Subgroups — #30826498

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570768	C₂	1	1	yes	maximal, characteristic
2	#22569306	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569287	C₂	1	1	yes	maximal, characteristic
4	#22559110	C₂	1	2	yes	maximal, complemented
5	#22559098	C₂	1	2	yes	maximal, complemented
6	#22559020	C₂	1	2	yes	maximal, complemented
7	#22559008	C₂	1	2	yes	maximal, complemented
8	#22558681	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558677	C₂	1	1	yes	maximal, characteristic
10	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863953	C₂²	1	1	yes	characteristic
13	#16863946	C₂²	1	1	yes	characteristic
14	#16863794	C₄	1	1	yes	characteristic
15	#16863769	C₄	1	1	yes	characteristic
16	#16863747	C₄	1	1	yes	characteristic
17	#16863743	C₄	1	1	yes	characteristic
18	#16863625	C₄	1	2	yes	—
19	#16863625	C₄	1	2	yes	—
20	#16863524	—	2	2	no	—
21	#16863523	C₂²	1	2	yes	—
22	#16863520	C₂²	1	2	yes	—
23	C₄×#11761617	C₂²	1	2	yes	—
24	C₄×#11761617	C₂²	1	2	yes	—
25	#16863473	C₂²	1	2	yes	—
26	#16863469	—	2	2	no	—
27	#16863461	C₂²	1	2	yes	—
28	#16863454	—	2	2	no	—
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863361	—	2	2	no	—
32	#16863126	C₂²	1	2	yes	—
33	#16863126	C₂²	1	2	yes	—
34	#16863123	C₂²	1	2	yes	—
35	#16863123	C₂²	1	2	yes	—
36	#16863117	C₂²	1	1	yes	characteristic
37	#16863117	C₂²	1	1	yes	characteristic
38	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#432×C₆₇	C₂²	1	1	yes	characteristic
41	#432×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
43	#335×C₆₇	C₂²	1	2	yes	—
44	#334×C₆₇	C₂²	1	2	yes	—
45	C₂×#11761628	—	2	1	no	—
46	C₂×#11761617	C₂×C₄	1	2	yes	—
47	C₂×#11761617	C₂×C₄	1	2	yes	—
48	C₂×#11761611	—	2	2	no	—
49	C₂×C₈×D₆₇	—	2	2	no	—
50	#14524043	C₂×C₄	1	1	yes	characteristic
51	#14524043	C₂×C₄	1	1	yes	characteristic
52	#14524043	C₂×C₄	1	1	yes	characteristic
53	#14524040	C₂×C₄	1	1	yes	characteristic
54	#14524035	—	2	1	no	—
55	#14524031	—	2	1	no	—
56	#14524022	—	2	1	no	—
57	#14524022	C₂×C₄	1	2	yes	—
58	#14524017	—	2	1	no	—
59	#14524017	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₃₄	—	2	1	no	—
61	C₄×Dic₁₃₄	C₂³	1	2	yes	—
62	C₄×D₂₆₈	—	2	1	no	—
63	C₄×D₂₆₈	C₂³	1	2	yes	—
64	#14524002	—	2	1	no	—
65	#14523975	C₂×C₄	1	2	yes	—
66	#14523975	C₂×C₄	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
69	Dic₆₇⋊C₈	—	2	2	no	—
70	Dic₆₇⋊C₈	—	2	2	no	—
71	#14523968	C₂×C₄	1	2	yes	—
72	#14523968	C₂×C₄	1	2	yes	—
73	#14523965	C₂×C₄	1	2	yes	—
74	#14523965	C₂×C₄	1	2	yes	—
75	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
76	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	#14523953	—	2	2	no	—
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
