Subgroups — #30826518

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 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	#22569354	C₂	1	2	yes	maximal, complemented
3	#22559110	C₂	1	2	yes	maximal, complemented
4	#22559105	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559093	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559020	C₂	1	2	yes	maximal, complemented
7	#22559015	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559003	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558672	C₂	1	2	yes	maximal, complemented
10	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1567	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	#16863523	C₂²	1	2	yes	—
20	#16863520	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	#16863473	C₂²	1	2	yes	—
25	#16863472	—	2	2	no	—
26	#16863465	—	2	2	no	—
27	#16863461	C₂²	1	2	yes	—
28	#16863456	—	2	2	no	—
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863421	—	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	#423×C₆₇	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
42	#335×C₆₇	C₂²	1	1	yes	characteristic
43	#335×C₆₇	C₂²	1	1	yes	characteristic
44	#335×C₆₇	C₂²	1	2	yes	—
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)	—	2	1	no	—
64	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
65	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	#14523968	C₂×C₄	1	2	yes	—
69	#14523968	C₂×C₄	1	2	yes	—
70	#14523965	C₂×C₄	1	2	yes	—
71	#14523965	C₂×C₄	1	2	yes	—
72	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
73	C₂²⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
74	C₂²⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	#14523953	—	2	2	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₆₇	—	2	2	no	—
83	C₂×M₁₆×C₆₇	—	2	2	no	—
84	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	—	2	2	no	—
88	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
95	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₆₇	—	2	2	no	—
97	C₂×C₄×D₆₇	—	2	2	no	—
98	Dic₂₆₈	—	2	2	no	—
99	#11761617	—	2	4	no	—
100	D₅₃₆	—	2	2	no	—
101	#11761611	—	8	2	no	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	—	2	1	no	—
104	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₈×D₆₇	—	8	2	no	—
109	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
111	Dic₆₇⋊C₄	—	2	2	no	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	Dic₆₇⋊C₄	—	2	2	no	—
114	Dic₆₇⋊C₄	—	2	2	no	—
115	C₂²⋊Dic₆₇	—	2	2	no	—
116	(C₂×D₆₇)⋊C₄	—	2	2	no	—
117	(C₂×D₆₇)⋊C₄	—	2	2	no	—
118	(C₂×D₆₇)⋊C₄	—	2	2	no	—
119	(C₂×D₆₇)⋊C₄	—	2	2	no	—
120	C₄×Dic₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
124	M₁₆×C₆₇	—	8	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	#41	1	1	yes	characteristic
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
138	C₂²×D₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₆₇	—	8	2	no	—
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	2	2	no	—
154	D₂₆₈	—	2	2	no	—
155	D₂₆₈	—	2	2	no	—
156	C₄×D₆₇	—	4	2	no	—
157	C₄×D₆₇	—	4	2	no	—
158	Dic₁₃₄	—	2	2	no	—
159	Dic₁₃₄	—	2	2	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	#131	1	2	yes	—
165	C₈×C₆₇	#131	1	2	yes	—
