Subgroups — C₆₇×#10262 #30689513

Order 17152.

742 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×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#460×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×SD₁₆×C₆₇	—	2	2	no	—
7	C₂²×D₈×C₆₇	—	2	2	no	—
8	C₂×#127×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	#493×C₆₇	—	2	4	no	—
11	#460×C₆₇	—	2	4	no	—
12	#460×C₆₇	—	2	4	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#102×C₆₇	—	2	2	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#336×C₆₇	—	2	2	no	—
18	#331×C₆₇	—	2	4	no	—
19	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
20	C₂²×D₄×C₆₇	—	2	2	no	—
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂²×D₄×C₆₇	—	2	2	no	—
23	C₂×SD₁₆×C₆₇	—	2	8	no	—
24	C₂×SD₁₆×C₆₇	—	4	4	no	—
25	C₂×SD₁₆×C₆₇	—	4	4	no	—
26	C₂×D₈×C₆₇	—	2	8	no	—
27	C₂×D₈×C₆₇	—	4	4	no	—
28	C₂×D₈×C₆₇	—	4	4	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	D₄	1	2	yes	—
32	#127×C₆₇	—	2	2	no	—
33	#127×C₆₇	—	2	2	no	—
34	#127×C₆₇	—	2	2	no	—
35	#127×C₆₇	D₄	1	4	yes	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	—	2	1	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
45	#102×C₆₇	—	2	8	no	—
46	C₄×C₈×C₆₇	—	2	2	no	—
47	C₂⁴×C₆₇	—	4	2	no	—
48	C₂×Q₈×C₆₇	—	2	1	no	—
49	C₂×Q₈×C₆₇	—	2	2	no	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	D₈	1	2	yes	complemented
52	C₂×D₄×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	2	4	no	—
54	C₂×D₄×C₆₇	—	2	4	no	—
55	C₂×D₄×C₆₇	—	4	2	no	—
56	C₂×D₄×C₆₇	—	4	2	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	4	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂×D₄×C₆₇	—	4	2	no	—
64	C₂×D₄×C₆₇	—	8	4	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	SD₁₆×C₆₇	—	4	16	no	—
71	D₈×C₆₇	—	4	16	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	4	2	no	—
82	C₄²×C₆₇	—	4	1	no	—
83	C₄²×C₆₇	—	4	1	no	—
84	C₄²×C₆₇	C₂×D₄	1	2	yes	—
85	C₄²×C₆₇	C₂×D₄	1	2	yes	—
86	C₂³×C₆₇	—	4	4	no	—
87	C₂³×C₆₇	—	8	4	no	—
88	C₂³×C₆₇	—	8	2	no	—
89	C₂³×C₆₇	—	8	2	no	—
90	C₂³×C₆₇	—	8	4	no	—
91	C₂³×C₆₇	#121	1	1	yes	characteristic
92	Q₈×C₆₇	—	2	4	no	—
93	Q₈×C₆₇	—	4	2	no	—
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	2	no	—
105	C₂×C₄×C₆₇	—	4	1	no	—
106	C₂×C₄×C₆₇	—	4	1	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	#120	1	2	yes	—
110	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#121	1	2	yes	—
112	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
115	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
116	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
118	C₈×C₆₇	—	2	4	no	—
119	C₈×C₆₇	—	4	4	no	—
120	C₈×C₆₇	—	8	1	no	—
121	C₂²×C₆₇	—	8	4	no	—
122	C₂²×C₆₇	—	8	4	no	—
123	C₂²×C₆₇	—	8	4	no	—
124	C₂²×C₆₇	—	16	4	no	—
125	C₂²×C₆₇	—	16	4	no	—
126	C₂²×C₆₇	#392	1	1	yes	characteristic
127	C₂²×C₆₇	#460	1	2	yes	—
128	C₂²×C₆₇	#466	1	1	yes	characteristic
129	C₂²×C₆₇	#469	1	1	yes	characteristic
130	C₂²×C₆₇	#472	1	2	yes	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	—	4	2	no	—
133	C₄×C₆₇	—	8	1	no	—
134	C₄×C₆₇	#447	1	2	yes	—
135	C₄×C₆₇	#449	1	2	yes	—
136	C₄×C₆₇	#450	1	2	yes	—
137	C₄×C₆₇	#492	1	1	yes	characteristic
138	C₄×C₆₇	#496	1	1	yes	characteristic
139	#10262	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₆₇	—	16	4	no	—
141	C₂×C₆₇	#1668	1	2	yes	—
142	C₂×C₆₇	#1688	1	2	yes	—
143	C₂×C₆₇	#1999	1	1	yes	characteristic
144	C₂×C₆₇	#2014	1	1	yes	characteristic
145	C₂×C₆₇	#2018	1	1	yes	characteristic
146	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
147	C₂×#460	C₂×C₆₇	1	2	yes	complemented
148	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
149	C₂×#331	C₂×C₆₇	1	2	yes	complemented
150	#1277	C₂×C₆₇	1	1	yes	characteristic, complemented