84	C₈×Dic₆₇	—	2	2	no	—
85	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	1	no	—
92	#97×C₆₇	—	2	1	no	—
93	#97×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	#11761628	—	8	2	no	—
99	C₂×Dic₁₃₄	—	2	1	no	—
100	C₂×Dic₁₃₄	—	2	1	no	—
101	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
102	C₂×D₂₆₈	—	2	1	no	—
103	C₂×D₂₆₈	—	2	1	no	—
104	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
105	C₂×C₂²⋊D₆₇	—	2	2	no	—
106	C₂×C₄×D₆₇	—	2	2	no	—
107	#11761617	—	2	4	no	—
108	#11761617	—	2	4	no	—
109	#11761611	—	8	2	no	—
110	C₂×C₆₇⋊C₈	—	2	1	no	—
111	C₂×C₆₇⋊C₈	—	2	1	no	—
112	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
113	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
114	C₈×D₆₇	—	8	2	no	—
115	C₄⋊Dic₆₇	—	2	1	no	—
116	C₄⋊Dic₆₇	—	2	1	no	—
117	C₄⋊Dic₆₇	—	2	1	no	—
118	C₄⋊Dic₆₇	—	2	1	no	—
119	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
120	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
121	Dic₆₇⋊C₄	—	2	2	no	—
122	Dic₆₇⋊C₄	—	2	2	no	—
123	C₂²⋊Dic₆₇	—	2	2	no	—
124	(C₂×D₆₇)⋊C₄	—	2	2	no	—
125	(C₂×D₆₇)⋊C₄	—	2	2	no	—
126	C₄×Dic₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₆₇	#41	1	1	yes	characteristic
141	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₆₇	—	8	2	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	2	2	no	—
157	C₂×Dic₆₇	—	2	2	no	—
158	D₂₆₈	—	2	2	no	—
159	D₂₆₈	—	2	2	no	—
160	D₂₆₈	—	8	1	no	—
161	D₂₆₈	—	8	1	no	—
162	C₄×D₆₇	—	4	2	no	—
163	C₄×D₆₇	—	4	2	no	—
164	Dic₁₃₄	—	2	2	no	—
165	Dic₁₃₄	—	2	2	no	—
166	Dic₁₃₄	—	8	1	no	—
167	Dic₁₃₄	—	8	1	no	—
168	C₈×C₆₇	—	2	1	no	—
169	C₈×C₆₇	—	2	1	no	—
170	C₈×C₆₇	—	4	2	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	#131	1	2	yes	—
173	C₈×C₆₇	#131	1	2	yes	—
174	C₆₇⋊C₈	—	2	2	no	—
175	C₆₇⋊C₈	—	2	2	no	—
176	C₆₇⋊C₈	—	4	1	no	—
177	C₆₇⋊C₈	—	4	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×D₆₇	—	4	2	no	—
183	C₂×D₆₇	—	4	2	no	—
184	C₂×D₆₇	—	4	2	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
190	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
191	C₄×C₆₇	#577	1	2	yes	—
192	Dic₆₇	—	4	2	no	—
193	Dic₆₇	—	4	2	no	—
194	Dic₆₇	—	4	2	no	—
195	#17971	—	67	1	no	maximal, Sylow
196	C₂×C₆₇	—	4	1	no	—
197	C₂×C₆₇	#2931	1	1	yes	characteristic
198	C₂×C₆₇	#2949	1	1	yes	characteristic
199	C₂×C₆₇	#2957	1	1	yes	characteristic
200	D₆₇	—	8	2	no	—
201	#3167	—	67	1	no	—
202	#2985	—	67	1	no	—
203	#2981	—	67	1	no	—
204	#1592	—	67	2	no	—
205	#1590	—	67	2	no	—
206	SD₁₆⋊C₈	—	67	2	no	—
207	C₈×SD₁₆	—	67	2	no	—
208	#1568	D₆₇	1	1	yes	characteristic, complemented
209	#1497	—	67	1	no	—
210	#1496	—	67	1	no	—
211	#1450	—	67	1	no	—
212	C₆₇	#17971	1	1	yes	characteristic, complemented, Sylow