166	C₆₇⋊C₈	—	2	1	no	—
167	C₆₇⋊C₈	—	2	1	no	—
168	C₆₇⋊C₈	—	2	1	no	—
169	C₆₇⋊C₈	—	2	1	no	—
170	C₆₇⋊C₈	—	4	1	no	—
171	C₆₇⋊C₈	—	4	1	no	—
172	C₂²×C₆₇	—	2	1	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₆₇	—	4	2	no	—
177	C₂×D₆₇	—	4	2	no	—
178	C₂×D₆₇	—	4	2	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
184	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
185	C₄×C₆₇	#576	1	1	yes	characteristic
186	C₄×C₆₇	#578	1	1	yes	characteristic
187	Dic₆₇	—	4	2	no	—
188	Dic₆₇	—	4	2	no	—
189	Dic₆₇	—	4	2	no	—
190	#17973	—	67	1	no	maximal, Sylow
191	C₂×C₆₇	—	4	1	no	—
192	C₂×C₆₇	#2931	1	1	yes	characteristic
193	C₂×C₆₇	#2951	1	1	yes	characteristic
194	C₂×C₆₇	#2957	1	1	yes	characteristic
195	D₆₇	—	8	2	no	—
196	#3168	—	67	1	no	—
197	#2993	—	67	2	no	—
198	#1592	—	67	2	no	—
199	#1591	—	67	1	no	—
200	#1589	—	67	1	no	—
201	SD₁₆⋊C₈	—	67	2	no	—
202	C₈×Q₁₆	—	67	1	no	—
203	C₈×D₈	—	67	1	no	—
204	#1567	D₆₇	1	1	yes	characteristic, complemented
205	#1495	—	67	2	no	—
206	#1450	—	67	1	no	—
207	C₆₇	#17973	1	1	yes	characteristic, complemented, Sylow
208	#529	—	67	2	no	—
209	#499	—	67	1	no	—
210	#495	—	67	1	no	—
211	#488	—	67	2	no	—
212	#467	—	67	1	no	—
213	#466	—	67	1	no	—
214	#465	—	67	2	no	—
215	#446	—	67	2	no	—
216	C₈×Q₈	—	67	2	no	—
217	C₄×Q₁₆	—	67	1	no	—
218	C₄×SD₁₆	—	67	2	no	—
219	C₄×D₈	—	67	1	no	—
220	#436	—	67	2	no	—
221	#436	—	134	2	no	—
222	#435	—	134	2	no	—
223	C₈×D₄	—	67	2	no	—
224	#433	—	134	2	no	—
225	#431	—	67	1	no	—
226	#431	—	67	1	no	—
227	#424	—	134	2	no	—
228	#423	C₂×D₆₇	1	2	yes	—
229	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
230	#335	C₂×D₆₇	1	1	yes	characteristic
231	#335	C₂×D₆₇	1	1	yes	characteristic
232	#335	C₂×D₆₇	1	2	yes	—
233	Q₈⋊C₈	—	67	2	no	—
234	Q₈⋊C₈	—	67	2	no	—
235	D₄⋊C₈	—	67	2	no	—
236	D₄⋊C₈	—	67	2	no	—
237	#322	—	67	1	no	—
238	#322	—	67	1	no	—
239	C₈²	—	67	1	no	—
240	C₈²	—	67	1	no	—
241	C₂×Q₁₆	—	67	1	no	—
242	C₂×SD₁₆	—	67	2	no	—
243	C₂×D₈	—	67	1	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	134	2	no	—
246	C₂²×C₈	—	134	2	no	—
247	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
248	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
249	#126	—	134	2	no	—
250	#125	—	67	2	no	—
251	#124	—	67	2	no	—
252	#123	—	134	2	no	—
253	C₂²⋊Q₈	—	67	2	no	—
254	#121	—	67	2	no	—
255	C₄×Q₈	—	67	2	no	—
256	C₄×D₄	—	67	2	no	—
257	C₄×D₄	—	134	2	no	—
258	#117	—	134	2	no	—
259	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
260	#107	—	67	1	no	—
261	#107	—	67	1	no	—
262	#106	—	67	2	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	67	1	no	—
265	C₄⋊C₈	—	67	1	no	—
266	C₄⋊C₈	—	134	2	no	—
267	C₄⋊C₈	—	134	2	no	—
268	C₄⋊C₈	—	134	1	no	—
269	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
270	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
271	#103	—	67	2	no	—
272	#103	—	67	2	no	—
273	#102	—	67	2	no	—
274	#102	—	67	2	no	—
275	C₂²⋊C₈	—	67	1	no	—
276	C₂²⋊C₈	—	67	1	no	—
277	C₂²⋊C₈	—	134	2	no	—
278	C₂²⋊C₈	—	134	2	no	—
279	C₂²⋊C₈	—	134	1	no	—
280	#97	—	134	2	no	—
281	C₄×C₈	—	67	1	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	—	67	1	no	—
285	C₄×C₈	—	134	2	no	—
