Subgroups — C₆₇×#33247 #30712498

Order 17152.

922 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×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#489×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2640	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1973	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1970	C₂	1	2	yes	maximal
7	C₆₇×#1969	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#124×C₆₇	C₂²	1	2	yes	—
14	#500×C₆₇	—	2	4	no	—
15	#495×C₆₇	—	2	4	no	—
16	#489×C₆₇	—	2	8	no	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	4	yes	—
20	C₂×#102×C₆₇	C₂²	1	4	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#397×C₆₇	—	2	4	no	—
26	#393×C₆₇	—	2	4	no	—
27	#391×C₆₇	C₂²	1	4	yes	—
28	#390×C₆₇	C₂²	1	4	yes	—
29	#376×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₆₇	—	2	2	no	—
31	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	4	no	—
35	C₂×SD₁₆×C₆₇	—	2	4	no	—
36	C₂×SD₁₆×C₆₇	—	2	8	no	—
37	C₂×D₈×C₆₇	—	2	2	no	—
38	C₂×D₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₇	—	2	4	no	—
43	C₄⋊Q₈×C₆₇	—	2	4	no	—
44	#124×C₆₇	—	2	4	no	—
45	#124×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
51	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	#107×C₆₇	—	2	4	no	—
54	#107×C₆₇	—	2	4	no	—
55	#103×C₆₇	—	2	8	no	—
56	#102×C₆₇	—	2	8	no	—
57	#97×C₆₇	—	2	4	no	—
58	#97×C₆₇	D₄	1	4	yes	—
59	C₄×C₈×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	D₄	1	4	yes	—
61	#95×C₆₇	—	2	8	no	—
62	#95×C₆₇	D₄	1	2	yes	—
63	#95×C₆₇	D₄	1	2	yes	—
64	C₂⁴×C₆₇	—	2	2	no	—
65	C₂×Q₈×C₆₇	—	2	2	no	—
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	—	2	4	no	—
68	C₂×Q₈×C₆₇	—	2	4	no	—
69	C₂×Q₈×C₆₇	—	4	4	no	—
70	C₂×Q₈×C₆₇	—	8	4	no	—
71	C₂×D₄×C₆₇	—	2	2	no	—
72	C₂×D₄×C₆₇	—	2	2	no	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₆₇	—	4	4	no	—
82	SD₁₆×C₆₇	—	4	8	no	—
83	D₈×C₆₇	—	4	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	#41	1	4	yes	—
90	C₂×C₈×C₆₇	#41	1	4	yes	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	—	4	4	no	—
94	C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₂²⋊C₄×C₆₇	—	4	4	no	—
96	C₂²⋊C₄×C₆₇	—	8	8	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	C₂×D₄	1	2	yes	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	4	no	—
103	C₂³×C₆₇	—	4	4	no	—
104	C₂³×C₆₇	—	4	4	no	—
105	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₆₇	—	4	2	no	—
107	Q₈×C₆₇	—	4	2	no	—
108	Q₈×C₆₇	—	4	4	no	—
109	Q₈×C₆₇	—	8	8	no	—
110	D₄×C₆₇	—	4	2	no	—
111	D₄×C₆₇	—	4	2	no	—
112	D₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	—	4	8	no	—
122	C₂×C₄×C₆₇	#127	1	2	yes	—
123	C₂×C₄×C₆₇	#127	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
128	C₈×C₆₇	—	2	4	no	—
129	C₈×C₆₇	#124	1	4	yes	—
130	C₈×C₆₇	#124	1	4	yes	—
131	C₂²×C₆₇	—	8	2	no	—
132	C₂²×C₆₇	—	8	2	no	—
133	C₂²×C₆₇	—	8	2	no	—
134	C₂²×C₆₇	—	8	8	no	—
135	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
136	C₂²×C₆₇	#532	1	2	yes	—
137	C₂²×C₆₇	#576	1	2	yes	—
138	C₂²×C₆₇	#578	1	2	yes	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	4	4	no	—
142	C₄×C₆₇	—	8	2	no	—
143	C₄×C₆₇	—	8	4	no	—
144	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
146	C₄×C₆₇	#532	1	2	yes	—
147	#33247	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₆₇	—	16	2	no	—
149	C₂×C₆₇	#2858	1	1	yes	characteristic
150	C₂×C₆₇	#3155	1	1	yes	characteristic
151	C₂×C₆₇	#3156	1	1	yes	characteristic
152	C₂×C₆₇	#3175	1	4	yes	—
153	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
154	C₂×#495	C₂×C₆₇	1	1	yes	characteristic
155	C₂×#489	C₂×C₆₇	1	2	yes	complemented
