Subgroups — #30772826

Order 17152.

572 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863206	C₂	1	2	yes	maximal, complemented
3	#22561820	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558314	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3050	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
7	C₂×#14523983	C₂²	1	2	yes	—
8	C₂×#14523975	—	2	4	no	—
9	C₂×#14523975	—	2	1	no	—
10	C₂×#14523975	C₂²	1	2	yes	—
11	#16863206	—	2	4	no	—
12	C₂²⋊(C₆₇⋊C₁₆)	—	2	2	no	—
13	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#525×C₆₇	—	2	1	no	—
16	#467×C₆₇	C₄	1	2	yes	complemented
17	#466×C₆₇	C₄	1	2	yes	complemented
18	#465×C₆₇	—	2	2	no	—
19	#427×C₆₇	C₂²	1	1	yes	characteristic
20	#416×C₆₇	—	2	1	no	—
21	C₂³×Dic₆₇	—	4	1	no	—
22	C₂×C₄⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	—	4	2	no	—
25	C₂×C₄⋊Dic₆₇	—	4	1	no	—
26	#14523983	—	2	2	no	—
27	#14523983	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	#14523975	—	2	4	no	—
30	#14523975	—	2	2	no	—
31	#14523975	D₄	1	4	yes	complemented
32	C₂×#41×C₆₇	—	2	1	no	—
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	#135×C₆₇	—	2	2	no	—
35	#135×C₆₇	—	2	2	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×SD₁₆×C₆₇	—	2	1	no	—
38	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	#123×C₆₇	—	4	1	no	—
46	C₂²⋊Q₈×C₆₇	—	2	2	no	—
47	#121×C₆₇	—	2	2	no	—
48	#120×C₆₇	—	4	1	no	—
49	C₄×D₄×C₆₇	—	4	2	no	—
50	#117×C₆₇	—	2	1	no	—
51	#107×C₆₇	C₂×C₄	1	2	yes	complemented
52	#106×C₆₇	—	2	1	no	—
53	#103×C₆₇	—	2	2	no	—
54	#102×C₆₇	—	2	2	no	—
55	C₂²×Dic₆₇	—	4	4	no	—
56	C₂²×Dic₆₇	—	4	2	no	—
57	C₂²×Dic₆₇	—	8	4	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	4	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₆₇⋊C₁₆	—	4	2	no	—
62	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
63	#41×C₆₇	—	4	1	no	—
64	#41×C₆₇	—	4	1	no	—
65	#41×C₆₇	—	4	1	no	—
66	#41×C₆₇	—	4	1	no	—
67	C₂×Q₈×C₆₇	—	2	1	no	—
68	C₂×D₄×C₆₇	—	2	1	no	—
69	C₂×D₄×C₆₇	—	4	2	no	—
70	C₂×D₄×C₆₇	—	4	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	4	1	no	—
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₆₇	—	2	2	no	—
81	SD₁₆×C₆₇	—	4	2	no	—
82	D₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	2	1	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	#41	1	2	yes	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	4	2	no	—
94	C₂²⋊C₄×C₆₇	—	4	2	no	—
95	C₂²⋊C₄×C₆₇	—	4	1	no	—
96	C₂²⋊C₄×C₆₇	—	4	2	no	—
97	C₄²×C₆₇	—	4	1	no	—
98	C₂³×C₆₇	—	2	4	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	—	4	1	no	—
102	C₂³×C₆₇	#102	1	2	yes	—
103	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₆₇	—	4	1	no	—
105	Q₈×C₆₇	—	4	1	no	—
106	D₄×C₆₇	—	4	1	no	—
107	D₄×C₆₇	—	4	1	no	—
108	D₄×C₆₇	—	8	4	no	—
109	D₄×C₆₇	—	8	1	no	—
110	D₄×C₆₇	—	8	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	8	1	no	—
120	C₂×C₄×C₆₇	—	8	1	no	—
121	C₂×C₄×C₆₇	#102	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	4	no	—
129	C₂×Dic₆₇	—	8	8	no	—
130	C₂×Dic₆₇	—	8	4	no	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	C₄×D₄	1	2	yes	—
133	C₈×C₆₇	#121	1	2	yes	—
134	C₂²×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	—	2	4	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	8	1	no	—
140	C₂²×C₆₇	—	8	2	no	—
141	C₂²×C₆₇	#360	1	4	yes	complemented
142	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
143	C₂²×C₆₇	#447	1	1	yes	characteristic
144	C₂²×C₆₇	#480	1	1	yes	characteristic
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	8	1	no	—
147	C₄×C₆₇	—	8	2	no	—
148	C₄×C₆₇	#386	1	1	yes	characteristic
149	C₄×C₆₇	#417	1	1	yes	characteristic
150	C₄×C₆₇	#448	1	1	yes	characteristic
151	C₄×C₆₇	#481	1	1	yes	characteristic
152	Dic₆₇	—	8	4	no	—
153	#13001	—	67	1	no	maximal, Sylow
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	—	2	4	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	16	1	no	—
158	C₂×C₆₇	#1874	1	1	yes	characteristic
159	C₂×C₆₇	C₂×#360	1	2	yes	—
160	#3050	D₆₇	1	1	yes	characteristic
161	C₂²×#106	—	67	1	no	—
162	C₂×#360	—	67	2	no	—
163	#2112	—	67	1	no	—
164	#1386	—	67	2	no	—
165	C₆₇	#13001	1	1	yes	characteristic, complemented, Sylow
166	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
167	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
168	#525	—	2	1	no	—
