Subgroups — #30743802

Order 17152.

444 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	C₄×#14524035	C₂	1	1	yes	maximal, characteristic
2	#22559955	C₂	1	1	yes	maximal, characteristic
3	#22558033	C₂	1	2	yes	maximal
4	#22558031	C₂	1	2	yes	maximal
5	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524035	C₄	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
8	#16863294	C₂²	1	1	yes	characteristic
9	#16863294	C₂²	1	1	yes	characteristic
10	#16863280	—	2	1	no	—
11	#16863280	—	2	1	no	—
12	#16863262	C₄	1	1	yes	characteristic
13	C₄×Dic₆₇⋊C₄	—	2	1	no	—
14	C₄×Dic₆₇⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
16	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
17	#16863151	—	2	1	no	—
18	#16863151	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	—	2	1	no	—
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	#336×C₆₇	—	2	1	no	—
22	#336×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	#14524035	—	2	2	no	—
25	#14524035	C₈	1	4	yes	complemented
26	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₆₇⋊C₄	—	2	1	no	—
31	C₂×Dic₆₇⋊C₄	—	2	1	no	—
32	C₂×Dic₆₇⋊C₄	—	2	1	no	—
33	C₂×Dic₆₇⋊C₄	—	2	1	no	—
34	C₂×C₄×Dic₆₇	—	2	1	no	—
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	C₄×C₆₇⋊C₈	—	2	4	no	—
37	C₆₇⋊#95	—	2	1	no	—
38	C₆₇⋊#95	—	2	1	no	—
39	C₆₇⋊#95	—	2	1	no	—
40	C₆₇⋊#95	—	2	1	no	—
41	C₆₇⋊#95	—	2	1	no	—
42	C₆₇⋊#95	—	2	1	no	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₇	—	2	4	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	2	1	no	—
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂²×Dic₆₇	—	2	1	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₄⋊Dic₆₇	—	2	1	no	—
60	C₄⋊Dic₆₇	—	2	1	no	—
61	C₄⋊Dic₆₇	—	4	1	no	—
62	C₄⋊Dic₆₇	—	4	1	no	—
63	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
64	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
65	Dic₆₇⋊C₄	—	2	4	no	—
66	Dic₆₇⋊C₄	—	2	4	no	—
67	Dic₆₇⋊C₄	—	4	2	no	—
68	Dic₆₇⋊C₄	—	4	2	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	C₄×Dic₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	4	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	2	2	no	—
83	C₄²×C₆₇	—	2	2	no	—
84	C₄²×C₆₇	—	2	2	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	4	1	no	—
87	C₄²×C₆₇	C₂×C₈	1	2	yes	—
88	C₄²×C₆₇	M₁₆	1	2	yes	—
89	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₆₇	#101	1	2	yes	—
106	C₂×C₄×C₆₇	#104	1	2	yes	—
107	C₂×C₄×C₆₇	#104	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#135	1	2	yes	—
111	C₂×C₄×C₆₇	#135	1	2	yes	—
112	C₂×Dic₆₇	—	2	2	no	—
113	C₂×Dic₆₇	—	2	2	no	—
114	C₂×Dic₆₇	—	4	1	no	—
115	C₂×Dic₆₇	—	4	1	no	—
116	C₂×Dic₆₇	—	4	1	no	—
117	C₂×Dic₆₇	—	4	1	no	—
118	C₂×Dic₆₇	—	4	1	no	—
119	C₂×Dic₆₇	—	4	1	no	—
120	C₂×Dic₆₇	—	4	1	no	—
121	C₂×Dic₆₇	—	4	1	no	—
122	C₂×Dic₆₇	—	4	1	no	—
123	C₂×Dic₆₇	—	4	1	no	—
124	C₈×C₆₇	—	4	4	no	—
125	C₆₇⋊C₈	—	4	4	no	—
126	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₆₇	#410	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
129	C₂²×C₆₇	#416	1	1	yes	characteristic
130	C₂²×C₆₇	#416	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	4	1	no	—
140	C₄×C₆₇	#330	1	4	yes	—
141	Dic₆₇	—	4	2	no	—
142	Dic₆₇	—	4	2	no	—
143	Dic₆₇	—	8	1	no	—
144	Dic₆₇	—	8	1	no	—
145	#7568	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	#1461	1	1	yes	characteristic
147	C₂×C₆₇	#1476	1	1	yes	characteristic
148	C₂×C₆₇	#1476	1	1	yes	characteristic
149	C₂×C₆₇	#1500	1	1	yes	characteristic
150	C₂×C₆₇	#1500	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#330	1	1	yes	characteristic
152	C₂×C₆₇	#1544	1	1	yes	characteristic
153	C₄×#125	—	67	1	no	—
154	#1744	—	67	1	no	—
