Subgroups — #30827676

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570779	C₂	1	1	yes	maximal, characteristic
2	#22569213	C₂	1	2	yes	maximal, complemented
3	#22559124	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559119	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559089	C₂	1	2	yes	maximal, complemented
6	#22559068	C₂	1	2	yes	maximal, complemented
7	#22559034	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559029	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558618	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863930	C₂²	1	2	yes	—
13	#16863798	C₄	1	1	yes	characteristic
14	#16863780	C₄	1	1	yes	characteristic
15	#16863751	C₄	1	2	yes	—
16	#16863637	C₄	1	1	yes	characteristic
17	#16863632	C₄	1	1	yes	characteristic
18	#16863625	C₄	1	2	yes	—
19	#16863526	C₂²	1	2	yes	—
20	C₈×Dic₁₃₄	C₂²	1	2	yes	—
21	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
22	C₄×#11761617	C₂²	1	2	yes	—
23	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
24	#16863475	C₂²	1	2	yes	—
25	#16863472	—	2	2	no	—
26	#16863465	—	2	2	no	—
27	C₈×D₂₆₈	C₂²	1	2	yes	—
28	#16863458	—	2	2	no	—
29	#16863449	C₂²	1	2	yes	—
30	#16863353	—	2	2	no	—
31	#16863149	C₂²	1	2	yes	—
32	#16863147	C₂²	1	1	yes	characteristic
33	#16863147	C₂²	1	1	yes	characteristic
34	#16863128	C₂²	1	2	yes	—
35	#16863128	C₂²	1	2	yes	—
36	#16863125	C₂²	1	2	yes	—
37	#16863125	C₂²	1	2	yes	—
38	#431×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#322×C₆₇	C₂²	1	1	yes	characteristic
42	#322×C₆₇	C₂²	1	1	yes	characteristic
43	#322×C₆₇	C₂²	1	1	yes	characteristic
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761617	C₂×C₄	1	2	yes	—
47	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
48	C₂×#11761611	—	2	2	no	—
49	C₂×C₈×D₆₇	—	2	2	no	—
50	#14524037	—	2	2	no	—
51	#14524035	C₂×C₄	1	2	yes	—
52	#14524031	C₂×C₄	1	2	yes	—
53	#14524026	—	2	2	no	—
54	#14524022	C₂×C₄	1	2	yes	—
55	#14524017	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃₄	C₂³	1	2	yes	—
57	C₄×D₂₆₈	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₆₇	—	2	2	no	—
59	#14524000	—	2	2	no	—
60	#14523977	C₂×C₄	1	1	yes	characteristic
61	#14523977	C₂×C₄	1	1	yes	characteristic
62	#14523975	C₂×C₄	1	2	yes	—
63	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
64	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₆₇⋊C₈	—	2	2	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	#14523968	C₂×C₄	1	2	yes	—
68	#14523968	C₂×C₄	1	2	yes	—
69	#14523965	C₂×C₄	1	2	yes	—
70	#14523965	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	2	no	—
74	#14523953	—	2	2	no	—
75	C₄×C₆₇⋊C₈	—	2	2	no	—
76	C₈×Dic₆₇	—	2	2	no	—
77	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₂²⋊C₈×C₆₇	—	2	1	no	—
83	#97×C₆₇	D₄	1	2	yes	—
84	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
93	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
94	C₂×C₂²⋊D₆₇	—	2	2	no	—
95	C₂×C₄×D₆₇	—	2	2	no	—
96	Dic₂₆₈	—	2	2	no	—
97	#11761617	—	4	2	no	—
98	D₅₃₆	—	2	2	no	—
99	#11761611	—	4	4	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₈×D₆₇	—	4	4	no	—
103	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
105	Dic₆₇⋊C₄	—	2	2	no	—
106	Dic₆₇⋊C₄	—	2	2	no	—
107	Dic₆₇⋊C₄	—	2	2	no	—
108	Dic₆₇⋊C₄	—	2	2	no	—
109	C₂²⋊Dic₆₇	—	2	2	no	—
110	(C₂×D₆₇)⋊C₄	—	2	2	no	—
