Subgroups — #30746587

Order 17152.

468 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569170	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523969	C₂	1	1	yes	maximal, characteristic
3	#22559954	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558592	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558041	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761628	C₂²	1	1	yes	characteristic
9	#16863523	C₄	1	2	yes	complemented
10	#16863473	C₄	1	2	yes	complemented
11	#16863464	—	2	2	no	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863447	—	2	1	no	—
14	C₂×#14523969	C₄	1	2	yes	—
15	C₄×#11761611	—	2	1	no	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863157	C₄	1	2	yes	—
19	#16863150	—	2	2	no	—
20	#16863128	C₄	1	2	yes	complemented
21	#16863125	C₄	1	2	yes	complemented
22	#16863117	—	2	1	no	—
23	C₈×C₆₇⋊C₈	—	2	1	no	—
24	#431×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	C₂×#11761611	—	2	2	no	—
29	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₆₇	—	2	1	no	—
35	C₄×C₂²⋊D₆₇	—	2	1	no	—
36	#14524002	C₂×C₄	1	1	yes	characteristic
37	#14524000	—	2	1	no	—
38	C₂×C₄×Dic₆₇	—	2	1	no	—
39	C₂×C₄×Dic₆₇	—	2	1	no	—
40	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
41	C₄²×D₆₇	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
43	Dic₆₇⋊C₈	—	2	2	no	—
44	#14523969	—	2	4	no	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
46	(C₂×D₆₇)⋊C₈	—	2	2	no	—
47	#14523953	—	2	2	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
50	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₆₇	—	2	1	no	—
57	#97×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	—	2	1	no	—
61	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
62	#11761628	—	2	2	no	—
63	#11761628	—	2	2	no	—
64	C₂×Dic₁₃₄	C₄²	1	1	yes	characteristic
65	C₂×D₂₆₈	C₄²	1	1	yes	characteristic
66	C₂×C₂²⋊D₆₇	—	2	1	no	—
67	C₂²×Dic₆₇	—	2	2	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	1	no	—
71	#11761611	—	4	4	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	4	2	no	—
74	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
76	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
77	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
78	Dic₆₇⋊C₄	—	2	1	no	—
79	Dic₆₇⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₆₇	—	2	1	no	—
81	(C₂×D₆₇)⋊C₄	—	2	1	no	—
82	(C₂×D₆₇)⋊C₄	—	2	1	no	—
83	C₄×Dic₆₇	—	2	4	no	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₄×Dic₆₇	—	2	1	no	—
87	C₄×Dic₆₇	—	2	2	no	—
88	C₄×Dic₆₇	—	2	2	no	—
89	C₄×Dic₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	4	2	no	—
99	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	#41	1	1	yes	characteristic
103	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₆₇	#131	1	2	yes	—
115	C₂²⋊D₆₇	—	4	2	no	—
116	C₂×Dic₆₇	—	2	4	no	—
117	C₂×Dic₆₇	—	2	1	no	—
118	C₂×Dic₆₇	—	2	1	no	—
119	C₂×Dic₆₇	—	2	1	no	—
120	C₂×Dic₆₇	—	4	2	no	—
121	C₂×Dic₆₇	—	4	2	no	—
122	D₂₆₈	—	2	1	no	—
123	D₂₆₈	—	2	1	no	—
124	C₄×D₆₇	—	4	2	no	—
125	C₄×D₆₇	—	4	2	no	—
126	C₄×D₆₇	—	4	2	no	—
127	Dic₁₃₄	—	2	1	no	—
128	Dic₁₃₄	—	2	1	no	—
129	C₈×C₆₇	—	2	4	no	—
130	C₈×C₆₇	—	4	2	no	—
131	C₈×C₆₇	—	4	2	no	—
132	C₆₇⋊C₈	—	2	2	no	—
133	C₆₇⋊C₈	—	2	2	no	—
134	C₆₇⋊C₈	—	4	2	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₆₇	#436	1	2	yes	—
139	C₂×D₆₇	—	4	1	no	—
140	C₂×D₆₇	—	4	1	no	—
141	C₂×D₆₇	—	4	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₆₇	#405	1	1	yes	characteristic
146	C₄×C₆₇	#421	1	2	yes	—
147	C₄×C₆₇	#435	1	2	yes	—
148	C₄×C₆₇	#443	1	2	yes	—
149	Dic₆₇	—	4	4	no	—
150	Dic₆₇	—	4	1	no	—
151	Dic₆₇	—	4	1	no	—
152	Dic₆₇	—	4	1	no	—
153	#8197	—	67	1	no	maximal, Sylow
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	#1753	1	1	yes	characteristic
156	C₂×C₆₇	#1759	1	1	yes	characteristic
157	C₂×C₆₇	#1764	1	1	yes	characteristic
158	D₆₇	—	8	1	no	—
159	#2965	—	67	1	no	—
160	C₄×#104	—	67	1	no	—
161	#1744	—	67	1	no	—
162	#1476	—	67	1	no	—
163	C₈×M₁₆	—	67	1	no	—
164	#1278	D₆₇	1	1	yes	characteristic, complemented
165	#1276	—	67	1	no	—
166	C₆₇	#8197	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	—	67	1	no	—
168	#446	—	67	2	no	—
169	#436	—	67	2	no	—
170	#435	—	134	2	no	—
171	#431	—	67	1	no	—
172	#431	—	134	1	no	—
173	#431	C₂×D₆₇	1	1	yes	characteristic
174	C₂×#104	—	67	2	no	—
