Subgroups — C₆₇×#33175 #30712426

Order 17152.

1370 conjugacy classes of subgroups.

The subgroup lattice has 306 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, complemented
2	C₂×#469×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2856	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1972	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1957	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₇	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	C₂²	1	2	yes	—
11	C₂×#121×C₆₇	—	2	4	no	—
12	C₂×#121×C₆₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#493×C₆₇	—	2	4	no	—
15	#469×C₆₇	—	2	16	no	—
16	C₂×#106×C₆₇	C₂²	1	4	yes	—
17	C₂×#102×C₆₇	C₂²	1	8	yes	—
18	C₄×M₁₆×C₆₇	—	2	4	no	—
19	C₂×#97×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#390×C₆₇	—	2	4	no	—
23	#390×C₆₇	C₂²	1	4	yes	—
24	#384×C₆₇	C₂²	1	2	yes	—
25	C₂²×D₄×C₆₇	—	2	4	no	—
26	C₂²×D₄×C₆₇	—	2	4	no	—
27	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
28	C₂³×C₄×C₆₇	D₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×D₈×C₆₇	—	2	8	no	—
31	C₂×D₈×C₆₇	—	2	8	no	—
32	C₂×D₈×C₆₇	—	2	8	no	—
33	C₂×M₁₆×C₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	—	2	4	no	—
35	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	#127×C₆₇	—	2	4	no	—
39	#127×C₆₇	—	2	4	no	—
40	#121×C₆₇	—	2	16	no	—
41	#121×C₆₇	—	4	16	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#106×C₆₇	—	2	8	no	—
52	#102×C₆₇	—	2	16	no	—
53	#97×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	—	2	2	no	—
55	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
56	C₂⁴×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₆₇	—	2	8	no	—
59	C₂×D₄×C₆₇	—	2	4	no	—
60	C₂×D₄×C₆₇	—	2	4	no	—
61	C₂×D₄×C₆₇	—	2	8	no	—
62	C₂×D₄×C₆₇	—	4	16	no	—
63	C₂×D₄×C₆₇	—	4	8	no	—
64	C₂×D₄×C₆₇	—	4	8	no	—
65	C₂×D₄×C₆₇	—	4	8	no	—
66	C₂×D₄×C₆₇	—	4	8	no	—
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	D₈×C₆₇	—	4	16	no	—
79	M₁₆×C₆₇	—	2	16	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₆₇	#41	1	4	yes	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	4	8	no	—
88	C₂²⋊C₄×C₆₇	—	4	16	no	—
89	C₂²⋊C₄×C₆₇	—	4	8	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	1	no	—
92	C₄²×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	C₂×D₄	1	2	yes	—
94	C₄²×C₆₇	#41	1	2	yes	—
95	C₂³×C₆₇	—	2	1	no	—
96	C₂³×C₆₇	—	2	1	no	—
97	C₂³×C₆₇	—	2	4	no	—
98	C₂³×C₆₇	—	4	4	no	—
99	C₂³×C₆₇	—	4	4	no	—
100	C₂³×C₆₇	—	4	4	no	—
101	C₂³×C₆₇	—	4	8	no	—
102	C₂³×C₆₇	—	4	8	no	—
103	C₂³×C₆₇	#127	1	2	yes	—
104	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	D₄×C₆₇	—	4	8	no	—
106	D₄×C₆₇	—	4	8	no	—
107	D₄×C₆₇	—	8	16	no	—
108	D₄×C₆₇	—	8	16	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	8	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	8	no	—
118	C₂×C₄×C₆₇	#121	1	4	yes	—
119	C₂×C₄×C₆₇	#127	1	2	yes	—
120	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	8	yes	—
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₈×C₆₇	—	2	4	no	—
126	C₈×C₆₇	#121	1	8	yes	complemented
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	8	4	no	—
132	C₂²×C₆₇	—	8	4	no	—
133	C₂²×C₆₇	—	8	4	no	—
134	C₂²×C₆₇	—	8	8	no	—
135	C₂²×C₆₇	—	8	8	no	—
136	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
138	C₂²×C₆₇	#532	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	4	2	no	—
142	C₄×C₆₇	—	8	4	no	—
143	C₄×C₆₇	#469	1	8	yes	complemented
144	C₄×C₆₇	C₂×#121	1	2	yes	—
145	C₄×C₆₇	#532	1	2	yes	—
146	#33175	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₆₇	—	4	2	no	—
148	C₂×C₆₇	—	16	4	no	—
149	C₂×C₆₇	#2856	1	1	yes	characteristic
150	C₂×C₆₇	C₂×#469	1	2	yes	—
151	C₂×C₆₇	#3151	1	2	yes	—
152	C₂×C₆₇	#3184	1	2	yes	—
153	C₂×#493	C₂×C₆₇	1	1	yes	characteristic, complemented
154	C₂×#469	C₂×C₆₇	1	4	yes	complemented
155	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
156	#2856	C₂×C₆₇	1	2	yes	complemented
157	#1972	C₂×C₆₇	1	4	yes	complemented
158	#1957	C₂×C₆₇	1	2	yes	complemented
159	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
160	C₆₇	#33175	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²×D₈	C₂²×C₆₇	1	4	yes	—
162	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
163	C₂×#127	C₂²×C₆₇	1	2	yes	—
164	C₂×#121	—	2	4	no	—
165	C₂×#121	C₂²×C₆₇	1	4	yes	—
166	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