213	#533	—	67	1	no	—
214	#532	—	67	1	no	—
215	#498	—	67	1	no	—
216	#492	—	67	1	no	—
217	#487	—	67	1	no	—
218	#486	—	67	1	no	—
219	#465	—	67	2	no	—
220	#465	—	67	2	no	—
221	#446	—	67	2	no	—
222	#446	—	134	2	no	—
223	C₈×Q₈	—	67	2	no	—
224	C₄×SD₁₆	—	67	2	no	—
225	C₄×SD₁₆	—	67	2	no	—
226	#436	—	67	2	no	—
227	#435	—	134	2	no	—
228	C₈×D₄	—	67	2	no	—
229	#433	—	134	2	no	—
230	#432	C₂×D₆₇	1	1	yes	characteristic
231	#432	C₂×D₆₇	1	1	yes	characteristic
232	#431	—	67	1	no	—
233	#431	—	67	1	no	—
234	#408	—	134	2	no	—
235	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
236	#335	C₂×D₆₇	1	2	yes	—
237	#334	C₂×D₆₇	1	2	yes	—
238	Q₈⋊C₈	—	67	2	no	—
239	Q₈⋊C₈	—	67	2	no	—
240	D₄⋊C₈	—	67	2	no	—
241	D₄⋊C₈	—	67	2	no	—
242	#322	—	67	1	no	—
243	#322	—	67	1	no	—
244	C₈²	—	67	1	no	—
245	C₈²	—	67	1	no	—
246	C₂×#41	—	134	1	no	—
247	C₂×SD₁₆	—	67	2	no	—
248	C₂×SD₁₆	—	67	2	no	—
249	C₂×M₁₆	—	134	2	no	—
250	C₂²×C₈	—	134	2	no	—
251	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
252	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
253	C₄⋊Q₈	—	67	1	no	—
254	C₄⋊Q₈	—	67	1	no	—
255	C₄⋊Q₈	—	67	1	no	—
256	#127	—	67	1	no	—
257	#125	—	134	1	no	—
258	#124	—	134	1	no	—
259	C₂²⋊Q₈	—	67	2	no	—
260	C₂²⋊Q₈	—	134	1	no	—
261	#121	—	67	2	no	—
262	#121	—	134	1	no	—
263	C₄×Q₈	—	67	2	no	—
264	C₄×Q₈	—	134	1	no	—
265	C₄×D₄	—	67	2	no	—
266	C₄×D₄	—	134	1	no	—
267	#117	—	134	1	no	—
268	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
269	#106	—	67	2	no	—
270	#106	—	67	2	no	—
271	C₄⋊C₈	—	67	2	no	—
272	C₄⋊C₈	—	134	2	no	—
273	C₄⋊C₈	—	134	2	no	—
274	C₄⋊C₈	—	134	1	no	—
275	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
276	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
277	#103	—	67	2	no	—
278	#103	—	67	2	no	—
279	#102	—	67	2	no	—
280	#102	—	67	2	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	67	2	no	—
284	C₂²⋊C₈	—	134	2	no	—
285	C₂²⋊C₈	—	134	2	no	—
286	C₂²⋊C₈	—	134	1	no	—
287	#97	—	2	1	no	—
288	#97	—	2	1	no	—
289	#97	—	134	2	no	—
290	C₄×C₈	—	67	1	no	—
291	C₄×C₈	—	67	1	no	—
292	C₄×C₈	—	67	1	no	—
293	C₄×C₈	—	67	1	no	—
294	C₄×C₈	—	134	2	no	—
295	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
296	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
298	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
299	#41	—	536	2	no	—
300	C₂×Q₈	—	67	2	no	—
301	C₂×Q₈	—	134	1	no	—
302	C₂×Q₈	—	134	1	no	—
303	C₂×D₄	—	67	2	no	—
304	C₂×D₄	—	134	2	no	—
305	C₂×D₄	—	134	1	no	—
306	C₂×D₄	—	134	1	no	—
307	C₂²×C₄	—	134	2	no	—
308	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
309	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
310	C₂²×C₄	#11761629	1	1	yes	characteristic
311	SD₁₆	—	134	4	no	—