175	C₄×M₁₆	—	134	1	no	—
176	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
177	C₂×C₄×C₈	—	67	1	no	—
178	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
179	C₄³	—	67	1	no	—
180	#339	—	67	2	no	—
181	#336	—	134	2	no	—
182	Q₈⋊C₈	—	67	2	no	—
183	D₄⋊C₈	—	67	2	no	—
184	#322	—	134	1	no	—
185	C₈²	—	134	1	no	—
186	C₂×#41	—	67	1	no	—
187	C₂×M₁₆	—	134	2	no	—
188	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	67	2	no	—
191	C₄×Q₈	—	67	1	no	—
192	C₄×Q₈	—	67	1	no	—
193	C₄×D₄	—	67	1	no	—
194	C₄×D₄	—	67	1	no	—
195	C₄×D₄	—	134	1	no	—
196	C₄×D₄	—	134	1	no	—
197	#117	—	67	1	no	—
198	#117	—	134	1	no	—
199	C₂×C₄²	—	67	2	no	—
200	C₂×C₄²	—	134	1	no	—
201	C₂×C₄²	—	134	1	no	—
202	C₂×C₄²	—	134	1	no	—
203	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
204	C₄⋊C₈	—	67	2	no	—
205	C₄⋊C₈	—	134	2	no	—
206	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
207	#104	—	134	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	134	2	no	—
210	C₂²⋊C₈	—	134	1	no	—
211	#97	—	2	1	no	—
212	#97	—	134	2	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	67	2	no	—
217	C₄×C₈	—	67	1	no	—
218	C₄×C₈	—	67	1	no	—
219	C₄×C₈	—	134	1	no	—
220	C₄×C₈	C₄×D₆₇	1	2	yes	—
221	#41	—	134	2	no	—
222	#41	—	134	2	no	—
223	C₂×Q₈	—	67	1	no	—
224	C₂×D₄	—	67	1	no	—
225	C₂×D₄	—	134	1	no	—
226	C₂²×C₄	—	134	2	no	—
227	C₂²×C₄	—	134	1	no	—
228	C₂²×C₄	—	134	1	no	—
229	C₂²×C₄	—	134	1	no	—
230	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	#11761629	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	—	268	4	no	—
235	M₁₆	#11761611	1	4	yes	complemented
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	268	2	no	—
241	C₂×C₈	—	67	2	no	—
242	C₂×C₈	—	67	2	no	—
243	C₂×C₈	—	134	2	no	—
244	C₄⋊C₄	—	67	1	no	—
245	C₄⋊C₄	—	67	1	no	—
246	C₄⋊C₄	—	134	1	no	—
247	C₄⋊C₄	—	134	1	no	—
248	C₂²⋊C₄	—	134	1	no	—
249	C₂²⋊C₄	—	134	1	no	—
250	C₂²⋊C₄	—	134	1	no	—
251	C₄²	—	268	2	no	—
252	C₄²	—	134	4	no	—
253	C₄²	—	134	2	no	—
254	C₄²	—	134	2	no	—
255	C₄²	—	134	1	no	—
256	C₄²	—	134	1	no	—
257	C₄²	—	134	1	no	—
258	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
260	C₄²	D₄×D₆₇	1	1	yes	characteristic
261	C₄²	#11761627	1	1	yes	characteristic
262	C₂³	—	134	1	no	—
263	C₂³	#14524005	1	1	yes	characteristic
264	Q₈	—	134	1	no	—
265	Q₈	—	134	1	no	—
266	D₄	—	268	2	no	—
267	D₄	—	134	1	no	—
268	D₄	—	134	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	268	2	no	—
272	C₂×C₄	—	268	2	no	—
273	C₂×C₄	—	268	2	no	—
274	C₂×C₄	—	268	2	no	—
275	C₂×C₄	—	268	2	no	—
276	C₂×C₄	—	134	4	no	—
277	C₂×C₄	—	134	1	no	—
278	C₂×C₄	—	134	1	no	—
279	C₂×C₄	—	134	1	no	—
280	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
282	C₂×C₄	#14524001	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#14524005	1	1	yes	characteristic
286	C₂×C₄	C₂×#11761611	1	2	yes	—
287	C₂×C₄	#14524051	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	268	2	no	—
292	C₈	—	134	2	no	—
293	C₈	—	134	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	268	1	no	—
297	C₂²	—	268	1	no	—
298	C₂²	—	268	1	no	—
299	C₂²	#16863243	1	1	yes	characteristic
300	C₂²	#16863473	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	268	4	no	—
304	C₄	—	268	1	no	—
305	C₄	—	268	1	no	—
306	C₄	—	268	1	no	—
307	C₄	C₄×#11761611	1	1	yes	characteristic
308	C₄	#16863353	1	1	yes	characteristic
309	C₄	#16863411	1	2	yes	—
310	C₄	#16863464	1	2	yes	—
311	C₄	#16863512	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	536	1	no	—
314	C₂	#22559989	1	1	yes	characteristic
315	C₂	#22560018	1	1	yes	characteristic
316	C₂	#22560045	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1278	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1278	D₆₇
hypocenter	C₆₇	#8197
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
3. #22559954	C₂	semidirect
5. M₁₆×C₆₇⋊C₈	C₂	semidirect
6. #22558041	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
9. #16863523	C₄	semidirect
10. #16863473	C₄	semidirect
20. #16863128	C₄	semidirect
21. #16863125	C₄	semidirect
94. M₁₆×C₆₇	M₁₆	semidirect
164. #1278	D₆₇	semidirect
166. C₆₇	#8197	semidirect
235. M₁₆	#11761611	semidirect

Maximal subgroups

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

subgroup	index
#22569170	2
C₄×#14523969	2
#22559954	2
#22558592	2
M₁₆×C₆₇⋊C₈	2
#22558041	2
C₆₇×#1278	2
#8197	67