Subgroups — #30827619

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 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	#22569216	C₂	1	2	yes	maximal, complemented
3	#22559105	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559050	C₂	1	2	yes	maximal, complemented
5	#22559042	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558956	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558614	C₂	1	2	yes	maximal, complemented
10	C₈×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1589	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	#16863523	C₂²	1	2	yes	—
20	C₈×Dic₁₃₄	C₂²	1	2	yes	—
21	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
22	C₄×#11761616	C₂²	1	2	yes	—
23	#16863472	—	2	2	no	—
24	#16863464	—	2	2	no	—
25	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
26	#16863454	—	2	2	no	—
27	#16863448	C₂²	1	2	yes	—
28	#16863447	C₂²	1	1	yes	characteristic
29	#16863354	—	2	2	no	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863149	C₂²	1	1	yes	characteristic
33	#16863146	C₂²	1	2	yes	—
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	#16863117	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
41	#436×C₆₇	C₂²	1	2	yes	—
42	#335×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761616	C₂×C₄	1	2	yes	—
47	C₂×#11761611	D₄	1	1	yes	characteristic
48	C₂×#11761611	D₄	1	1	yes	characteristic
49	C₂²×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
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	1	no	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	#14523968	C₂×C₄	1	2	yes	—
69	#14523966	C₂×C₄	1	2	yes	—
70	#14523964	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	1	no	—
74	#14523953	—	2	2	no	—
75	#14523953	D₄	1	1	yes	characteristic
76	#14523953	D₄	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×Dic₆₇	—	2	2	no	—
81	C₈×Dic₆₇	—	2	2	no	—
82	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	C₂³	1	2	yes	—
87	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₄×C₈×C₆₇	C₂³	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×C₂²⋊D₆₇	—	2	2	no	—
96	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
99	Dic₂₆₈	—	2	2	no	—
100	#11761616	—	4	2	no	—
101	#11761611	—	2	2	no	—
102	#11761611	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	1	no	—
105	C₂×C₆₇⋊C₈	—	4	4	no	—
106	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₈×D₆₇	—	2	2	no	—
111	C₈×D₆₇	—	4	1	no	—
112	C₄⋊Dic₆₇	—	2	2	no	—
113	C₄⋊Dic₆₇	—	2	2	no	—
114	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
115	Dic₆₇⋊C₄	—	2	2	no	—
116	Dic₆₇⋊C₄	—	2	2	no	—
117	C₂²⋊Dic₆₇	—	2	2	no	—
118	C₂²⋊Dic₆₇	—	2	2	no	—
119	C₂²⋊Dic₆₇	—	2	2	no	—
120	(C₂×D₆₇)⋊C₄	—	2	2	no	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₆₇	#41	1	1	yes	characteristic
125	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	D₈×C₆₇	—	2	2	no	—
128	M₁₆×C₆₇	—	4	4	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂²⋊C₄×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂³×C₆₇	—	2	2	no	—
139	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
140	D₄×C₆₇	—	2	2	no	—
141	D₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₆₇	—	8	2	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
154	C₄×D₆₇	—	2	1	no	—
155	C₄×D₆₇	—	2	1	no	—
