Subgroups — #30826401

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570777	C₂	1	1	yes	maximal, characteristic
2	#22569187	C₂	1	2	yes	maximal, complemented
3	#22559119	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559111	C₂	1	2	yes	maximal, complemented
5	#22559033	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559027	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559018	C₂	1	2	yes	maximal, complemented
8	#22558969	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558666	C₂	1	2	yes	maximal, complemented
10	#322×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1594	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863928	C₂²	1	2	yes	—
13	#16863797	C₄	1	1	yes	characteristic
14	#16863779	C₄	1	1	yes	characteristic
15	#16863749	C₄	1	2	yes	—
16	#16863638	C₄	1	1	yes	characteristic
17	#16863629	C₄	1	1	yes	characteristic
18	#16863624	C₄	1	2	yes	—
19	#16863526	C₂²	1	2	yes	—
20	#16863520	C₂²	1	2	yes	—
21	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
22	C₄×#11761616	C₂²	1	2	yes	—
23	#16863465	—	2	2	no	—
24	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
25	(C₂²⋊D₆₇)⋊C₈	—	2	2	no	—
26	#16863454	—	2	2	no	—
27	#16863452	C₂²	1	2	yes	—
28	#16863354	—	2	2	no	—
29	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	2	yes	—
33	#16863147	C₂²	1	1	yes	characteristic
34	#16863128	C₂²	1	2	yes	—
35	#16863126	C₂²	1	2	yes	—
36	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863118	C₂²	1	1	yes	characteristic
39	#16863117	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
41	#436×C₆₇	C₂²	1	2	yes	—
42	#334×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761616	C₂×C₄	1	2	yes	—
47	C₂²×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₈×D₆₇	—	2	1	no	—
49	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	D₄	1	2	yes	—
52	#14524036	—	2	2	no	—
53	#14524034	C₂×C₄	1	2	yes	—
54	#14524030	C₂×C₄	1	2	yes	—
55	#14524025	—	2	2	no	—
56	#14524023	C₂×C₄	1	2	yes	—
57	#14524014	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₃₄	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₆₇	—	2	2	no	—
60	#14524002	—	2	2	no	—
61	C₄²×D₆₇	C₂³	1	1	yes	characteristic
62	#14523977	C₂×C₄	1	1	yes	characteristic
63	#14523974	C₂×C₄	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	#14523966	C₂×C₄	1	2	yes	—
69	#14523964	C₂×C₄	1	2	yes	—
70	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	#14523953	—	2	2	no	—
73	#14523953	—	2	2	no	—
74	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₆₇	—	2	2	no	—
78	C₈×Dic₆₇	—	2	1	no	—
79	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₇	D₄	1	2	yes	—
82	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×M₁₆×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	C₂³	1	2	yes	—
85	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
87	#102×C₆₇	C₂×C₄	1	2	yes	—
88	C₂²⋊C₈×C₆₇	—	2	2	no	—
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×C₂²⋊D₆₇	—	2	2	no	—
94	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
97	Dic₂₆₈	—	2	2	no	—
98	#11761616	—	2	4	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	—	4	4	no	—
102	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
105	C₈×D₆₇	—	2	2	no	—
106	C₈×D₆₇	—	2	2	no	—
107	C₈×D₆₇	—	4	2	no	—
108	C₈×D₆₇	—	4	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	C₄⋊Dic₆₇	—	2	2	no	—
111	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	Dic₆₇⋊C₄	—	2	2	no	—
114	C₂²⋊Dic₆₇	—	2	2	no	—
115	C₂²⋊Dic₆₇	—	2	2	no	—
116	C₂²⋊Dic₆₇	—	2	2	no	—