286	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
287	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₂×Q₈	—	67	2	no	—
291	C₂×D₄	—	67	2	no	—
292	C₂×D₄	—	134	2	no	—
293	C₂²×C₄	—	134	2	no	—
294	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
296	C₂²×C₄	#11761629	1	1	yes	characteristic
297	Q₁₆	—	134	2	no	—
298	SD₁₆	—	134	4	no	—
299	D₈	—	134	2	no	—
300	M₁₆	—	8	2	no	—
301	M₁₆	—	536	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	536	2	no	—
307	C₂×C₈	—	67	1	no	—
308	C₂×C₈	—	67	1	no	—
309	C₂×C₈	—	67	1	no	—
310	C₂×C₈	—	67	1	no	—
311	C₂×C₈	—	134	1	no	—
312	C₂×C₈	—	134	1	no	—
313	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
317	C₄⋊C₄	—	67	2	no	—
318	C₄⋊C₄	—	67	2	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	—	134	2	no	—
321	C₄⋊C₄	—	134	2	no	—
322	C₄⋊C₄	—	134	2	no	—
323	C₂²⋊C₄	—	134	2	no	—
324	C₂²⋊C₄	—	134	2	no	—
325	C₂²⋊C₄	—	134	2	no	—
326	C₂²⋊C₄	—	134	2	no	—
327	C₂²⋊C₄	—	134	2	no	—
328	C₄²	—	134	2	no	—
329	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
330	C₄²	D₄×D₆₇	1	1	yes	characteristic
331	C₄²	#11761629	1	1	yes	characteristic
332	C₄²	C₂³×D₆₇	1	1	yes	characteristic
333	C₂³	—	134	2	no	—
334	C₂³	#14524005	1	1	yes	characteristic
335	Q₈	—	134	2	no	—
336	Q₈	—	134	2	no	—
337	D₄	—	536	2	no	—
338	D₄	—	134	2	no	—
339	D₄	—	134	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	268	2	no	—
346	C₂×C₄	—	268	2	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	2	no	—
349	C₂×C₄	—	134	2	no	—
350	C₂×C₄	#14524005	1	1	yes	characteristic
351	C₂×C₄	#14524005	1	1	yes	characteristic
352	C₂×C₄	#14524005	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	268	1	no	—
361	C₈	—	268	1	no	—
362	C₈	—	134	1	no	—
363	C₈	—	134	1	no	—
364	C₈	—	134	1	no	—
365	C₈	—	134	1	no	—
366	C₈	#14524051	1	2	yes	—
367	C₈	#14524051	1	2	yes	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	268	2	no	—
372	C₂²	—	268	2	no	—
373	C₂²	—	268	2	no	—
374	C₂²	C₂×#14524005	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	268	2	no	—
380	C₄	—	268	2	no	—
381	C₄	—	268	2	no	—
382	C₄	C₂×#14524051	1	1	yes	characteristic
383	C₄	C₂×#14524051	1	1	yes	characteristic
384	C₄	#16864197	1	1	yes	characteristic
385	C₄	#16864209	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	536	2	no	—
388	C₂	#22568920	1	1	yes	characteristic
389	C₂	#22569083	1	1	yes	characteristic
390	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×#1567	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863632	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#17973
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
2. #22569354	C₂	semidirect
3. #22559110	C₂	semidirect
4. #22559105	C₂	semidirect
5. #22559093	C₂	semidirect
6. #22559020	C₂	semidirect
7. #22559015	C₂	semidirect
8. #22559003	C₂	semidirect
9. #22558672	C₂	semidirect
10. #22558492	C₂	semidirect
11. C₆₇×#1567	C₂	semidirect
204. #1567	D₆₇	semidirect
207. C₆₇	#17973	semidirect

Maximal subgroups

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

subgroup	index
#22570779	2
#22569354	2
#22559110	2
#22559105	2
#22559093	2
#22559020	2
#22559015	2
#22559003	2
#22558672	2
#22558492	2
C₆₇×#1567	2
#17973	67