Subgroups — #30755824

Order 17152.

460 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#22560062	C₂	1	1	yes	maximal, characteristic
3	#22558049	C₂	1	2	yes	maximal
4	#22558049	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524035	C₄	1	1	yes	characteristic
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14523954	—	2	1	no	—
9	#16863294	C₄	1	1	yes	characteristic
10	#16863294	C₄	1	1	yes	characteristic
11	#16863280	—	2	1	no	—
12	#16863280	—	2	1	no	—
13	#16863262	C₄	1	1	yes	characteristic
14	C₄×Dic₆₇⋊C₄	—	2	1	no	—
15	C₄×Dic₆₇⋊C₄	—	2	1	no	—
16	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863151	—	2	1	no	—
19	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	#336×C₆₇	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	#14524035	—	2	2	no	—
25	#14524035	—	2	2	no	—
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	2	no	—
37	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
38	#14523954	—	4	2	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	C₆₇⋊#95	—	2	1	no	—
44	C₆₇⋊#95	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	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	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	1	no	—
56	C₂²×Dic₆₇	—	2	1	no	—
57	C₂²×Dic₆₇	—	2	1	no	—
58	C₂²×Dic₆₇	—	2	1	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₄⋊Dic₆₇	—	2	1	no	—
63	C₄⋊Dic₆₇	—	2	1	no	—
64	C₄⋊Dic₆₇	—	2	2	no	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	M₁₆	1	2	yes	—
67	C₄⋊Dic₆₇	M₁₆	1	2	yes	—
68	Dic₆₇⋊C₄	—	2	4	no	—
69	Dic₆₇⋊C₄	—	2	4	no	—
70	Dic₆₇⋊C₄	—	4	2	no	—
71	Dic₆₇⋊C₄	—	4	2	no	—
72	C₄×Dic₆₇	—	4	2	no	—
73	C₄×Dic₆₇	—	4	1	no	—
74	C₄×Dic₆₇	—	4	1	no	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	—	4	2	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	—	2	1	no	—
92	C₄²×C₆₇	—	4	1	no	—
93	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₆₇	#41	1	2	yes	—
95	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	#104	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#131	1	2	yes	—
117	C₂×C₄×C₆₇	#135	1	2	yes	—
118	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
120	C₂×Dic₆₇	—	2	2	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	C₂×Dic₆₇	—	4	1	no	—
123	C₂×Dic₆₇	—	4	1	no	—
124	C₂×Dic₆₇	—	4	1	no	—
125	C₂×Dic₆₇	—	4	1	no	—
126	C₂×Dic₆₇	—	4	1	no	—
127	C₂×Dic₆₇	—	4	1	no	—
128	C₂×Dic₆₇	—	4	1	no	—
129	C₂×Dic₆₇	—	4	1	no	—
130	C₂×Dic₆₇	—	4	1	no	—
131	C₂×Dic₆₇	—	4	1	no	—
132	C₈×C₆₇	—	4	4	no	—
133	C₆₇⋊C₈	—	4	4	no	—
134	C₂²×C₆₇	#391	1	1	yes	characteristic
135	C₂²×C₆₇	#407	1	1	yes	characteristic
136	C₂²×C₆₇	#408	1	1	yes	characteristic
137	C₂²×C₆₇	#416	1	1	yes	characteristic
138	C₂²×C₆₇	#417	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
140	C₂²×C₆₇	#421	1	1	yes	characteristic
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	4	1	no	—
145	C₄×C₆₇	—	4	1	no	—
146	C₄×C₆₇	—	4	1	no	—
147	C₄×C₆₇	#446	1	2	yes	—
148	C₄×C₆₇	#446	1	2	yes	—
149	C₄×C₆₇	#478	1	2	yes	—
150	C₄×C₆₇	#479	1	2	yes	—
151	Dic₆₇	—	4	2	no	—
152	Dic₆₇	—	4	2	no	—
153	Dic₆₇	—	8	1	no	—
154	Dic₆₇	—	8	1	no	—
155	#9867	—	67	1	no	maximal, Sylow
156	C₂×C₆₇	#1488	1	1	yes	characteristic
157	C₂×C₆₇	#1489	1	1	yes	characteristic
158	C₂×C₆₇	#1494	1	1	yes	characteristic
159	C₂×C₆₇	#1495	1	1	yes	characteristic
160	C₂×C₆₇	#1985	1	1	yes	characteristic
161	C₂×C₆₇	#1986	1	1	yes	characteristic
162	C₂×C₆₇	#1989	1	1	yes	characteristic
163	C₄×#125	—	67	1	no	—
164	#1767	—	67	1	no	—
165	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
166	#1279	—	67	2	no	—
167	#1279	—	67	2	no	—
168	C₆₇	#9867	1	1	yes	characteristic, complemented, Sylow
169	C₂×#125	—	67	1	no	—
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×C₄⋊C₈	—	67	2	no	—