117	(C₂×D₆₇)⋊C₄	—	2	2	no	—
118	C₄×Dic₆₇	—	2	2	no	—
119	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₆₇	#41	1	1	yes	characteristic
122	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	D₈×C₆₇	—	2	2	no	—
125	M₁₆×C₆₇	—	4	4	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂³×C₆₇	—	2	2	no	—
135	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₆₇	—	4	4	no	—
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
150	C₄×D₆₇	—	2	2	no	—
151	C₄×D₆₇	—	2	1	no	—
152	C₄×D₆₇	—	2	1	no	—
153	C₄×D₆₇	—	2	2	no	—
154	Dic₁₃₄	—	2	2	no	—
155	Dic₁₃₄	—	2	2	no	—
156	C₈×C₆₇	—	2	1	no	—
157	C₈×C₆₇	—	2	2	no	—
158	C₈×C₆₇	—	4	1	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	#131	1	2	yes	—
161	C₆₇⋊C₈	—	2	1	no	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₆₇⋊C₈	—	4	1	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₆₇⋊C₈	#131	1	2	yes	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	4	2	no	—
169	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₆₇	—	2	1	no	—
171	C₂×D₆₇	—	2	1	no	—
172	C₂×D₆₇	#444	1	2	yes	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	4	2	no	—
175	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
176	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
177	C₄×C₆₇	#575	1	2	yes	—
178	Dic₆₇	—	2	1	no	—
179	Dic₆₇	—	2	1	no	—
180	Dic₆₇	—	2	1	no	—
181	Dic₆₇	—	4	2	no	—
182	Dic₆₇	—	4	2	no	—
183	#17963	—	67	1	no	maximal, Sylow
184	C₂×C₆₇	—	8	2	no	—
185	C₂×C₆₇	#2931	1	1	yes	characteristic
186	C₂×C₆₇	#2943	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
188	D₆₇	—	2	2	no	—
189	#3168	—	67	1	no	—
190	#2966	—	67	2	no	—
191	#1594	D₆₇	1	1	yes	characteristic, complemented
192	#1593	—	67	1	no	—
193	#1592	—	67	2	no	—
194	#1580	—	67	1	no	—
195	#1579	—	67	1	no	—
196	SD₁₆⋊C₈	—	67	2	no	—
197	#1568	—	67	1	no	—
198	#1494	—	67	2	no	—
199	C₂×#322	—	67	1	no	—
200	C₆₇	#17963	1	1	yes	characteristic, complemented, Sylow
201	#529	—	67	2	no	—
202	#499	—	67	1	no	—
203	#495	—	67	1	no	—
204	#488	—	67	2	no	—
205	#467	—	67	1	no	—
206	#466	—	67	1	no	—
207	#465	—	67	2	no	—
208	#446	—	67	2	no	—
209	C₈×Q₈	—	67	2	no	—
210	C₄×Q₁₆	—	67	1	no	—
211	C₄×SD₁₆	—	67	2	no	—
212	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
213	#436	C₂×D₆₇	1	2	yes	—
214	#435	—	134	2	no	—
215	C₈×D₄	—	67	2	no	—
216	C₈×D₄	—	134	2	no	—
217	#433	—	134	2	no	—
218	#432	—	67	2	no	—
219	#405	—	134	2	no	—
220	C₂×C₄×C₈	—	67	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	C₂×C₄×C₈	—	67	1	no	—
223	#335	—	67	2	no	—
224	#334	—	67	1	no	—
225	#334	C₂×D₆₇	1	1	yes	characteristic
226	Q₈⋊C₈	—	67	2	no	—
227	Q₈⋊C₈	—	67	2	no	—
228	D₄⋊C₈	—	67	2	no	—
229	D₄⋊C₈	C₂×D₆₇	1	2	yes	—
230	#322	—	67	1	no	—
231	#322	—	67	1	no	—
232	#322	—	67	1	no	—
233	#322	C₂×D₆₇	1	1	yes	characteristic
234	C₂×Q₁₆	—	67	1	no	—
235	C₂×SD₁₆	—	67	2	no	—
236	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
237	C₂×M₁₆	—	2	2	no	—
238	C₂²×C₈	—	67	2	no	—
239	C₂²×C₈	—	67	1	no	—
240	C₂²×C₈	—	67	1	no	—
241	C₂²×C₈	—	134	1	no	—
242	C₂²×C₈	—	134	2	no	—
243	#126	—	134	2	no	—
244	#125	—	67	2	no	—
245	#124	—	67	2	no	—
246	#123	—	134	2	no	—
247	C₂²⋊Q₈	—	67	2	no	—
248	#121	—	67	2	no	—
249	C₄×Q₈	—	67	2	no	—
250	C₄×D₄	—	134	2	no	—
251	C₄×D₄	C₂²×D₆₇	1	2	yes	—
252	#117	—	134	2	no	—
253	C₂×C₄²	—	67	1	no	—
254	#107	—	67	1	no	—
255	#107	C₄×D₆₇	1	1	yes	characteristic
256	#106	—	67	2	no	—
257	C₄⋊C₈	—	67	2	no	—
258	C₄⋊C₈	—	134	2	no	—
259	C₄⋊C₈	—	134	2	no	—
260	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
