Subgroups — #30746766

Order 17152.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559958	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863138	C₂	1	2	yes	maximal
4	#22558090	C₂	1	2	yes	maximal
5	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523954	—	2	1	no	—
7	C₂×#14523954	C₂²	1	2	yes	—
8	C₂×#14523954	C₂²	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	#16863151	—	2	1	no	—
12	#16863138	C₄	1	8	yes	complemented
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	#389×C₆₇	C₄	1	1	yes	characteristic
15	#389×C₆₇	C₄	1	1	yes	characteristic
16	#387×C₆₇	—	2	1	no	—
17	#387×C₆₇	—	2	1	no	—
18	#382×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	#14523954	—	2	4	no	—
27	#14523954	—	2	2	no	—
28	#14523954	—	2	2	no	—
29	#14523954	C₂×C₄	1	4	yes	—
30	#14523954	C₂×C₄	1	4	yes	—
31	C₄×C₆₇⋊C₈	—	2	4	no	—
32	#125×C₆₇	—	2	2	no	—
33	#125×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	—	2	1	no	—
43	C₂×C₄²×C₆₇	—	2	1	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	#95×C₆₇	—	2	1	no	—
52	#95×C₆₇	—	2	1	no	—
53	#95×C₆₇	—	2	1	no	—
54	#95×C₆₇	—	2	1	no	—
55	#95×C₆₇	—	2	1	no	—
56	#95×C₆₇	—	2	1	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	1	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	—	4	1	no	—
79	C₄⋊C₄×C₆₇	—	4	1	no	—
80	C₄⋊C₄×C₆₇	—	4	2	no	—
81	C₄⋊C₄×C₆₇	—	4	2	no	—
82	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
83	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
84	C₄²×C₆₇	—	2	2	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	2	2	no	—
87	C₄²×C₆₇	—	4	1	no	—
88	C₄²×C₆₇	—	4	1	no	—
89	C₄²×C₆₇	—	4	1	no	—
90	C₄²×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₆₇	#41	1	2	yes	—
94	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	—	4	1	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	#101	1	2	yes	—
123	C₂×C₄×C₆₇	#104	1	2	yes	—
124	C₂×C₄×C₆₇	#104	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
129	C₆₇⋊C₈	—	2	8	no	—
130	C₆₇⋊C₈	—	4	4	no	—
131	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
133	C₂²×C₆₇	#413	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
136	C₂²×C₆₇	#421	1	1	yes	characteristic
137	C₂²×C₆₇	#421	1	1	yes	characteristic
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	4	2	no	—
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	—	4	1	no	—
148	C₄×C₆₇	—	8	1	no	—
149	C₄×C₆₇	—	8	1	no	—
150	#8227	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	#1528	1	1	yes	characteristic
152	C₂×C₆₇	#1529	1	1	yes	characteristic
153	C₂×C₆₇	#1531	1	1	yes	characteristic
154	C₂×C₆₇	#1532	1	1	yes	characteristic
155	C₂×C₆₇	C₄×#101	1	1	yes	characteristic
156	C₂×C₆₇	C₄×#104	1	1	yes	characteristic
157	C₂×C₆₇	#1759	1	1	yes	characteristic
158	C₄×#125	D₆₇	1	1	yes	characteristic
159	#1745	—	67	1	no	—
160	C₄×#97	—	67	1	no	—
161	C₂×#330	—	67	2	no	—
162	#1295	—	67	2	no	—
163	C₆₇	#8227	1	1	yes	characteristic, complemented, Sylow
164	C₂×#125	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#97	—	67	2	no	—
166	C₂×#97	—	67	2	no	—
167	C₂×#97	—	134	1	no	—
168	C₂×C₄×C₈	—	134	1	no	—
169	#389	Dic₆₇	1	1	yes	characteristic
170	#389	Dic₆₇	1	1	yes	characteristic
171	#387	—	2	1	no	—
172	#387	—	2	1	no	—
173	#382	C₂×D₆₇	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
177	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
178	C₄³	C₂×D₆₇	1	1	yes	characteristic
179	#336	—	134	1	no	—
180	#336	—	134	1	no	—
181	#330	—	67	8	no	—
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	—	134	2	no	—
184	#125	—	2	2	no	—
185	#125	C₄×D₆₇	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	#97	—	67	4	no	—
204	#97	—	67	4	no	—
205	#97	—	134	4	no	—
206	#97	—	134	2	no	—
207	#97	—	134	2	no	—
208	C₄×C₈	—	134	4	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₂×C₈	—	268	2	no	—
227	C₂×C₈	—	268	2	no	—
228	C₂×C₈	—	134	4	no	—
229	C₂×C₈	—	134	4	no	—
230	C₂×C₈	—	134	4	no	—
231	C₂×C₈	—	134	4	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₄×Dic₆₇	1	2	yes	—
241	C₄⋊C₄	C₄×Dic₆₇	1	2	yes	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×C₄×D₆₇	1	2	yes	—
251	C₄²	#11761628	1	2	yes	—
252	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	#14523962	1	2	yes	—
281	C₂×C₄	#14523970	1	2	yes	—
282	C₂×C₄	#14523970	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
284	C₂×C₄	#14524002	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
286	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
287	C₈	—	268	4	no	—
288	C₈	—	134	8	no	—
289	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
290	C₂²	C₂×#14523962	1	1	yes	characteristic
291	C₂²	#16863380	1	1	yes	characteristic
292	C₂²	C₂×#14523970	1	1	yes	characteristic
293	C₂²	C₂×#14523970	1	1	yes	characteristic
294	C₂²	#16863413	1	1	yes	characteristic
295	C₂²	#16863413	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#22558811	1	1	yes	characteristic
309	C₂	#22558816	1	1	yes	characteristic
310	C₂	#22558824	1	1	yes	characteristic
311	C₂	#22558829	1	1	yes	characteristic
312	C₂	C₄×#14523962	1	1	yes	characteristic
313	C₂	C₄×#14523970	1	1	yes	characteristic
314	C₂	#22560019	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#125×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#125	D₆₇
hypocenter	C₆₇	#8227
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
12. #16863138	C₄	semidirect
163. C₆₇	#8227	semidirect

Maximal subgroups

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

subgroup	index
#22559958	2
C₄×#14523954	2
C₂×#16863138	2
#22558090	2
C₄×#125×C₆₇	2
#8227	67