Subgroups — #30827300

Order 17152.

652 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569229	C₂	1	2	yes	maximal, complemented
2	#22559094	C₂	1	2	yes	maximal, complemented
3	#22559053	C₂	1	4	yes	maximal, complemented
4	Q₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558643	C₂	1	2	yes	maximal, complemented
7	C₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
10	#16863474	—	2	4	no	—
11	#16863474	C₂²	1	4	yes	—
12	#16863453	—	2	2	no	—
13	#16863453	C₂²	1	2	yes	—
14	#16863449	C₂²	1	1	yes	characteristic
15	C₄×#11761612	—	2	2	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863149	C₂²	1	2	yes	—
18	#16863147	C₂²	1	2	yes	—
19	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	#16863118	C₂²	1	2	yes	—
22	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
23	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
24	C₄×#41×C₆₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₆₇	C₄	1	2	yes	—
28	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
30	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
31	#427×C₆₇	C₄	1	1	yes	characteristic
32	#416×C₆₇	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	4	no	—
35	C₂×#11761612	D₄	1	2	yes	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
42	#14523954	—	2	2	no	—
43	#14523954	D₄	1	1	yes	characteristic
44	#14523954	D₄	1	1	yes	characteristic
45	C₄×C₆₇⋊C₈	—	2	4	no	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
49	#135×C₆₇	—	2	4	no	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₆₇	C₂³	1	2	yes	—
61	#117×C₆₇	—	2	2	no	—
62	#117×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#106×C₆₇	C₂×C₄	1	2	yes	—
68	#103×C₆₇	C₂×C₄	1	2	yes	—
69	#103×C₆₇	C₂×C₄	1	2	yes	—
70	#102×C₆₇	C₂×C₄	1	2	yes	—
71	#102×C₆₇	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	#11761612	—	2	4	no	—
77	#11761612	—	4	8	no	—
78	C₂×C₆₇⋊C₈	—	2	4	no	—
79	C₂×C₆₇⋊C₈	—	2	2	no	—
80	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
81	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
82	#41×C₆₇	—	2	4	no	—
83	#41×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	Q₁₆×C₆₇	M₁₆	1	4	yes	complemented
94	SD₁₆×C₆₇	—	2	4	no	—
95	D₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	4	1	no	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂²⋊C₄×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	#41	1	1	yes	characteristic
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
120	D₄×C₆₇	—	2	2	no	—
121	D₄×C₆₇	—	2	2	no	—
122	D₄×C₆₇	—	4	4	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	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	8	2	no	—
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
143	C₆₇⋊C₈	—	2	2	no	—
144	C₆₇⋊C₈	—	4	2	no	—
145	C₆₇⋊C₈	—	4	4	no	—
146	C₆₇⋊C₈	#135	1	4	yes	complemented
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	4	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	4	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
162	C₄×C₆₇	#568	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#135	1	2	yes	—
164	#18039	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	4	1	no	—
166	C₂×C₆₇	—	8	2	no	—
167	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
168	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
169	C₂×C₆₇	#2957	1	1	yes	characteristic
170	#2971	—	67	2	no	—
171	C₄×#135	D₆₇	1	1	yes	characteristic
172	#1589	—	67	2	no	—
173	#1583	—	67	4	no	—
174	C₈×Q₁₆	—	67	2	no	—
175	#1568	—	67	1	no	—
176	#1488	—	67	2	no	—
177	C₈×M₁₆	—	67	1	no	—
178	C₆₇	#18039	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	Dic₆₇	1	1	yes	characteristic