156	#2640	C₂×C₆₇	1	2	yes	complemented
157	#1973	C₂×C₆₇	1	4	yes	complemented
158	#1970	C₂×C₆₇	1	2	yes	—
159	#1969	C₂×C₆₇	1	2	yes	complemented
160	#1726	C₂×C₆₇	1	1	yes	characteristic, complemented
161	C₆₇	#33247	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
163	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
164	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
166	C₂×#124	C₂²×C₆₇	1	2	yes	—
167	#500	—	2	4	no	—
168	#495	—	2	4	no	—
169	#489	—	2	8	no	—
170	C₂×#107	C₂²×C₆₇	1	2	yes	—
171	C₂×#107	C₂²×C₆₇	1	2	yes	—
172	C₂×#103	C₂²×C₆₇	1	4	yes	—
173	C₂×#102	C₂²×C₆₇	1	4	yes	—
174	C₂×#97	C₂²×C₆₇	1	2	yes	—
175	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
178	#397	—	2	4	no	—
179	#393	—	2	4	no	—
180	#391	C₂²×C₆₇	1	4	yes	—
181	#390	C₂²×C₆₇	1	4	yes	—
182	#376	C₂²×C₆₇	1	1	yes	characteristic
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
185	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂×SD₁₆	—	2	4	no	—
189	C₂×SD₁₆	—	2	8	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
193	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	4	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	#124	—	2	4	no	—
198	#124	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	2	8	no	—
204	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#103	—	2	8	no	—
209	#102	—	2	8	no	—
210	#97	—	2	4	no	—
211	#97	D₄×C₆₇	1	4	yes	—
212	C₄×C₈	—	2	4	no	—
213	C₄×C₈	D₄×C₆₇	1	4	yes	—
214	#95	—	2	8	no	—
215	#95	D₄×C₆₇	1	2	yes	—
216	#95	D₄×C₆₇	1	2	yes	—
217	C₂⁴	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	4	no	—
223	C₂×Q₈	—	8	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
231	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
234	Q₁₆	—	4	4	no	—
235	SD₁₆	—	4	8	no	—
236	D₈	—	4	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
240	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
242	C₂×C₈	#41×C₆₇	1	4	yes	—
243	C₂×C₈	#41×C₆₇	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	8	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂×D₄×C₆₇	1	2	yes	—
253	C₄²	C₂×D₄×C₆₇	1	2	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	8	8	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	#127×C₆₇	1	2	yes	—
276	C₂×C₄	#127×C₆₇	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
280	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	#124×C₆₇	1	4	yes	—
283	C₈	#124×C₆₇	1	4	yes	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	8	no	—
288	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
289	C₂²	#532×C₆₇	1	2	yes	—
290	C₂²	#576×C₆₇	1	2	yes	—
291	C₂²	#578×C₆₇	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
298	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
299	C₄	#532×C₆₇	1	2	yes	—
300	C₂	—	16	2	no	—
301	C₂	C₆₇×#2858	1	1	yes	characteristic
302	C₂	C₆₇×#3155	1	1	yes	characteristic
303	C₂	C₆₇×#3156	1	1	yes	characteristic
304	C₂	C₆₇×#3175	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₆₇	C₂	semidirect
3. C₂×#489×C₆₇	C₂	semidirect
4. C₆₇×#2640	C₂	semidirect
5. C₆₇×#1973	C₂	semidirect
7. C₆₇×#1969	C₂	semidirect
8. C₆₇×#1726	C₂	semidirect
147. #33247	C₆₇	direct (N × H)
153. C₂×#500	C₂×C₆₇	semidirect
155. C₂×#489	C₂×C₆₇	semidirect
156. #2640	C₂×C₆₇	semidirect
157. #1973	C₂×C₆₇	semidirect
159. #1969	C₂×C₆₇	semidirect
160. #1726	C₂×C₆₇	semidirect
161. C₆₇	#33247	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#495×C₆₇	2
C₂×#489×C₆₇	2
C₆₇×#2640	2
C₆₇×#1973	2
C₆₇×#1970	2
C₆₇×#1969	2
C₆₇×#1726	2
#33247	67

Subgroups — C67×#33247 #30712498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33247 #30712498