Subgroups — #30748995

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560229	C₂	1	2	yes	maximal
2	C₄×#14523975	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558111	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523975	—	2	1	no	—
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523970	—	2	2	no	—
9	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	#16863159	—	2	1	no	—
11	#16863155	—	2	2	no	—
12	#16863147	—	2	1	no	—
13	#16863147	—	2	1	no	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
17	C₈×D₄×C₆₇	—	2	1	no	—
18	C₈×D₄×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
20	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#431×C₆₇	—	2	1	no	—
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄×Dic₆₇	—	2	2	no	—
28	#14523975	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	#14523970	—	4	4	no	—
32	C₆₇⋊#95	—	2	2	no	—
33	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₆₇	—	2	1	no	—
41	C₄×D₄×C₆₇	—	2	1	no	—
42	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₆₇	Q₈	1	1	yes	characteristic
44	#117×C₆₇	—	2	1	no	—
45	#117×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	—	2	1	no	—
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₆₇	—	2	1	no	—
53	C₂²⋊C₈×C₆₇	—	2	1	no	—
54	C₂²⋊C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₂²×Dic₆₇	—	4	2	no	—
63	#11761612	—	4	2	no	—
64	#11761612	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₄⋊Dic₆₇	—	2	4	no	—
67	C₄⋊Dic₆₇	—	4	2	no	—
68	C₄×Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	#41×C₆₇	—	2	2	no	—
71	#41×C₆₇	SD₁₆	1	4	yes	—
72	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₄×C₆₇	—	2	1	no	—
74	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	1	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	4	1	no	—
85	C₂×C₈×C₆₇	—	4	2	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₆₇	—	2	1	no	—
96	C₂²⋊C₄×C₆₇	—	2	1	no	—
97	C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	2	1	no	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂³×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	#121	1	1	yes	characteristic
107	Q₈×C₆₇	—	2	1	no	—
108	Q₈×C₆₇	#106	1	2	yes	complemented
109	D₄×C₆₇	—	2	4	no	—
110	D₄×C₆₇	—	2	1	no	—
111	D₄×C₆₇	#106	1	2	yes	complemented
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₆₇	#135	1	2	yes	—
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	8	2	no	—
132	C₂×Dic₆₇	—	8	2	no	—
133	C₈×C₆₇	—	2	1	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	4	1	no	—
137	C₈×C₆₇	—	4	1	no	—
138	C₈×C₆₇	#104	1	2	yes	complemented
139	C₆₇⋊C₈	—	8	2	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	4	1	no	—
144	C₂²×C₆₇	—	4	1	no	—
145	C₂²×C₆₇	#380	1	1	yes	characteristic
146	C₂²×C₆₇	#465	1	2	yes	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	2	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	4	1	no	—
152	C₄×C₆₇	—	4	1	no	—
153	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
154	C₄×C₆₇	#421	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
156	C₄×C₆₇	#427	1	1	yes	characteristic
157	C₄×C₆₇	#465	1	2	yes	—
158	Dic₆₇	—	8	4	no	—
159	#8666	—	67	1	no	maximal, Sylow
160	C₂×C₆₇	—	2	2	no	—
161	C₂×C₆₇	—	4	2	no	—
162	C₂×C₆₇	#1806	1	1	yes	characteristic
163	C₂×C₆₇	#1807	1	1	yes	characteristic
164	C₂×C₆₇	#1814	1	1	yes	characteristic
165	C₈×#41	D₆₇	1	1	yes	characteristic
166	#1806	—	67	2	no	—
167	C₄×#106	—	67	1	no	—
168	#1566	—	67	1	no	—
169	#1303	—	67	2	no	—
170	C₆₇	#8666	1	1	yes	characteristic, complemented, Sylow
171	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
172	C₈×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
173	C₈×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
174	C₈×D₄	—	2	1	no	—
175	C₈×D₄	—	2	1	no	—
176	C₈×D₄	Dic₆₇	1	1	yes	characteristic, complemented
177	C₈×D₄	Dic₆₇	1	1	yes	characteristic, complemented
178	#431	—	2	1	no	—