111	(C₂×D₆₇)⋊C₄	—	2	2	no	—
112	(C₂×D₆₇)⋊C₄	—	2	2	no	—
113	(C₂×D₆₇)⋊C₄	—	2	2	no	—
114	C₄×Dic₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
118	M₁₆×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	4	1	no	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₆₇	#41	1	1	yes	characteristic
132	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂²×D₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂²⋊D₆₇	—	8	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	2	2	no	—
149	D₂₆₈	—	2	2	no	—
150	D₂₆₈	—	2	2	no	—
151	C₄×D₆₇	—	4	2	no	—
152	C₄×D₆₇	—	4	2	no	—
153	Dic₁₃₄	—	2	2	no	—
154	Dic₁₃₄	—	2	2	no	—
155	C₈×C₆₇	—	2	1	no	—
156	C₈×C₆₇	—	2	1	no	—
157	C₈×C₆₇	—	2	1	no	—
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	4	1	no	—
161	C₈×C₆₇	—	4	1	no	—
162	C₈×C₆₇	#131	1	2	yes	—
163	C₆₇⋊C₈	—	4	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₆₇	—	4	2	no	—
170	C₂×D₆₇	—	4	2	no	—
171	C₂×D₆₇	—	4	2	no	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
177	C₄×C₆₇	#568	1	1	yes	characteristic
178	C₄×C₆₇	#575	1	2	yes	—
179	Dic₆₇	—	4	2	no	—
180	Dic₆₇	—	4	2	no	—
181	Dic₆₇	—	4	2	no	—
182	#18075	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	—	4	1	no	—
184	C₂×C₆₇	#2935	1	1	yes	characteristic
185	C₂×C₆₇	#2943	1	1	yes	characteristic
186	C₂×C₆₇	#2957	1	1	yes	characteristic
187	D₆₇	—	8	2	no	—
188	#3168	—	67	1	no	—
189	#2969	—	67	2	no	—
190	#1594	—	67	1	no	—
191	#1593	—	67	1	no	—
192	#1588	—	67	2	no	—
193	#1585	—	67	2	no	—
194	#1580	—	67	1	no	—
195	#1579	—	67	1	no	—
196	#1564	—	67	1	no	—
197	#1482	—	67	2	no	—
198	#1451	D₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#18075	1	1	yes	characteristic, complemented, Sylow
200	#529	—	67	2	no	—
201	#499	—	67	1	no	—
202	#495	—	67	1	no	—
203	#488	—	67	2	no	—
204	#467	—	67	1	no	—
205	#466	—	67	1	no	—
206	#465	—	67	2	no	—
207	#446	—	67	2	no	—
208	C₈×Q₈	—	67	2	no	—
209	C₄×Q₁₆	—	67	1	no	—
210	C₄×SD₁₆	—	67	2	no	—
211	C₄×D₈	—	67	1	no	—
212	#436	—	67	2	no	—
213	#436	—	134	2	no	—
214	#435	—	134	2	no	—
215	C₈×D₄	—	67	2	no	—
216	#433	—	134	2	no	—
217	#431	—	67	2	no	—
218	#431	C₂×D₆₇	1	1	yes	characteristic
219	#405	—	134	2	no	—
220	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
222	#335	—	67	2	no	—
223	#334	—	67	1	no	—
224	#334	—	67	1	no	—
225	Q₈⋊C₈	—	67	2	no	—
226	Q₈⋊C₈	—	67	2	no	—
227	D₄⋊C₈	—	67	2	no	—
228	D₄⋊C₈	—	67	2	no	—
229	#322	C₂×D₆₇	1	1	yes	characteristic
230	#322	C₂×D₆₇	1	1	yes	characteristic
231	#322	C₂×D₆₇	1	1	yes	characteristic
232	#322	C₂×D₆₇	1	1	yes	characteristic
233	C₂×Q₁₆	—	67	1	no	—
234	C₂×SD₁₆	—	67	2	no	—
235	C₂×D₈	—	67	1	no	—
236	C₂×M₁₆	—	134	2	no	—
237	C₂×M₁₆	D₂₆₈	1	2	yes	—
238	C₂²×C₈	—	134	2	no	—
239	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
240	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
241	#126	—	134	2	no	—
242	#125	—	67	2	no	—
243	#124	—	67	2	no	—
244	#123	—	134	2	no	—
245	C₂²⋊Q₈	—	67	2	no	—
246	#121	—	67	2	no	—
247	C₄×Q₈	—	67	2	no	—
248	C₄×D₄	—	67	2	no	—
249	C₄×D₄	—	134	2	no	—
250	#117	—	134	2	no	—
251	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
252	#107	—	67	1	no	—
253	#107	—	67	1	no	—
254	#106	—	67	2	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	67	2	no	—
257	C₄⋊C₈	—	67	2	no	—
258	C₄⋊C₈	—	134	2	no	—
259	C₄⋊C₈	—	134	2	no	—