172	C₂×#97	—	134	1	no	—
173	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
174	#389	—	67	1	no	—
175	#389	—	67	1	no	—
176	#387	—	134	1	no	—
177	#387	—	134	1	no	—
178	#382	—	67	1	no	—
179	C₄×C₄⋊C₄	—	67	1	no	—
180	C₄×C₄⋊C₄	—	67	1	no	—
181	C₄×C₄⋊C₄	—	134	1	no	—
182	C₄×C₄⋊C₄	—	134	1	no	—
183	C₄³	C₂×D₆₇	1	1	yes	characteristic
184	#336	—	2	1	no	—
185	#336	—	134	1	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	134	2	no	—
188	#125	—	134	2	no	—
189	#125	—	134	2	no	—
190	C₂×C₄⋊C₄	—	67	1	no	—
191	C₂×C₄⋊C₄	—	67	1	no	—
192	C₂×C₄⋊C₄	—	67	1	no	—
193	C₂×C₄⋊C₄	—	67	1	no	—
194	C₂×C₄⋊C₄	—	134	1	no	—
195	C₂×C₄⋊C₄	—	134	1	no	—
196	C₂×C₄⋊C₄	—	134	1	no	—
197	C₂×C₄⋊C₄	—	134	1	no	—
198	C₂×C₄²	—	134	1	no	—
199	C₂×C₄²	—	134	1	no	—
200	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
202	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
203	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	67	4	no	—
209	C₄⋊C₈	—	134	2	no	—
210	#97	—	268	2	no	—
211	C₄×C₈	—	2	4	no	—
212	#95	—	134	1	no	—
213	#95	—	134	1	no	—
214	#95	—	134	1	no	—
215	#95	—	134	1	no	—
216	#95	—	134	1	no	—
217	#95	—	134	1	no	—
218	C₂²×C₄	—	134	1	no	—
219	C₂²×C₄	—	134	1	no	—
220	C₂²×C₄	—	134	1	no	—
221	C₂²×C₄	—	134	1	no	—
222	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	268	2	no	—
233	C₂×C₈	—	268	2	no	—
234	C₂×C₈	—	134	4	no	—
235	C₄⋊C₄	—	268	2	no	—
236	C₄⋊C₄	—	268	2	no	—
237	C₄⋊C₄	—	67	2	no	—
238	C₄⋊C₄	—	67	2	no	—
239	C₄⋊C₄	—	134	4	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₄⋊C₄	—	134	2	no	—
242	C₄⋊C₄	—	134	2	no	—
243	C₄⋊C₄	—	134	1	no	—
244	C₄⋊C₄	—	134	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	268	2	no	—
254	C₄²	—	268	1	no	—
255	C₄²	—	268	1	no	—
256	C₄²	Q₈×D₆₇	1	2	yes	—
257	C₄²	#11761627	1	2	yes	—
258	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
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	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	268	1	no	—
273	C₂×C₄	—	268	1	no	—
274	C₂×C₄	—	268	1	no	—
275	C₂×C₄	—	268	1	no	—
276	C₂×C₄	—	268	1	no	—
277	C₂×C₄	—	268	1	no	—
278	C₂×C₄	—	268	1	no	—
279	C₂×C₄	—	268	1	no	—
280	C₂×C₄	—	268	1	no	—
281	C₂×C₄	—	268	1	no	—
282	C₂×C₄	—	134	2	no	—
283	C₂×C₄	—	134	2	no	—
284	C₂×C₄	#14523971	1	2	yes	—
285	C₂×C₄	#14524008	1	2	yes	—
286	C₂×C₄	#14524023	1	2	yes	—
287	C₂×C₄	#14524030	1	1	yes	characteristic
288	C₂×C₄	#14524041	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
290	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
291	C₂×C₄	#14524049	1	2	yes	—
292	C₂×C₄	#14524065	1	2	yes	—
293	C₂×C₄	#14524071	1	1	yes	characteristic
294	C₂×C₄	#14524076	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	268	4	no	—
297	C₂²	#16863301	1	1	yes	characteristic
298	C₂²	#16863359	1	1	yes	characteristic
299	C₂²	#16863362	1	1	yes	characteristic
300	C₂²	#16863393	1	1	yes	characteristic
301	C₂²	#16863398	1	1	yes	characteristic
302	C₂²	C₂×#14523971	1	1	yes	characteristic
303	C₂²	#16863414	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	268	2	no	—
311	C₄	—	268	2	no	—
312	C₄	—	536	1	no	—
313	C₄	—	536	1	no	—
314	C₄	#16863525	1	2	yes	—
315	C₄	#16863525	1	2	yes	—
316	C₄	#16863697	1	2	yes	—
317	C₄	#16863704	1	2	yes	—
318	C₂	#22558644	1	1	yes	characteristic
319	C₂	#22558649	1	1	yes	characteristic
320	C₂	#22558670	1	1	yes	characteristic
321	C₂	#22558675	1	1	yes	characteristic
322	C₂	#22561148	1	1	yes	characteristic
323	C₂	#22561152	1	1	yes	characteristic
324	C₂	#22561164	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#9867
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
168. C₆₇	#9867	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
#22560062	2
#22558049	2
C₄×C₄⋊C₈×C₆₇	2
#9867	67