈×C₆₇	—	4	2	no	—
27	C₂×D₈×C₆₇	—	4	2	no	—
28	C₂×D₈×C₆₇	—	4	2	no	—
29	C₂²×C₈×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	#127×C₆₇	—	2	2	no	—
34	#127×C₆₇	—	2	2	no	—
35	#127×C₆₇	—	2	2	no	—
36	#127×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
38	C₂×C₄²×C₆₇	—	2	1	no	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	1	no	—
43	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
44	#102×C₆₇	—	4	8	no	—
45	C₄×C₈×C₆₇	—	4	2	no	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	D₄	1	2	yes	—
48	#95×C₆₇	—	2	1	no	—
49	C₂⁴×C₆₇	—	4	2	no	—
50	C₂×D₄×C₆₇	—	2	4	no	—
51	C₂×D₄×C₆₇	—	2	4	no	—
52	C₂×D₄×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	4	2	no	—
55	C₂×D₄×C₆₇	—	4	2	no	—
56	C₂×D₄×C₆₇	—	4	4	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂×D₄×C₆₇	—	4	4	no	—
60	C₂×D₄×C₆₇	—	4	2	no	—
61	C₂×D₄×C₆₇	—	4	2	no	—
62	C₂×D₄×C₆₇	—	8	4	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	D₈×C₆₇	—	4	8	no	—
68	D₈×C₆₇	—	4	8	no	—
69	C₂×C₈×C₆₇	—	2	1	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	4	1	no	—
73	C₂×C₈×C₆₇	—	4	1	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	4	4	no	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	D₈	1	2	yes	—
79	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	D₈	1	4	yes	complemented
82	C₄²×C₆₇	—	4	1	no	—
83	C₄²×C₆₇	—	4	1	no	—
84	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₆₇	#41	1	1	yes	characteristic
88	C₂³×C₆₇	—	4	4	no	—
89	C₂³×C₆₇	—	8	2	no	—
90	C₂³×C₆₇	—	8	2	no	—
91	C₂³×C₆₇	—	8	4	no	—
92	C₂³×C₆₇	—	8	4	no	—
93	C₂³×C₆₇	#121	1	1	yes	characteristic
94	D₄×C₆₇	—	4	4	no	—
95	D₄×C₆₇	—	4	4	no	—
96	D₄×C₆₇	—	4	8	no	—
97	D₄×C₆₇	—	8	4	no	—
98	D₄×C₆₇	—	8	4	no	—
99	D₄×C₆₇	—	8	2	no	—
100	D₄×C₆₇	—	8	2	no	—
101	D₄×C₆₇	—	8	2	no	—
102	D₄×C₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#135	1	2	yes	—
117	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈×C₆₇	—	2	2	no	—
120	C₈×C₆₇	—	4	2	no	—
121	C₈×C₆₇	—	8	2	no	—
122	C₈×C₆₇	#102	1	4	yes	complemented
123	C₂²×C₆₇	—	8	4	no	—
124	C₂²×C₆₇	—	8	4	no	—
125	C₂²×C₆₇	—	8	4	no	—
126	C₂²×C₆₇	—	16	4	no	—
127	C₂²×C₆₇	—	16	4	no	—
128	C₂²×C₆₇	#390	1	1	yes	characteristic
129	C₂²×C₆₇	#459	1	1	yes	characteristic
130	C₂²×C₆₇	#460	1	1	yes	characteristic
131	C₂²×C₆₇	#466	1	2	yes	—
132	C₂²×C₆₇	#472	1	2	yes	—
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
137	C₄×C₆₇	#418	1	1	yes	characteristic
138	C₄×C₆₇	C₄×D₈	1	2	yes	—
139	C₄×C₆₇	#459	1	1	yes	characteristic
140	C₄×C₆₇	#460	1	1	yes	characteristic
141	C₄×C₆₇	#493	1	1	yes	characteristic
142	C₄×C₆₇	#495	1	1	yes	characteristic
143	#9734	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₆₇	—	16	4	no	—
145	C₂×C₆₇	#1667	1	2	yes	—
146	C₂×C₆₇	#1668	1	2	yes	—
147	C₂×C₆₇	#1970	1	1	yes	characteristic
148	C₂×C₆₇	#1978	1	1	yes	characteristic
149	C₂×C₆₇	#1982	1	1	yes	characteristic
150	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
151	#1982	C₂×C₆₇	1	2	yes	complemented
152	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
153	C₂×#335	C₂×C₆₇	1	1	yes	characteristic, complemented
154	C₂×#331	C₂×C₆₇	1	2	yes	complemented
155	C₆₇	#9734	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×D₈	—	2	1	no	—
157	C₂²×D₈	—	2	1	no	—
158	C₂×#127	C₂²×C₆₇	1	2	yes	—
159	#493	—	2	2	no	—
160	#493	C₄×C₆₇	1	4	yes	complemented
161	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
