Subgroups — #30748633

Order 17152.

664 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic, complemented
2	#22560326	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863140	C₂	1	2	yes	maximal
4	#22558092	C₂	1	2	yes	maximal, complemented
5	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863151	—	2	1	no	—
11	#16863151	—	2	1	no	—
12	#16863140	—	2	4	no	—
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	—	2	1	no	—
15	C₂×C₄×D₄×C₆₇	—	2	1	no	—
16	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
17	#390×C₆₇	—	2	1	no	—
18	#390×C₆₇	—	2	1	no	—
19	#390×C₆₇	C₄	1	2	yes	—
20	#384×C₆₇	C₂²	1	1	yes	characteristic
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₆₇	—	2	2	no	—
34	#127×C₆₇	—	2	1	no	—
35	#127×C₆₇	—	2	1	no	—
36	#127×C₆₇	D₄	1	2	yes	—
37	#127×C₆₇	Q₈	1	2	yes	—
38	C₄×D₄×C₆₇	—	4	4	no	—
39	C₄×D₄×C₆₇	—	4	4	no	—
40	C₄×D₄×C₆₇	—	4	2	no	—
41	C₄×D₄×C₆₇	—	4	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	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	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂⁴×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	2	2	no	—
63	C₂×D₄×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	—	2	2	no	—
66	C₂×D₄×C₆₇	—	2	2	no	—
67	C₂×D₄×C₆₇	—	4	2	no	—
68	C₂×D₄×C₆₇	—	4	2	no	—
69	C₂×D₄×C₆₇	—	4	1	no	—
70	C₂×D₄×C₆₇	—	4	1	no	—
71	C₂×D₄×C₆₇	—	4	2	no	—
72	C₂×D₄×C₆₇	—	4	2	no	—
73	C₂×D₄×C₆₇	—	4	1	no	—
74	C₂×D₄×C₆₇	—	4	1	no	—
75	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	4	4	no	—
78	C₂²×C₄×C₆₇	—	4	4	no	—
79	C₂²×C₄×C₆₇	—	4	4	no	—
80	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	—	4	2	no	—
88	C₄⋊C₄×C₆₇	—	4	1	no	—
89	C₄⋊C₄×C₆₇	—	4	1	no	—
90	C₄⋊C₄×C₆₇	—	4	2	no	—
91	C₂²⋊C₄×C₆₇	—	4	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₂²⋊C₄×C₆₇	—	4	4	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	4	1	no	—
100	C₄²×C₆₇	D₈	1	2	yes	—
101	C₄²×C₆₇	SD₁₆	1	2	yes	—
102	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₆₇	#41	1	1	yes	characteristic
105	C₄²×C₆₇	#41	1	1	yes	characteristic
106	C₂³×C₆₇	—	2	4	no	—
107	C₂³×C₆₇	—	4	2	no	—
108	C₂³×C₆₇	—	4	2	no	—
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	—	4	2	no	—
111	C₂³×C₆₇	—	4	2	no	—
112	C₂³×C₆₇	—	4	2	no	—
113	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	D₄×C₆₇	—	4	4	no	—
115	D₄×C₆₇	—	4	4	no	—
116	D₄×C₆₇	—	4	2	no	—
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	4	no	—
119	D₄×C₆₇	—	4	2	no	—
120	D₄×C₆₇	—	4	2	no	—
121	D₄×C₆₇	—	4	4	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	8	8	no	—
137	C₂×C₄×C₆₇	#100	1	2	yes	—
138	C₂×C₄×C₆₇	#102	1	2	yes	—
139	C₂×C₄×C₆₇	#102	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#135	1	2	yes	—
149	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
152	C₆₇⋊C₈	—	8	2	no	—
153	C₆₇⋊C₈	—	8	2	no	—
154	C₂²×C₆₇	—	4	4	no	—
155	C₂²×C₆₇	—	4	4	no	—
156	C₂²×C₆₇	—	4	4	no	—
157	C₂²×C₆₇	—	8	2	no	—
158	C₂²×C₆₇	—	8	2	no	—
159	C₂²×C₆₇	—	8	2	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	#380	1	1	yes	characteristic
162	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
163	C₂²×C₆₇	#413	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
165	C₂²×C₆₇	#416	1	1	yes	characteristic
166	C₂²×C₆₇	#417	1	1	yes	characteristic
167	C₂²×C₆₇	#418	1	1	yes	characteristic
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	4	1	no	—
174	C₄×C₆₇	—	8	2	no	—
175	C₄×C₆₇	#459	1	1	yes	characteristic
176	C₄×C₆₇	#461	1	1	yes	characteristic
177	C₄×C₆₇	#463	1	2	yes	—
178	C₄×C₆₇	#474	1	1	yes	characteristic
179	C₄×C₆₇	#476	1	1	yes	characteristic
180	C₄×C₆₇	#478	1	2	yes	—
181	#8588	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	8	4	no	—
183	C₂×C₆₇	#1540	1	1	yes	characteristic
184	C₂×C₆₇	#1545	1	1	yes	characteristic
185	C₂×C₆₇	#1551	1	1	yes	characteristic
186	C₂×C₆₇	#1557	1	1	yes	characteristic
187	C₂×C₆₇	#1800	1	1	yes	characteristic
188	C₂×C₆₇	#1802	1	1	yes	characteristic
189	C₂×C₆₇	#1804	1	1	yes	characteristic