261	#103	—	67	2	no	—
262	#103	—	67	2	no	—
263	#102	—	67	2	no	—
264	#102	C₄×D₆₇	1	2	yes	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	—	134	2	no	—
267	C₂²⋊C₈	—	134	2	no	—
268	#97	—	134	2	no	—
269	#97	—	134	2	no	—
270	C₄×C₈	—	67	1	no	—
271	C₄×C₈	—	67	1	no	—
272	C₄×C₈	—	67	2	no	—
273	C₄×C₈	—	67	1	no	—
274	C₄×C₈	—	67	1	no	—
275	C₄×C₈	—	67	1	no	—
276	C₄×C₈	—	134	1	no	—
277	C₄×C₈	—	134	2	no	—
278	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
281	C₂×Q₈	—	67	2	no	—
282	C₂×D₄	—	134	2	no	—
283	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	67	1	no	—
286	C₂²×C₄	—	67	1	no	—
287	C₂²×C₄	—	67	1	no	—
288	Q₁₆	—	134	2	no	—
289	SD₁₆	—	134	4	no	—
290	D₈	—	2	2	no	—
291	M₁₆	—	4	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	268	2	no	—
295	C₂×C₈	—	268	2	no	—
296	C₂×C₈	—	268	4	no	—
297	C₂×C₈	—	67	1	no	—
298	C₂×C₈	—	67	1	no	—
299	C₂×C₈	—	67	2	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	—	134	1	no	—
303	C₂×C₈	—	134	2	no	—
304	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
306	C₂×C₈	D₄×D₆₇	1	2	yes	—
307	C₄⋊C₄	—	67	2	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₄⋊C₄	—	134	2	no	—
310	C₄⋊C₄	—	134	2	no	—
311	C₄⋊C₄	—	134	2	no	—
312	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₂²⋊C₄	—	134	2	no	—
317	C₂²⋊C₄	—	134	2	no	—
318	C₄²	—	67	1	no	—
319	C₄²	—	67	1	no	—
320	C₄²	—	67	1	no	—
321	C₄²	—	134	2	no	—
322	C₄²	C₂³×D₆₇	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	—	67	1	no	—
325	Q₈	—	134	2	no	—
326	Q₈	—	134	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	268	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	67	1	no	—
334	C₂×C₄	—	67	1	no	—
335	C₂×C₄	—	67	1	no	—
336	C₂×C₄	—	134	2	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	—	134	1	no	—
340	C₂×C₄	—	134	1	no	—
341	C₂×C₄	—	134	2	no	—
342	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
344	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	268	1	no	—
350	C₈	—	268	2	no	—
351	C₈	—	67	2	no	—
352	C₈	—	134	1	no	—
353	C₈	—	134	2	no	—
354	C₈	#14524051	1	2	yes	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	67	2	no	—
359	C₂²	—	134	1	no	—
360	C₂²	—	134	1	no	—
361	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	268	2	no	—
365	C₄	—	268	2	no	—
366	C₄	—	134	1	no	—
367	C₄	—	134	1	no	—
368	C₄	—	134	1	no	—
369	C₄	#131×D₆₇	1	1	yes	characteristic
370	C₄	C₂×#14524051	1	1	yes	characteristic
371	C₄	#16864192	1	2	yes	—
372	C₂	—	8	2	no	—
373	C₂	—	134	2	no	—
374	C₂	#22568918	1	1	yes	characteristic
375	C₂	#22569006	1	1	yes	characteristic
376	C₂	#444×D₆₇	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×#1594	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863629	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1594	D₆₇
hypocenter	C₆₇	#17963
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1594

Complements

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

N	H ≅ G/N	type
2. #22569187	C₂	semidirect
3. #22559119	C₂	semidirect
4. #22559111	C₂	semidirect
5. #22559033	C₂	semidirect
6. #22559027	C₂	semidirect
7. #22559018	C₂	semidirect
8. #22558969	C₂	semidirect
9. #22558666	C₂	semidirect
10. #322×D₆₇	C₂	semidirect
11. C₆₇×#1594	C₂	semidirect
191. #1594	D₆₇	semidirect
200. C₆₇	#17963	semidirect

Maximal subgroups

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

subgroup	index
#22570777	2
#22569187	2
#22559119	2
#22559111	2
#22559033	2
#22559027	2
#22559018	2
#22558969	2
#22558666	2
#322×D₆₇	2
C₆₇×#1594	2
#17963	67