155	#1744	D₆₇	1	1	yes	characteristic
156	#1271	—	67	2	no	—
157	#1271	—	67	2	no	—
158	C₆₇	#7568	1	1	yes	characteristic, complemented, Sylow
159	C₂×#125	—	67	1	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	—	134	1	no	—
162	#389	—	67	1	no	—
163	#389	—	67	1	no	—
164	#387	—	134	1	no	—
165	#387	—	134	1	no	—
166	#382	—	67	1	no	—
167	C₄×C₄⋊C₄	—	67	1	no	—
168	C₄×C₄⋊C₄	—	67	1	no	—
169	C₄×C₄⋊C₄	—	134	1	no	—
170	C₄×C₄⋊C₄	—	134	1	no	—
171	C₄³	C₂×D₆₇	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	#336	—	67	2	no	—
174	#336	—	134	1	no	—
175	#336	C₂×D₆₇	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	134	2	no	—
178	#125	—	67	4	no	—
179	#125	—	134	2	no	—
180	C₂×C₄⋊C₄	—	67	1	no	—
181	C₂×C₄⋊C₄	—	67	1	no	—
182	C₂×C₄⋊C₄	—	67	1	no	—
183	C₂×C₄⋊C₄	—	67	1	no	—
184	C₂×C₄⋊C₄	—	134	1	no	—
185	C₂×C₄⋊C₄	—	134	1	no	—
186	C₂×C₄⋊C₄	—	134	1	no	—
187	C₂×C₄⋊C₄	—	134	1	no	—
188	C₂×C₄²	—	134	1	no	—
189	C₂×C₄²	—	134	1	no	—
190	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
191	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
192	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
193	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	134	4	no	—
199	#95	—	134	1	no	—
200	#95	—	134	1	no	—
201	#95	—	134	1	no	—
202	#95	—	134	1	no	—
203	#95	—	134	1	no	—
204	#95	—	134	1	no	—
205	C₂²×C₄	—	134	1	no	—
206	C₂²×C₄	—	134	1	no	—
207	C₂²×C₄	—	134	1	no	—
208	C₂²×C₄	—	134	1	no	—
209	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	268	2	no	—
220	C₂×C₈	—	268	2	no	—
221	C₂×C₈	—	134	4	no	—
222	C₄⋊C₄	—	268	1	no	—
223	C₄⋊C₄	—	268	1	no	—
224	C₄⋊C₄	—	268	2	no	—
225	C₄⋊C₄	—	268	2	no	—
226	C₄⋊C₄	—	67	2	no	—
227	C₄⋊C₄	—	67	2	no	—
228	C₄⋊C₄	—	134	4	no	—
229	C₄⋊C₄	—	134	4	no	—
230	C₄⋊C₄	—	134	1	no	—
231	C₄⋊C₄	—	134	1	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	268	2	no	—
239	C₄²	—	268	2	no	—
240	C₄²	C₈×D₆₇	1	2	yes	—
241	C₄²	#11761611	1	2	yes	—
242	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	268	1	no	—
257	C₂×C₄	—	268	1	no	—
258	C₂×C₄	—	268	1	no	—
259	C₂×C₄	—	268	1	no	—
260	C₂×C₄	—	268	1	no	—
261	C₂×C₄	—	268	1	no	—
262	C₂×C₄	—	268	1	no	—
263	C₂×C₄	—	268	1	no	—
264	C₂×C₄	—	268	1	no	—
265	C₂×C₄	—	268	1	no	—
266	C₂×C₄	—	134	2	no	—
267	C₂×C₄	—	134	2	no	—
268	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
269	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
270	C₂×C₄	#14523961	1	2	yes	—
271	C₂×C₄	#14523969	1	2	yes	—
272	C₂×C₄	#14523971	1	2	yes	—
273	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
275	C₂×C₄	#14524063	1	2	yes	—
276	C₂×C₄	#14524065	1	2	yes	—
277	C₈	—	4	4	no	—
278	C₈	—	268	4	no	—
279	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
280	C₂²	#16863369	1	1	yes	characteristic
281	C₂²	C₂×#14523961	1	1	yes	characteristic
282	C₂²	#16863390	1	1	yes	characteristic
283	C₂²	#16863393	1	1	yes	characteristic
284	C₂²	C₂×#14523969	1	1	yes	characteristic
285	C₂²	C₂×#14523971	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	268	2	no	—
294	C₄	—	268	2	no	—
295	C₄	—	536	1	no	—
296	C₄	—	536	1	no	—
297	C₄	#16863137	1	4	yes	—
298	C₂	#22558532	1	1	yes	characteristic
299	C₂	#22558589	1	1	yes	characteristic
300	C₂	#22558592	1	1	yes	characteristic
301	C₂	#22558693	1	1	yes	characteristic
302	C₂	#22558696	1	1	yes	characteristic
303	C₂	C₂×#16863137	1	1	yes	characteristic
304	C₂	#22558873	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#7568
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
25. #14524035	C₈	semidirect
158. C₆₇	#7568	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524035	2
#22559955	2
#22558033	2
#22558031	2
C₆₇×#1744	2
#7568	67