312	SD₁₆	—	134	4	no	—
313	M₁₆	—	536	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	536	2	no	—
319	C₂×C₈	—	67	2	no	—
320	C₂×C₈	—	67	2	no	—
321	C₂×C₈	—	134	1	no	—
322	C₂×C₈	—	134	1	no	—
323	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
326	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
327	C₄⋊C₄	—	67	2	no	—
328	C₄⋊C₄	—	67	2	no	—
329	C₄⋊C₄	—	134	1	no	—
330	C₄⋊C₄	—	134	1	no	—
331	C₄⋊C₄	—	134	1	no	—
332	C₄⋊C₄	—	134	2	no	—
333	C₄⋊C₄	—	134	1	no	—
334	C₄⋊C₄	—	134	2	no	—
335	C₂²⋊C₄	—	134	2	no	—
336	C₂²⋊C₄	—	134	2	no	—
337	C₂²⋊C₄	—	134	2	no	—
338	C₄²	—	134	2	no	—
339	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
340	C₄²	D₄×D₆₇	1	1	yes	characteristic
341	C₄²	#11761629	1	1	yes	characteristic
342	C₄²	C₂³×D₆₇	1	1	yes	characteristic
343	C₂³	—	134	2	no	—
344	C₂³	#14524005	1	1	yes	characteristic
345	Q₈	—	536	1	no	—
346	Q₈	—	536	1	no	—
347	Q₈	—	134	2	no	—
348	Q₈	—	134	2	no	—
349	D₄	—	536	1	no	—
350	D₄	—	536	1	no	—
351	D₄	—	536	2	no	—
352	D₄	—	134	2	no	—
353	D₄	—	134	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	268	2	no	—
359	C₂×C₄	—	268	2	no	—
360	C₂×C₄	—	134	2	no	—
361	C₂×C₄	—	134	2	no	—
362	C₂×C₄	—	134	2	no	—
363	C₂×C₄	#14524005	1	1	yes	characteristic
364	C₂×C₄	#14524005	1	1	yes	characteristic
365	C₂×C₄	#14524005	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	268	1	no	—
374	C₈	—	268	1	no	—
375	C₈	—	134	2	no	—
376	C₈	—	134	2	no	—
377	C₈	#14524051	1	2	yes	—
378	C₈	#14524051	1	2	yes	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	268	2	no	—
383	C₂²	—	268	2	no	—
384	C₂²	—	268	2	no	—
385	C₂²	C₂×#14524005	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	268	2	no	—
391	C₄	—	268	2	no	—
392	C₄	—	268	2	no	—
393	C₄	C₂×#14524051	1	1	yes	characteristic
394	C₄	C₂×#14524051	1	1	yes	characteristic
395	C₄	#16864203	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	536	2	no	—
398	C₂	#22568920	1	1	yes	characteristic
399	C₂	#22569067	1	1	yes	characteristic
400	C₂	#22569125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1568	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#17971
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. #22569306	C₂	semidirect
4. #22559110	C₂	semidirect
5. #22559098	C₂	semidirect
6. #22559020	C₂	semidirect
7. #22559008	C₂	semidirect
8. #22558681	C₂	semidirect
10. #22558492	C₂	semidirect
11. C₆₇×#1568	C₂	semidirect
208. #1568	D₆₇	semidirect
212. C₆₇	#17971	semidirect

Maximal subgroups

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

subgroup	index
#22570768	2
#22569306	2
#22569287	2
#22559110	2
#22559098	2
#22559020	2
#22559008	2
#22558681	2
#22558677	2
#22558492	2
C₆₇×#1568	2
#17971	67