190	C₄×#127	D₆₇	1	1	yes	characteristic
191	#1842	—	67	1	no	—
192	#1831	—	67	1	no	—
193	C₂×#331	—	67	2	no	—
194	#1296	—	67	2	no	—
195	C₆₇	#8588	1	1	yes	characteristic, complemented, Sylow
196	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
197	C₂×C₄×D₄	—	2	1	no	—
198	C₂×C₄×D₄	—	2	1	no	—
199	C₂×C₄×D₄	Dic₆₇	1	2	yes	—
200	C₂×C₄⋊C₈	—	67	2	no	—
201	C₂×C₄⋊C₈	—	67	2	no	—
202	C₂×C₄⋊C₈	—	134	1	no	—
203	C₂×C₄⋊C₈	—	134	1	no	—
204	#390	—	2	1	no	—
205	#390	—	2	1	no	—
206	#390	Dic₆₇	1	2	yes	—
207	#384	C₂×D₆₇	1	1	yes	characteristic
208	C₄³	C₂×D₆₇	1	1	yes	characteristic
209	#336	—	134	1	no	—
210	#336	—	134	1	no	—
211	#331	—	134	4	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	2	1	no	—
214	C₂²×D₄	C₂×Dic₆₇	1	2	yes	—
215	C₂³×C₄	—	2	2	no	—
216	C₂²×C₈	—	134	2	no	—
217	C₂²×C₈	—	134	2	no	—
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	Dic₁₃₄	1	2	yes	—
221	#127	D₂₆₈	1	2	yes	—
222	C₄×D₄	—	4	4	no	—
223	C₄×D₄	—	4	4	no	—
224	C₄×D₄	—	4	2	no	—
225	C₄×D₄	—	4	2	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
239	C₄⋊C₈	—	268	2	no	—
240	C₄⋊C₈	—	268	2	no	—
241	C₄⋊C₈	—	134	2	no	—
242	C₄⋊C₈	—	134	2	no	—
243	C₄⋊C₈	—	134	2	no	—
244	C₄⋊C₈	—	134	2	no	—
245	C₂⁴	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	C₄⋊Dic₆₇	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	4	no	—
263	C₂²×C₄	—	4	4	no	—
264	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
271	C₂×C₈	—	268	2	no	—
272	C₂×C₈	—	268	2	no	—
273	C₂×C₈	—	268	2	no	—
274	C₂×C₈	—	268	2	no	—
275	C₂×C₈	—	268	2	no	—
276	C₂×C₈	—	268	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	#11761613	1	2	yes	—
291	C₄²	#11761616	1	2	yes	—
292	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
293	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
294	C₄²	#11761628	1	1	yes	characteristic
295	C₄²	#11761628	1	1	yes	characteristic
296	C₂³	—	2	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	8	no	—
327	C₂×C₄	#14523960	1	2	yes	—
328	C₂×C₄	#14523964	1	2	yes	—
329	C₂×C₄	#14523964	1	2	yes	—
330	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
331	C₂×C₄	#14524002	1	1	yes	characteristic
332	C₂×C₄	#14524017	1	1	yes	characteristic
333	C₂×C₄	#14524017	1	1	yes	characteristic
334	C₂×C₄	#14524022	1	1	yes	characteristic
335	C₂×C₄	#14524022	1	1	yes	characteristic
336	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
337	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
338	C₂×C₄	#14524063	1	2	yes	—
339	C₂×C₄	#14524068	1	1	yes	characteristic
340	C₂×C₄	#14524069	1	2	yes	—
341	C₂×C₄	#14524073	1	1	yes	characteristic
342	C₈	—	536	2	no	—
343	C₈	—	536	2	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	4	4	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	#16863256	1	1	yes	characteristic
352	C₂²	C₂×#14523960	1	1	yes	characteristic
353	C₂²	#16863380	1	1	yes	characteristic
354	C₂²	C₂×#14523964	1	1	yes	characteristic
355	C₂²	#16863392	1	1	yes	characteristic
356	C₂²	#16863397	1	1	yes	characteristic
357	C₂²	#16863401	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	8	2	no	—
365	C₄	#16863591	1	1	yes	characteristic
366	C₄	#16863601	1	1	yes	characteristic
367	C₄	#16863611	1	2	yes	—
368	C₄	#16863676	1	1	yes	characteristic
369	C₄	#16863686	1	1	yes	characteristic
370	C₄	#16863696	1	2	yes	—
371	C₂	—	8	4	no	—
372	C₂	#22558858	1	1	yes	characteristic
373	C₂	#22558877	1	1	yes	characteristic
374	C₂	#22558904	1	1	yes	characteristic
375	C₂	#22558931	1	1	yes	characteristic
376	C₂	#22560201	1	1	yes	characteristic
377	C₂	#22560209	1	1	yes	characteristic
378	C₂	#22560219	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#127×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#127	D₆₇
hypocenter	C₆₇	#8588
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #22560361	C₂	semidirect
2. #22560326	C₂	semidirect
4. #22558092	C₂	semidirect
195. C₆₇	#8588	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
#22560326	2
C₂×#16863140	2
#22558092	2
C₄×#127×C₆₇	2
#8588	67