163	C₂×#102	—	2	4	no	—
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
166	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#335	—	2	2	no	—
169	#331	—	2	4	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	—	4	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
183	#127	—	2	2	no	—
184	#127	—	2	2	no	—
185	#127	—	2	2	no	—
186	#127	C₂×C₄×C₆₇	1	4	yes	—
187	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
194	#102	—	4	8	no	—
195	C₄×C₈	—	4	2	no	—
196	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
197	C₄×C₈	D₄×C₆₇	1	2	yes	—
198	#95	—	2	1	no	—
199	C₂⁴	—	4	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
217	D₈	—	4	8	no	—
218	D₈	—	4	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
227	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
228	C₂×C₈	D₈×C₆₇	1	2	yes	—
229	C₂×C₈	SD₁₆×C₆₇	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	D₈×C₆₇	1	4	yes	complemented
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
235	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₄²	#41×C₆₇	1	1	yes	characteristic
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	4	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#121×C₆₇	1	1	yes	characteristic
244	D₄	—	4	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	8	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
260	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
261	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
262	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
265	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
266	C₂×C₄	#135×C₆₇	1	2	yes	—
267	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
268	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	#102×C₆₇	1	4	yes	complemented
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	16	4	no	—
277	C₂²	—	16	4	no	—
278	C₂²	#390×C₆₇	1	1	yes	characteristic
279	C₂²	#459×C₆₇	1	1	yes	characteristic
280	C₂²	#460×C₆₇	1	1	yes	characteristic
281	C₂²	#466×C₆₇	1	2	yes	—
282	C₂²	#472×C₆₇	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	4	no	—
286	C₄	C₂×#102×C₆₇	1	1	yes	characteristic
287	C₄	#418×C₆₇	1	1	yes	characteristic
288	C₄	C₄×D₈×C₆₇	1	2	yes	—
289	C₄	#459×C₆₇	1	1	yes	characteristic
290	C₄	#460×C₆₇	1	1	yes	characteristic
291	C₄	#493×C₆₇	1	1	yes	characteristic
292	C₄	#495×C₆₇	1	1	yes	characteristic
293	C₂	—	16	4	no	—
294	C₂	C₆₇×#1667	1	2	yes	—
295	C₂	C₆₇×#1668	1	2	yes	—
296	C₂	C₆₇×#1970	1	1	yes	characteristic
297	C₂	C₆₇×#1978	1	1	yes	characteristic
298	C₂	C₆₇×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#493×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1982	C₂	semidirect
3. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₂×#335×C₆₇	C₂	semidirect
5. C₂×#331×C₆₇	C₂	semidirect
10. #493×C₆₇	C₄	semidirect
81. C₄⋊C₄×C₆₇	D₈	semidirect
122. C₈×C₆₇	#102	semidirect
143. #9734	C₆₇	direct (N × H)
151. #1982	C₂×C₆₇	semidirect
152. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
153. C₂×#335	C₂×C₆₇	semidirect
154. C₂×#331	C₂×C₆₇	semidirect
155. C₆₇	#9734	direct (N × H)
160. #493	C₄×C₆₇	semidirect
231. C₄⋊C₄	D₈×C₆₇	semidirect
272. C₈	#102×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₆₇	2
C₆₇×#1982	2
C₈×C₄⋊C₄×C₆₇	2
C₂×#335×C₆₇	2
C₂×#331×C₆₇	2
#9734	67

Subgroups — C67×#9734 #30688985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9734 #30688985