151	C₆₇	#10262	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×SD₁₆	—	2	2	no	—
153	C₂²×D₈	—	2	2	no	—
154	C₂×#127	C₂²×C₆₇	1	2	yes	—
155	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
156	#493	—	2	4	no	—
157	#460	—	2	4	no	—
158	#460	—	2	4	no	—
159	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
161	C₂×#102	—	2	2	no	—
162	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
163	#336	—	2	2	no	—
164	#331	—	2	4	no	—
165	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
166	C₂²×D₄	—	2	2	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂×SD₁₆	—	2	8	no	—
170	C₂×SD₁₆	—	4	4	no	—
171	C₂×SD₁₆	—	4	4	no	—
172	C₂×D₈	—	2	8	no	—
173	C₂×D₈	—	4	4	no	—
174	C₂×D₈	—	4	4	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	D₄×C₆₇	1	2	yes	—
178	#127	—	2	2	no	—
179	#127	—	2	2	no	—
180	#127	—	2	2	no	—
181	#127	D₄×C₆₇	1	4	yes	—
182	C₄×Q₈	—	2	2	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	D₄×C₆₇	1	4	yes	—
191	#102	—	2	8	no	—
192	C₄×C₈	—	2	2	no	—
193	C₂⁴	—	4	2	no	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	D₈×C₆₇	1	2	yes	complemented
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	8	4	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
216	SD₁₆	—	4	16	no	—
217	D₈	—	4	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	SD₁₆×C₆₇	1	4	yes	complemented
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	C₂×D₄×C₆₇	1	2	yes	—
231	C₄²	C₂×D₄×C₆₇	1	2	yes	—
232	C₂³	—	4	4	no	—
233	C₂³	—	8	4	no	—
234	C₂³	—	8	2	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	4	no	—
237	C₂³	#121×C₆₇	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	4	2	no	—
240	D₄	—	4	4	no	—
241	D₄	—	4	8	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	4	no	—
244	D₄	—	8	4	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	#120×C₆₇	1	2	yes	—
256	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
257	C₂×C₄	#121×C₆₇	1	2	yes	—
258	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
260	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
261	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
262	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
263	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	4	4	no	—
266	C₈	—	8	1	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	16	4	no	—
271	C₂²	—	16	4	no	—
272	C₂²	#392×C₆₇	1	1	yes	characteristic
273	C₂²	#460×C₆₇	1	2	yes	—
274	C₂²	#466×C₆₇	1	1	yes	characteristic
275	C₂²	#469×C₆₇	1	1	yes	characteristic
276	C₂²	#472×C₆₇	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	4	2	no	—
279	C₄	—	8	1	no	—
280	C₄	#447×C₆₇	1	2	yes	—
281	C₄	#449×C₆₇	1	2	yes	—
282	C₄	#450×C₆₇	1	2	yes	—
283	C₄	#492×C₆₇	1	1	yes	characteristic
284	C₄	#496×C₆₇	1	1	yes	characteristic
285	C₂	—	16	4	no	—
286	C₂	C₆₇×#1668	1	2	yes	—
287	C₂	C₆₇×#1688	1	2	yes	—
288	C₂	C₆₇×#1999	1	1	yes	characteristic
289	C₂	C₆₇×#2014	1	1	yes	characteristic
290	C₂	C₆₇×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#493×C₆₇	C₂
LCS	G ▷ C₂×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. C₂×#460×C₆₇	C₂	semidirect
3. C₆₇×#1984	C₂	semidirect
4. C₂×#331×C₆₇	C₂	semidirect
5. C₆₇×#1277	C₂	semidirect
51. C₂×Q₈×C₆₇	D₈	semidirect
79. C₂×C₈×C₆₇	SD₁₆	semidirect
139. #10262	C₆₇	direct (N × H)
147. C₂×#460	C₂×C₆₇	semidirect
148. #1984	C₂×C₆₇	semidirect
149. C₂×#331	C₂×C₆₇	semidirect
150. #1277	C₂×C₆₇	semidirect
151. C₆₇	#10262	direct (N × H)
197. C₂×Q₈	D₈×C₆₇	semidirect
225. C₂×C₈	SD₁₆×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
C₂×#460×C₆₇	2
C₆₇×#1984	2
C₂×#331×C₆₇	2
C₆₇×#1277	2
#10262	67

Subgroups — C67×#10262 #30689513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10262 #30689513