260	#103	—	67	2	no	—
261	#103	—	67	2	no	—
262	#102	—	67	2	no	—
263	#102	—	67	2	no	—
264	C₂²⋊C₈	—	2	1	no	—
265	C₂²⋊C₈	—	134	2	no	—
266	C₂²⋊C₈	—	134	2	no	—
267	C₂²⋊C₈	—	134	2	no	—
268	#97	—	134	2	no	—
269	#97	D₂₆₈	1	2	yes	—
270	C₄×C₈	—	134	2	no	—
271	C₄×C₈	—	134	2	no	—
272	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
273	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
280	C₂×Q₈	—	67	2	no	—
281	C₂×D₄	—	67	2	no	—
282	C₂×D₄	—	134	2	no	—
283	C₂²×C₄	—	134	2	no	—
284	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
286	C₂²×C₄	#11761628	1	1	yes	characteristic
287	Q₁₆	—	134	2	no	—
288	SD₁₆	—	268	2	no	—
289	D₈	—	134	2	no	—
290	M₁₆	—	4	2	no	—
291	M₁₆	—	268	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	268	4	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	—	134	2	no	—
299	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
304	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
305	C₄⋊C₄	—	67	2	no	—
306	C₄⋊C₄	—	67	2	no	—
307	C₄⋊C₄	—	134	2	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₄⋊C₄	—	134	2	no	—
310	C₄⋊C₄	—	134	2	no	—
311	C₂²⋊C₄	—	134	2	no	—
312	C₂²⋊C₄	—	134	2	no	—
313	C₂²⋊C₄	—	134	2	no	—
314	C₂²⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₄²	—	134	2	no	—
317	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
318	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
319	C₄²	#11761628	1	1	yes	characteristic
320	C₄²	C₂³×D₆₇	1	1	yes	characteristic
321	C₂³	—	134	2	no	—
322	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
323	Q₈	—	134	2	no	—
324	Q₈	—	134	2	no	—
325	D₄	—	536	2	no	—
326	D₄	—	134	2	no	—
327	D₄	—	134	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	268	2	no	—
333	C₂×C₄	—	268	2	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	—	134	2	no	—
336	C₂×C₄	—	134	2	no	—
337	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
338	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
339	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	1	no	—
350	C₈	—	268	2	no	—
351	C₈	—	268	2	no	—
352	C₈	#14524051	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	268	2	no	—
357	C₂²	—	268	2	no	—
358	C₂²	—	268	2	no	—
359	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	268	2	no	—
365	C₄	—	268	2	no	—
366	C₄	—	268	2	no	—
367	C₄	C₂×#14524051	1	1	yes	characteristic
368	C₄	#16864162	1	1	yes	characteristic
369	C₄	#16864196	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	536	2	no	—
372	C₂	#22568955	1	1	yes	characteristic
373	C₂	#22569011	1	1	yes	characteristic
374	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863632	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#18075
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #22569213	C₂	semidirect
3. #22559124	C₂	semidirect
4. #22559119	C₂	semidirect
5. #22559089	C₂	semidirect
6. #22559068	C₂	semidirect
7. #22559034	C₂	semidirect
8. #22559029	C₂	semidirect
9. #22558957	C₂	semidirect
10. #22558618	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
198. #1451	D₆₇	semidirect
199. C₆₇	#18075	semidirect

Maximal subgroups

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

subgroup	index
#22570779	2
#22569213	2
#22559124	2
#22559119	2
#22559089	2
#22559068	2
#22559034	2
#22559029	2
#22558957	2
#22558618	2
C₆₇×#1451	2
#18075	67