Subgroups — #30748705

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558596	C₂	1	2	yes	maximal
4	#22558060	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863149	—	2	1	no	—
9	#16863147	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	2	no	—
13	D₄⋊(C₆₇⋊C₈)	—	2	2	no	—
14	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
15	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
16	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#405×C₆₇	—	2	1	no	—
18	#405×C₆₇	—	2	1	no	—
19	#405×C₆₇	C₄	1	1	yes	characteristic
20	#405×C₆₇	C₄	1	1	yes	characteristic
21	C₄×M₁₆×C₆₇	—	2	1	no	—
22	C₄×M₁₆×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	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
32	#131×C₆₇	—	2	2	no	—
33	#131×C₆₇	—	2	2	no	—
34	#131×C₆₇	—	2	2	no	—
35	#131×C₆₇	C₈	1	4	yes	complemented
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	—	2	1	no	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₆₇	—	2	1	no	—
47	C₄×D₄×C₆₇	—	2	1	no	—
48	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
49	C₄×D₄×C₆₇	Q₈	1	1	yes	characteristic
50	#117×C₆₇	—	2	1	no	—
51	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#97×C₆₇	—	2	1	no	—
55	#97×C₆₇	—	2	1	no	—
56	#97×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	8	2	no	—
67	#41×C₆₇	—	2	1	no	—
68	#41×C₆₇	—	2	1	no	—
69	#41×C₆₇	C₂×C₈	1	2	yes	—
70	#41×C₆₇	M₁₆	1	2	yes	—
71	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂×D₄×C₆₇	—	2	1	no	—
73	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	2	1	no	—
82	M₁₆×C₆₇	—	2	1	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	C₂×C₈	1	2	yes	—
85	M₁₆×C₆₇	M₁₆	1	2	yes	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
97	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
98	C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₆₇	—	2	1	no	—
103	C₂²⋊C₄×C₆₇	—	2	1	no	—
104	C₂²⋊C₄×C₆₇	—	2	1	no	—
105	C₄²×C₆₇	—	2	1	no	—
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
112	C₂³×C₆₇	—	2	1	no	—
113	C₂³×C₆₇	#117	1	1	yes	characteristic
114	Q₈×C₆₇	—	2	1	no	—
115	Q₈×C₆₇	C₄⋊C₈	1	2	yes	—
116	D₄×C₆₇	—	2	2	no	—
117	D₄×C₆₇	—	2	2	no	—
118	D₄×C₆₇	—	2	1	no	—
119	D₄×C₆₇	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
144	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
145	C₆₇⋊C₈	—	8	2	no	—
146	C₆₇⋊C₈	—	8	2	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	4	1	no	—
151	C₂²×C₆₇	—	4	1	no	—
152	C₂²×C₆₇	#380	1	1	yes	characteristic
153	C₂²×C₆₇	#431	1	1	yes	characteristic
154	C₂²×C₆₇	#432	1	1	yes	characteristic
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	4	1	no	—
160	C₄×C₆₇	—	4	1	no	—
161	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₆₇	#407	1	1	yes	characteristic
163	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₄×C₆₇	#423	1	1	yes	characteristic
165	C₄×C₆₇	#471	1	2	yes	—
166	#8604	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	—	2	1	no	—
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	4	2	no	—
170	C₂×C₆₇	#1797	1	1	yes	characteristic
171	C₂×C₆₇	#1814	1	1	yes	characteristic
172	C₂×C₆₇	#1815	1	1	yes	characteristic
173	C₄×#131	D₆₇	1	1	yes	characteristic
174	#1564	—	67	1	no	—
175	C₂×#334	—	67	1	no	—
176	#1477	—	67	2	no	—
177	#1283	—	67	2	no	—
178	C₆₇	#8604	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	Dic₆₇	1	1	yes	characteristic
180	C₂×#131	Dic₆₇	1	1	yes	characteristic
181	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
182	#431	—	67	2	no	—
183	C₂×C₄⋊C₈	—	67	2	no	—
184	#405	—	2	1	no	—
185	#405	—	2	1	no	—
186	#405	Dic₆₇	1	1	yes	characteristic
187	#405	Dic₆₇	1	1	yes	characteristic
188	C₄×M₁₆	—	2	1	no	—
189	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
190	C₂×C₄×C₈	—	2	1	no	—
191	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
192	#335	—	134	1	no	—
193	#334	—	134	1	no	—
194	#334	—	134	1	no	—
195	#334	—	134	1	no	—
196	Q₈⋊C₈	—	134	2	no	—
197	D₄⋊C₈	—	134	2	no	—
198	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	C₆₇⋊C₈	1	4	yes	complemented
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	134	2	no	—
210	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
212	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
213	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	Dic₁₃₄	1	1	yes	characteristic
217	C₄×D₄	D₂₆₈	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	#117	C₂×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
222	C₄⋊C₈	—	134	2	no	—
223	C₄⋊C₈	—	134	2	no	—
224	C₄⋊C₈	—	134	2	no	—
225	C₂²⋊C₈	—	134	2	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	#97	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	134	2	no	—
233	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
235	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	C₂×C₆₇⋊C₈	1	2	yes	—
240	#41	#11761612	1	2	yes	—
241	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
248	C₂²×C₄	#11761628	1	1	yes	characteristic
249	C₂²×C₄	#11761628	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	C₂×C₆₇⋊C₈	1	2	yes	—
255	M₁₆	#11761612	1	2	yes	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	268	2	no	—
263	C₂×C₈	—	268	2	no	—
264	C₂×C₈	—	536	2	no	—
265	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
270	C₂×C₈	#11761612	1	2	yes	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
283	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
284	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	#14524002	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	#14524017	1	1	yes	characteristic
305	C₂×C₄	#14524017	1	1	yes	characteristic
306	C₂×C₄	#14524022	1	1	yes	characteristic
307	C₂×C₄	#14524022	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
309	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
310	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
311	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	—	536	2	no	—
317	C₈	—	536	2	no	—
318	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
319	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#16863256	1	1	yes	characteristic
326	C₂²	#16863449	1	1	yes	characteristic
327	C₂²	#16863453	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
335	C₄	#16863360	1	1	yes	characteristic
336	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
337	C₄	#16863420	1	1	yes	characteristic
338	C₄	#16863659	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#22560189	1	1	yes	characteristic
343	C₂	#22560263	1	1	yes	characteristic
344	C₂	#22560268	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#8604
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #22558957	C₂	semidirect
2. C₂×#16863147	C₂	semidirect
4. #22558060	C₂	semidirect
35. #131×C₆₇	C₈	semidirect
178. C₆₇	#8604	semidirect
202. #131	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22558957	2
C₂×#16863147	2
#22558596	2
#22558060	2
C₄×#131×C₆₇	2
#8604	67