156	C₄×D₆₇	—	2	1	no	—
157	C₄×D₆₇	—	2	1	no	—
158	C₄×D₆₇	—	4	1	no	—
159	Dic₁₃₄	—	2	2	no	—
160	Dic₁₃₄	—	2	2	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	4	1	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₈×C₆₇	#131	1	2	yes	—
166	C₈×C₆₇	#135	1	2	yes	—
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	2	no	—
172	C₆₇⋊C₈	#135	1	2	yes	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₆₇	—	2	1	no	—
178	C₂×D₆₇	—	2	1	no	—
179	C₂×D₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
183	C₄×C₆₇	#568	1	1	yes	characteristic
184	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
185	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
186	Dic₆₇	—	2	1	no	—
187	Dic₆₇	—	2	1	no	—
188	Dic₆₇	—	2	1	no	—
189	Dic₆₇	—	4	2	no	—
190	Dic₆₇	—	4	2	no	—
191	#18069	—	67	1	no	maximal, Sylow
192	C₂×C₆₇	—	8	2	no	—
193	C₂×C₆₇	#2935	1	1	yes	characteristic
194	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
195	C₂×C₆₇	#2957	1	1	yes	characteristic
196	D₆₇	—	4	1	no	—
197	#3168	—	67	1	no	—
198	#2969	—	67	2	no	—
199	#1591	—	67	1	no	—
200	#1589	D₆₇	1	1	yes	characteristic, complemented
201	#1583	—	67	2	no	—
202	#1582	—	67	2	no	—
203	C₈×Q₁₆	—	67	1	no	—
204	C₈×D₈	—	67	1	no	—
205	#1564	—	67	1	no	—
206	#1482	—	67	2	no	—
207	C₈×M₁₆	—	67	1	no	—
208	C₆₇	#18069	1	1	yes	characteristic, complemented, Sylow
209	#529	—	67	2	no	—
210	#499	—	67	1	no	—
211	#495	—	67	1	no	—
212	#488	—	67	2	no	—
213	#467	—	67	1	no	—
214	#466	—	67	1	no	—
215	#465	—	67	2	no	—
216	#446	—	67	2	no	—
217	C₈×Q₈	—	67	2	no	—
218	C₄×Q₁₆	—	67	1	no	—
219	C₄×SD₁₆	—	67	2	no	—
220	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
221	#436	—	134	2	no	—
222	#436	C₂×D₆₇	1	2	yes	—
223	#435	—	134	2	no	—
224	C₈×D₄	—	67	2	no	—
225	#433	—	134	2	no	—
226	#431	—	67	1	no	—
227	#431	—	67	2	no	—
228	#405	—	134	2	no	—
229	C₄×M₁₆	—	67	1	no	—
230	C₂×C₄×C₈	—	67	1	no	—
231	#335	—	67	1	no	—
232	#335	C₂×D₆₇	1	1	yes	characteristic
233	#334	—	67	2	no	—
234	Q₈⋊C₈	—	67	2	no	—
235	Q₈⋊C₈	—	67	2	no	—
236	D₄⋊C₈	—	67	2	no	—
237	D₄⋊C₈	C₂×D₆₇	1	2	yes	—
238	#322	—	67	1	no	—
239	#322	—	67	1	no	—
240	C₈²	—	67	1	no	—
241	C₈²	C₂×D₆₇	1	1	yes	characteristic
242	C₂×Q₁₆	—	67	1	no	—
243	C₂×SD₁₆	—	67	2	no	—
244	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	67	1	no	—
247	C₂×M₁₆	—	67	1	no	—
248	C₂²×C₈	—	67	1	no	—
249	C₂²×C₈	—	67	1	no	—
250	C₂²×C₈	—	134	2	no	—
251	#126	—	134	2	no	—
252	#125	—	67	2	no	—
253	#124	—	67	2	no	—
254	#123	—	134	2	no	—
255	C₂²⋊Q₈	—	67	2	no	—
256	#121	—	67	2	no	—
257	C₄×Q₈	—	67	2	no	—
258	C₄×D₄	—	134	2	no	—
259	C₄×D₄	C₂²×D₆₇	1	2	yes	—
260	#117	—	134	2	no	—
261	C₂×C₄²	—	67	1	no	—
262	#107	—	67	1	no	—
263	#107	C₄×D₆₇	1	1	yes	characteristic
264	#106	—	67	2	no	—
265	C₄⋊C₈	—	67	2	no	—
266	C₄⋊C₈	—	134	1	no	—
267	C₄⋊C₈	—	134	2	no	—
268	C₄⋊C₈	—	134	2	no	—
269	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
270	#103	—	67	2	no	—
271	#103	—	67	2	no	—
272	#102	—	67	2	no	—
273	#102	C₄×D₆₇	1	2	yes	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	134	1	no	—
276	C₂²⋊C₈	—	134	2	no	—
277	C₂²⋊C₈	—	134	2	no	—
278	#97	—	67	1	no	—
279	#97	—	67	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	—	67	1	no	—
286	C₄×C₈	—	134	2	no	—
287	C₄×C₈	—	134	2	no	—