169	C₂²×C₄⋊C₄	—	134	1	no	—
170	C₂×M₃₂	—	67	2	no	—
171	#467	Dic₆₇	1	2	yes	complemented
172	#466	Dic₆₇	1	2	yes	complemented
173	#465	—	2	2	no	—
174	#427	C₂×D₆₇	1	1	yes	characteristic
175	C₂×#106	—	67	2	no	—
176	C₂×#106	—	134	4	no	—
177	C₂×#106	—	134	1	no	—
178	#416	—	2	1	no	—
179	#360	—	134	4	no	—
180	C₂²⋊C₁₆	—	134	2	no	—
181	C₂×#41	—	2	1	no	—
182	C₂³×C₄	—	268	1	no	—
183	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
184	#135	—	2	2	no	—
185	#135	—	2	2	no	—
186	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic, complemented
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
194	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	#123	—	4	1	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	#121	—	2	2	no	—
198	#120	—	4	1	no	—
199	C₄×D₄	—	4	2	no	—
200	#117	—	2	1	no	—
201	C₂×C₄⋊C₄	—	268	2	no	—
202	C₂×C₄⋊C₄	—	268	1	no	—
203	C₂×C₄⋊C₄	—	134	4	no	—
204	C₂×C₄⋊C₄	—	134	2	no	—
205	M₃₂	—	134	2	no	—
206	M₃₂	—	134	2	no	—
207	C₂×C₁₆	—	134	2	no	—
208	#107	C₂×Dic₆₇	1	2	yes	complemented
209	#106	—	2	1	no	—
210	#106	—	67	4	no	—
211	#106	—	134	4	no	—
212	#106	—	134	2	no	—
213	#103	—	2	2	no	—
214	#102	—	2	2	no	—
215	C₂⁴	#11761613	1	1	yes	characteristic
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	#41	—	4	1	no	—
219	#41	—	4	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	268	4	no	—
230	C₂²×C₄	—	268	2	no	—
231	C₂²×C₄	—	536	4	no	—
232	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
233	C₂²×C₄	#11761613	1	1	yes	characteristic
234	C₂²×C₄	#11761616	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	Q₁₆	—	2	2	no	—
237	SD₁₆	—	4	2	no	—
238	D₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
244	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
245	C₂×C₈	D₄×D₆₇	1	2	yes	—
246	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
247	C₂×C₈	#11761627	1	2	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	268	4	no	—
251	C₄⋊C₄	—	268	2	no	—
252	C₄⋊C₄	—	134	4	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	—	4	1	no	—
257	C₁₆	—	268	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	1	no	—
262	C₂³	#14523964	1	2	yes	—
263	C₂³	C₂×#11761613	1	1	yes	characteristic
264	Q₈	—	4	1	no	—
265	Q₈	—	4	1	no	—
266	D₄	—	4	1	no	—
267	D₄	—	4	1	no	—
268	D₄	—	8	1	no	—
269	D₄	—	8	1	no	—
270	D₄	—	8	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	—	268	4	no	—
282	C₂×C₄	—	536	8	no	—
283	C₂×C₄	—	536	4	no	—
284	C₂×C₄	#14523964	1	2	yes	—
285	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
286	C₂×C₄	#14524011	1	1	yes	characteristic
287	C₂×C₄	#14524024	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
289	C₂×C₄	#14524068	1	1	yes	characteristic
290	C₂×C₄	#14524073	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	D₄×Dic₆₇	1	2	yes	—
293	C₈	#14524015	1	2	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	2	no	—
301	C₂²	#16863206	1	4	yes	complemented
302	C₂²	C₂×#14523964	1	1	yes	characteristic
303	C₂²	#16863529	1	1	yes	characteristic
304	C₂²	#16863707	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	536	4	no	—
309	C₄	#16863276	1	1	yes	characteristic
310	C₄	#16863397	1	1	yes	characteristic
311	C₄	#16863534	1	1	yes	characteristic
312	C₄	#16863712	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	4	no	—
316	C₂	—	16	1	no	—
317	C₂	#22560527	1	1	yes	characteristic
318	C₂	C₂×#16863206	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3050	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3050	D₆₇
hypocenter	C₆₇	#13001
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #3050

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523975	C₂	semidirect
2. C₂×#16863206	C₂	semidirect
3. #22561820	C₂	semidirect
4. #22558314	C₂	semidirect
16. #467×C₆₇	C₄	semidirect
17. #466×C₆₇	C₄	semidirect
31. #14523975	D₄	semidirect
36. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
38. C₂×D₈×C₆₇	C₂×C₄	semidirect
51. #107×C₆₇	C₂×C₄	semidirect
141. C₂²×C₆₇	#360	semidirect
165. C₆₇	#13001	semidirect
171. #467	Dic₆₇	semidirect
172. #466	Dic₆₇	semidirect
186. C₂×Q₁₆	C₂×Dic₆₇	semidirect
188. C₂×D₈	C₂×Dic₆₇	semidirect
208. #107	C₂×Dic₆₇	semidirect
301. C₂²	#16863206	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523975	2
C₂×#16863206	2
#22561820	2
#22558314	2
C₆₇×#3050	2
#13001	67