179	#431	C₂×D₆₇	1	1	yes	characteristic
180	C₂×#106	—	134	1	no	—
181	#423	—	67	2	no	—
182	C₂×#104	—	134	2	no	—
183	C₂×C₄×C₈	—	2	1	no	—
184	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
185	C₄×C₄⋊C₄	—	67	2	no	—
186	#340	—	134	1	no	—
187	#339	—	134	2	no	—
188	#334	—	134	1	no	—
189	#334	—	134	1	no	—
190	C₂×#41	D₂₆₈	1	1	yes	characteristic
191	C₂×M₁₆	—	134	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
197	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	Dic₁₃₄	1	1	yes	characteristic
201	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	#117	D₂₆₈	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	134	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	134	2	no	—
207	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
208	#106	—	134	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	134	2	no	—
212	C₄⋊C₈	—	134	2	no	—
213	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
214	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
215	#104	—	268	4	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
223	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
226	#95	—	134	2	no	—
227	#41	—	2	2	no	—
228	#41	#11761617	1	4	yes	—
229	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	268	2	no	—
236	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
238	C₂²×C₄	#11761628	1	1	yes	characteristic
239	M₁₆	—	268	2	no	—
240	M₁₆	—	268	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	268	2	no	—
248	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	268	2	no	—
255	C₄⋊C₄	—	134	4	no	—
256	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	268	2	no	—
265	C₄²	—	268	2	no	—
266	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
268	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
269	C₄²	#11761628	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	#14524017	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	#14523975	1	2	yes	complemented
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	#14523975	1	2	yes	complemented
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	268	4	no	—
288	C₂×C₄	—	536	2	no	—
289	C₂×C₄	—	536	2	no	—
290	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
292	C₂×C₄	#14524002	1	1	yes	characteristic
293	C₂×C₄	#14524017	1	1	yes	characteristic
294	C₂×C₄	#14524022	1	1	yes	characteristic
295	C₂×C₄	#14524022	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761617	1	2	yes	—
297	C₂×C₄	#14524065	1	2	yes	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	536	2	no	—
304	C₈	#14523970	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#16863256	1	1	yes	characteristic
311	C₂²	#16863625	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	536	4	no	—
319	C₄	C₂×#14523970	1	1	yes	characteristic
320	C₄	#16863413	1	1	yes	characteristic
321	C₄	C₂×#14523975	1	1	yes	characteristic
322	C₄	#16863434	1	1	yes	characteristic
323	C₄	#16863625	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	#22560229	1	1	yes	characteristic
327	C₂	#22560234	1	1	yes	characteristic
328	C₂	#22560263	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#8666
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14523975	C₂	semidirect
3. #22558965	C₂	semidirect
4. #22558111	C₂	semidirect
15. C₈×Q₈×C₆₇	C₄	semidirect
16. C₈×Q₈×C₆₇	C₄	semidirect
19. C₈×D₄×C₆₇	C₄	semidirect
20. C₈×D₄×C₆₇	C₄	semidirect
108. Q₈×C₆₇	#106	semidirect
111. D₄×C₆₇	#106	semidirect
138. C₈×C₆₇	#104	semidirect
170. C₆₇	#8666	semidirect
172. C₈×Q₈	Dic₆₇	semidirect
173. C₈×Q₈	Dic₆₇	semidirect
176. C₈×D₄	Dic₆₇	semidirect
177. C₈×D₄	Dic₆₇	semidirect
273. Q₈	#14523975	semidirect
276. D₄	#14523975	semidirect
304. C₈	#14523970	semidirect

Maximal subgroups

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

subgroup	index
#22560229	2
C₄×#14523975	2
#22558965	2
#22558111	2
C₈×#41×C₆₇	2
#8666	67