180	C₄×#41	C₂×D₆₇	1	2	yes	—
181	C₈×Q₈	—	67	4	no	—
182	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
183	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
184	C₄×SD₁₆	Dic₆₇	1	2	yes	—
185	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
186	C₄×D₈	Dic₆₇	1	1	yes	characteristic
187	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
188	#436	—	67	4	no	—
189	#436	—	134	4	no	—
190	#432	—	67	2	no	—
191	#432	—	134	2	no	—
192	#431	—	67	1	no	—
193	#427	Dic₆₇	1	1	yes	characteristic
194	#416	Dic₆₇	1	2	yes	—
195	C₄×M₁₆	—	67	1	no	—
196	C₄×M₁₆	—	134	2	no	—
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	#335	—	67	2	no	—
199	#334	—	67	2	no	—
200	Q₈⋊C₈	—	67	4	no	—
201	D₄⋊C₈	—	67	4	no	—
202	#322	—	67	2	no	—
203	C₈²	—	67	2	no	—
204	C₂×#41	C₂×Dic₆₇	1	2	yes	—
205	#135	—	2	4	no	—
206	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
208	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
209	C₂×M₁₆	—	67	2	no	—
210	C₂×M₁₆	—	134	4	no	—
211	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
213	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
214	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
218	C₄×D₄	C₂²×D₆₇	1	2	yes	—
219	#117	—	2	2	no	—
220	#117	C₂×Dic₆₇	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	#107	C₂×Dic₆₇	1	1	yes	characteristic
224	#107	C₂×Dic₆₇	1	1	yes	characteristic
225	#106	C₂×Dic₆₇	1	2	yes	—
226	C₄⋊C₈	—	67	4	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	1	no	—
229	#103	C₂×Dic₆₇	1	2	yes	—
230	#103	C₂×Dic₆₇	1	2	yes	—
231	#102	C₂×Dic₆₇	1	2	yes	—
232	#102	C₂×Dic₆₇	1	2	yes	—
233	C₂²⋊C₈	—	134	4	no	—
234	C₂²⋊C₈	—	134	2	no	—
235	C₂²⋊C₈	—	134	1	no	—
236	#97	—	67	1	no	—
237	#97	—	67	1	no	—
238	#97	—	134	2	no	—
239	C₄×C₈	—	67	2	no	—
240	C₄×C₈	—	67	2	no	—
241	C₄×C₈	—	134	4	no	—
242	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
246	#41	—	2	4	no	—
247	#41	—	4	2	no	—
248	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
256	C₂²×C₄	#11761627	1	1	yes	characteristic
257	Q₁₆	#11761612	1	4	yes	complemented
258	SD₁₆	—	2	4	no	—
259	D₈	—	2	2	no	—
260	M₁₆	—	268	8	no	—
261	M₁₆	—	134	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	134	4	no	—
267	C₂×C₈	—	134	2	no	—
268	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
275	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₄²	#11761627	1	1	yes	characteristic
285	C₄²	C₂³×D₆₇	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	C₂×#11761612	1	4	yes	—
290	D₄	—	2	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	268	2	no	—
313	C₈	—	268	4	no	—
314	C₈	—	67	4	no	—
315	C₈	—	134	2	no	—
316	C₈	C₂×#11761612	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	C₂²×#11761612	1	1	yes	characteristic
332	C₄	#16864163	1	1	yes	characteristic
333	C₄	#135×D₆₇	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	8	2	no	—
336	C₂	D₄×#11761612	1	1	yes	characteristic
337	C₂	#135×Dic₆₇	1	1	yes	characteristic
338	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	#18039
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22569229	C₂	semidirect
2. #22559094	C₂	semidirect
3. #22559053	C₂	semidirect
4. Q₁₆×C₆₇⋊C₈	C₂	semidirect
5. #22558972	C₂	semidirect
6. #22558643	C₂	semidirect
7. C₈×#11761612	C₂	semidirect
93. Q₁₆×C₆₇	M₁₆	semidirect
146. C₆₇⋊C₈	#135	semidirect
178. C₆₇	#18039	semidirect
257. Q₁₆	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22569229	2
#22559094	2
#22559053	2
Q₁₆×C₆₇⋊C₈	2
#22558972	2
#22558643	2
C₈×#11761612	2
C₄×#135×C₆₇	2
#18039	67