Subgroups — #30827338

Order 17152.

644 conjugacy classes of subgroups.

The subgroup lattice has 334 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	#22559123	C₂	1	2	yes	maximal, complemented
3	#22559088	C₂	1	4	yes	maximal, complemented
4	#22559028	C₂	1	2	yes	maximal, complemented
5	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558643	C₂	1	2	yes	maximal, complemented
7	#22558498	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	#16863117	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	2	yes	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
48	#135×C₆₇	—	2	4	no	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₆₇	C₂³	1	2	yes	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#106×C₆₇	C₂×C₄	1	2	yes	—
67	#103×C₆₇	C₂×C₄	1	2	yes	—
68	#103×C₆₇	C₂×C₄	1	2	yes	—
69	#102×C₆₇	C₂×C₄	1	2	yes	—
70	#102×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#11761612	—	4	2	no	—
76	#11761612	—	4	8	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
80	#41×C₆₇	—	2	4	no	—
81	#41×C₆₇	—	4	2	no	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	Q₁₆×C₆₇	M₁₆	1	4	yes	complemented
92	SD₁₆×C₆₇	—	2	4	no	—
93	D₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	4	1	no	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	#41	1	1	yes	characteristic
113	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₇	—	2	2	no	—
115	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
118	D₄×C₆₇	—	2	2	no	—
119	D₄×C₆₇	—	2	2	no	—
120	D₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	4	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	8	2	no	—
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
141	C₆₇⋊C₈	—	2	4	no	—
142	C₆₇⋊C₈	—	4	2	no	—
143	C₆₇⋊C₈	—	4	4	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	4	2	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	4	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×C₆₇	#568	1	1	yes	characteristic
160	C₄×C₆₇	#575	1	2	yes	—
161	#18043	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	4	1	no	—
163	C₂×C₆₇	—	8	2	no	—
164	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
165	C₂×C₆₇	#2943	1	1	yes	characteristic
166	C₂×C₆₇	#2957	1	1	yes	characteristic
167	#2971	—	67	2	no	—
168	C₄×#135	D₆₇	1	1	yes	characteristic
169	#1594	—	67	2	no	—
170	#1588	—	67	4	no	—
171	#1579	—	67	2	no	—
172	#1568	—	67	1	no	—
173	#1488	—	67	2	no	—
174	#1451	—	67	1	no	—
175	C₆₇	#18043	1	1	yes	characteristic, complemented, Sylow
176	C₂×#135	Dic₆₇	1	1	yes	characteristic
177	C₄×#41	C₂×D₆₇	1	2	yes	—
178	C₈×Q₈	—	67	4	no	—
179	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
180	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
181	C₄×SD₁₆	Dic₆₇	1	2	yes	—
182	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
183	C₄×D₈	Dic₆₇	1	1	yes	characteristic
184	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
185	#436	—	67	4	no	—
186	#436	—	134	4	no	—
187	#432	—	67	2	no	—
188	#432	—	134	2	no	—
189	#431	—	67	1	no	—
190	#427	Dic₆₇	1	1	yes	characteristic
191	#416	Dic₆₇	1	2	yes	—
192	C₄×M₁₆	—	67	1	no	—
193	C₄×M₁₆	—	134	2	no	—
194	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
195	#335	—	67	2	no	—
196	#334	—	67	2	no	—
197	Q₈⋊C₈	—	67	4	no	—
198	D₄⋊C₈	—	67	4	no	—
199	#322	—	67	2	no	—
200	#322	—	67	2	no	—
201	C₂×#41	C₂×Dic₆₇	1	2	yes	—
202	#135	—	2	4	no	—
203	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
205	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂×M₁₆	—	67	2	no	—
207	C₂×M₁₆	—	134	4	no	—
208	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
210	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
211	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
215	C₄×D₄	C₂²×D₆₇	1	2	yes	—
216	#117	—	2	2	no	—
217	#117	C₂×Dic₆₇	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
220	#107	C₂×Dic₆₇	1	1	yes	characteristic
221	#107	C₂×Dic₆₇	1	1	yes	characteristic
222	#106	C₂×Dic₆₇	1	2	yes	—
223	C₄⋊C₈	—	67	4	no	—
224	C₄⋊C₈	—	134	4	no	—
225	C₄⋊C₈	—	134	1	no	—
226	#103	C₂×Dic₆₇	1	2	yes	—
227	#103	C₂×Dic₆₇	1	2	yes	—
228	#102	C₂×Dic₆₇	1	2	yes	—
229	#102	C₂×Dic₆₇	1	2	yes	—
230	C₂²⋊C₈	—	134	4	no	—
231	C₂²⋊C₈	—	134	2	no	—
232	C₂²⋊C₈	—	134	1	no	—
233	#97	—	67	2	no	—
234	#97	—	134	2	no	—
235	C₄×C₈	—	67	2	no	—
236	C₄×C₈	—	67	2	no	—
237	C₄×C₈	—	134	4	no	—
238	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
242	#41	—	2	4	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
252	C₂²×C₄	#11761627	1	1	yes	characteristic
253	Q₁₆	#11761612	1	4	yes	complemented
254	SD₁₆	—	2	4	no	—
255	D₈	—	2	2	no	—
256	M₁₆	—	268	8	no	—
257	M₁₆	—	268	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	67	4	no	—
261	C₂×C₈	—	134	4	no	—
262	C₂×C₈	—	134	2	no	—
263	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
270	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
278	C₄²	D₄×D₆₇	1	1	yes	characteristic
279	C₄²	#11761627	1	1	yes	characteristic
280	C₄²	C₂³×D₆₇	1	1	yes	characteristic
281	C₂³	—	2	2	no	—
282	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	C₂×#11761612	1	4	yes	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	4	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
299	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	268	2	no	—
308	C₈	—	268	4	no	—
309	C₈	—	134	4	no	—
310	C₈	C₂×#11761612	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	C₂²×#11761612	1	1	yes	characteristic
326	C₄	#16864163	1	1	yes	characteristic
327	C₄	#16864193	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	8	2	no	—
330	C₂	D₄×#11761612	1	1	yes	characteristic
331	C₂	#22569010	1	1	yes	characteristic
332	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	#18043
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. #22559123	C₂	semidirect
3. #22559088	C₂	semidirect
4. #22559028	C₂	semidirect
5. #22558972	C₂	semidirect
6. #22558643	C₂	semidirect
7. #22558498	C₂	semidirect
91. Q₁₆×C₆₇	M₁₆	semidirect
175. C₆₇	#18043	semidirect
253. 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
#22559123	2
#22559088	2
#22559028	2
#22558972	2
#22558643	2
#22558498	2
C₄×#135×C₆₇	2
#18043	67