288	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂×Q₈	—	67	2	no	—
292	C₂×D₄	—	134	2	no	—
293	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	67	1	no	—
296	C₂²×C₄	—	67	1	no	—
297	C₂²×C₄	—	67	1	no	—
298	Q₁₆	—	134	2	no	—
299	SD₁₆	—	268	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	4	4	no	—
302	M₁₆	—	134	2	no	—
303	M₁₆	—	134	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	268	1	no	—
307	C₂×C₈	—	268	4	no	—
308	C₂×C₈	—	67	1	no	—
309	C₂×C₈	—	67	1	no	—
310	C₂×C₈	—	67	1	no	—
311	C₂×C₈	—	67	1	no	—
312	C₂×C₈	—	134	2	no	—
313	C₂×C₈	—	134	1	no	—
314	C₂×C₈	—	134	2	no	—
315	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
317	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
318	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
319	C₄⋊C₄	—	67	2	no	—
320	C₄⋊C₄	—	134	2	no	—
321	C₄⋊C₄	—	134	2	no	—
322	C₄⋊C₄	—	134	2	no	—
323	C₄⋊C₄	—	134	2	no	—
324	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	134	2	no	—
327	C₂²⋊C₄	—	134	2	no	—
328	C₂²⋊C₄	—	134	2	no	—
329	C₂²⋊C₄	—	134	2	no	—
330	C₄²	—	67	1	no	—
331	C₄²	—	67	1	no	—
332	C₄²	—	67	1	no	—
333	C₄²	—	134	2	no	—
334	C₄²	C₂³×D₆₇	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	—	67	1	no	—
337	Q₈	—	134	2	no	—
338	Q₈	—	134	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	536	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	268	1	no	—
346	C₂×C₄	—	67	1	no	—
347	C₂×C₄	—	67	1	no	—
348	C₂×C₄	—	67	1	no	—
349	C₂×C₄	—	134	2	no	—
350	C₂×C₄	—	134	2	no	—
351	C₂×C₄	—	134	1	no	—
352	C₂×C₄	—	134	1	no	—
353	C₂×C₄	—	134	1	no	—
354	C₂×C₄	—	134	1	no	—
355	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
357	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	268	1	no	—
363	C₈	—	268	2	no	—
364	C₈	—	67	2	no	—
365	C₈	—	134	1	no	—
366	C₈	—	134	1	no	—
367	C₈	—	134	1	no	—
368	C₈	#14524051	1	2	yes	—
369	C₈	#14524062	1	2	yes	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	134	1	no	—
374	C₂²	—	134	1	no	—
375	C₂²	—	134	1	no	—
376	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	268	2	no	—
380	C₄	—	268	2	no	—
381	C₄	—	134	1	no	—
382	C₄	—	134	1	no	—
383	C₄	—	134	1	no	—
384	C₄	C₂×#14524051	1	1	yes	characteristic
385	C₄	#16864161	1	1	yes	characteristic
386	C₄	#135×D₆₇	1	1	yes	characteristic
387	C₄	C₂×#14524062	1	1	yes	characteristic
388	C₂	—	8	2	no	—
389	C₂	—	268	1	no	—
390	C₂	#22568954	1	1	yes	characteristic
391	C₂	C₄×#14524062	1	1	yes	characteristic
392	C₂	#22569121	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×#1589	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863629	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1589	D₆₇
hypocenter	C₆₇	#18069
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
2. #22569216	C₂	semidirect
3. #22559105	C₂	semidirect
4. #22559050	C₂	semidirect
5. #22559042	C₂	semidirect
6. C₈×Dic₂₆₈	C₂	semidirect
7. D₈×C₆₇⋊C₈	C₂	semidirect
8. #22558956	C₂	semidirect
9. #22558614	C₂	semidirect
10. C₈×#11761611	C₂	semidirect
11. C₆₇×#1589	C₂	semidirect
200. #1589	D₆₇	semidirect
208. C₆₇	#18069	semidirect

Maximal subgroups

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

subgroup	index
#22570777	2
#22569216	2
#22559105	2
#22559050	2
#22559042	2
C₈×Dic₂₆₈	2
D₈×C₆₇⋊C₈	2
#22558956	2
#22558614	2
C₈×#11761611	2
C₆₇×#1589	2
#18069	67