167	#493	—	2	4	no	—
168	#469	—	2	16	no	—
169	C₂×#106	C₂²×C₆₇	1	4	yes	—
170	C₂×#102	C₂²×C₆₇	1	8	yes	—
171	C₄×M₁₆	—	2	4	no	—
172	C₂×#97	C₂²×C₆₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
175	#390	—	2	4	no	—
176	#390	C₂²×C₆₇	1	4	yes	—
177	#384	C₂²×C₆₇	1	2	yes	—
178	C₂²×D₄	—	2	4	no	—
179	C₂²×D₄	—	2	4	no	—
180	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
181	C₂³×C₄	D₄×C₆₇	1	2	yes	—
182	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
183	C₂×D₈	—	2	8	no	—
184	C₂×D₈	—	2	8	no	—
185	C₂×D₈	—	2	8	no	—
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
189	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
190	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
191	#127	—	2	4	no	—
192	#127	—	2	4	no	—
193	#121	—	2	16	no	—
194	#121	—	4	16	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	8	no	—
198	C₂×C₂²⋊C₄	—	2	4	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	D₄×C₆₇	1	2	yes	—
202	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
204	#106	—	2	8	no	—
205	#102	—	2	16	no	—
206	#97	—	2	4	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
209	C₂⁴	—	2	4	no	—
210	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
211	C₂×D₄	—	2	8	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	8	no	—
215	C₂×D₄	—	4	16	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂×D₄	—	4	8	no	—
218	C₂×D₄	—	4	8	no	—
219	C₂×D₄	—	4	8	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
229	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
231	D₈	—	4	16	no	—
232	M₁₆	—	2	16	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
236	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
237	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
238	C₂×C₈	#41×C₆₇	1	4	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	4	8	no	—
241	C₂²⋊C₄	—	4	16	no	—
242	C₂²⋊C₄	—	4	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	C₂×D₄×C₆₇	1	2	yes	—
247	C₄²	#41×C₆₇	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	4	8	no	—
255	C₂³	—	4	8	no	—
256	C₂³	#127×C₆₇	1	2	yes	—
257	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
258	D₄	—	4	8	no	—
259	D₄	—	4	8	no	—
260	D₄	—	8	16	no	—
261	D₄	—	8	16	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	#121×C₆₇	1	4	yes	—
272	C₂×C₄	#127×C₆₇	1	2	yes	—
273	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	8	yes	—
274	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
276	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
277	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	#121×C₆₇	1	8	yes	complemented
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	8	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	8	4	no	—
288	C₂²	—	8	8	no	—
289	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
290	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
291	C₂²	#532×C₆₇	1	1	yes	characteristic
292	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	4	no	—
296	C₄	#469×C₆₇	1	8	yes	complemented
297	C₄	C₂×#121×C₆₇	1	2	yes	—
298	C₄	#532×C₆₇	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	—	16	4	no	—
301	C₂	C₆₇×#2856	1	1	yes	characteristic
302	C₂	C₂×#469×C₆₇	1	2	yes	—
303	C₂	C₆₇×#3151	1	2	yes	—
304	C₂	C₆₇×#3184	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#493×C₆₇	C₂	semidirect
2. C₂×#469×C₆₇	C₂	semidirect
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#2856	C₂	semidirect
5. C₆₇×#1972	C₂	semidirect
6. C₆₇×#1957	C₂	semidirect
7. C₆₇×#1844	C₂	semidirect
35. C₂×M₁₆×C₆₇	D₄	semidirect
55. C₄×C₈×C₆₇	D₄	semidirect
126. C₈×C₆₇	#121	semidirect
143. C₄×C₆₇	#469	semidirect
146. #33175	C₆₇	direct (N × H)
153. C₂×#493	C₂×C₆₇	semidirect
154. C₂×#469	C₂×C₆₇	semidirect
155. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
156. #2856	C₂×C₆₇	semidirect
157. #1972	C₂×C₆₇	semidirect
158. #1957	C₂×C₆₇	semidirect
159. #1844	C₂×C₆₇	semidirect
160. C₆₇	#33175	direct (N × H)
188. C₂×M₁₆	D₄×C₆₇	semidirect
208. C₄×C₈	D₄×C₆₇	semidirect
279. C₈	#121×C₆₇	semidirect
296. C₄	#469×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×#469×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2856	2
C₆₇×#1972	2
C₆₇×#1957	2
C₆₇×#1844	2
#33175	67

Subgroups — C67×#33175 #30712426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33175 #30712426