Subgroups — #30869345

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570555	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic
4	#22561190	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561041	C₂	1	1	yes	maximal, characteristic
6	#22560858	C₂	1	4	yes	maximal, complemented
7	#22560487	C₂	1	4	yes	maximal, complemented
8	#22560384	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524049	C₂²	1	2	yes	—
11	#16863953	C₂²	1	1	yes	characteristic
12	#16863946	C₂²	1	1	yes	characteristic
13	#16863765	C₄	1	2	yes	complemented
14	#16863761	C₄	1	2	yes	complemented
15	#16863756	C₄	1	2	yes	complemented
16	#16863756	C₄	1	2	yes	complemented
17	#16863526	—	2	1	no	—
18	#16863526	—	2	1	no	—
19	C₈×Dic₁₃₄	—	2	1	no	—
20	C₈×Dic₁₃₄	—	2	1	no	—
21	#16863475	—	2	1	no	—
22	#16863475	—	2	1	no	—
23	C₈×D₂₆₈	—	2	1	no	—
24	C₈×D₂₆₈	—	2	1	no	—
25	C₂×#14523979	C₂²	1	4	yes	—
26	#16863422	C₂²	1	2	yes	—
27	#16863420	C₂²	1	1	yes	characteristic
28	#16863420	C₂²	1	1	yes	characteristic
29	C₂×#14523971	C₂²	1	4	yes	—
30	#16863393	C₂²	1	4	yes	—
31	#16863362	C₂²	1	2	yes	—
32	#16863351	C₂²	1	2	yes	—
33	#16863156	C₂²	1	4	yes	—
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	#376×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	C₂×#11761628	C₂³	1	2	yes	—
40	#14524049	—	4	4	no	—
41	C₂×#11761612	C₂³	1	2	yes	—
42	C₂×#11761612	C₂³	1	2	yes	—
43	C₂×#11761611	—	2	2	no	—
44	C₂×C₈×D₆₇	—	2	2	no	—
45	#14524043	C₂×C₄	1	1	yes	characteristic
46	#14524043	C₂×C₄	1	1	yes	characteristic
47	#14524043	C₂×C₄	1	1	yes	characteristic
48	#14524040	C₂×C₄	1	1	yes	characteristic
49	#14524035	C₂×C₄	1	2	yes	—
50	#14524031	C₂×C₄	1	2	yes	—
51	#14524022	—	2	1	no	—
52	#14524022	—	2	1	no	—
53	#14524017	—	2	1	no	—
54	#14524017	—	2	1	no	—
55	C₄×Dic₁₃₄	—	2	1	no	—
56	C₄×Dic₁₃₄	—	2	1	no	—
57	C₄×D₂₆₈	—	2	1	no	—
58	C₄×D₂₆₈	—	2	1	no	—
59	#14524002	C₂³	1	2	yes	—
60	#14523979	—	4	4	no	—
61	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
62	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
63	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
65	Dic₆₇⋊C₈	—	2	2	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	#14523971	—	4	4	no	—
68	#14523968	C₂×C₄	1	4	yes	—
69	#14523968	C₂×C₄	1	4	yes	—
70	#14523965	C₂×C₄	1	4	yes	—
71	#14523965	C₂×C₄	1	4	yes	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	—	2	2	no	—
74	#14523953	—	2	2	no	—
75	C₈×Dic₆₇	—	2	2	no	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	#97×C₆₇	C₂×C₄	1	2	yes	—
82	#97×C₆₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
86	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	1	no	—
89	#11761628	—	2	4	no	—
90	#11761628	—	2	4	no	—
91	C₂×Dic₁₃₄	—	2	1	no	—
92	C₂×Dic₁₃₄	—	2	1	no	—
93	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
94	C₂×D₂₆₈	—	2	1	no	—
95	C₂×D₂₆₈	—	2	1	no	—
96	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
97	C₂×C₂²⋊D₆₇	—	2	2	no	—
98	C₂×C₄×D₆₇	—	2	2	no	—
99	#11761612	—	2	4	no	—
100	#11761612	—	2	4	no	—
101	#11761611	—	4	4	no	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₈×D₆₇	—	4	4	no	—
105	C₄⋊Dic₆₇	—	2	1	no	—
106	C₄⋊Dic₆₇	—	2	1	no	—
107	C₄⋊Dic₆₇	—	2	1	no	—
108	C₄⋊Dic₆₇	—	2	1	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	C₂²⋊Dic₆₇	—	2	2	no	—
114	(C₂×D₆₇)⋊C₄	—	2	2	no	—
115	(C₂×D₆₇)⋊C₄	—	2	2	no	—
116	C₄×Dic₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×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	C₂×D₄	1	1	yes	characteristic
136	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₂²×D₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
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	H(5,𝔽₂)	1	2	yes	—
150	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
151	C₂²⋊D₆₇	—	4	4	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	2	2	no	—
154	C₂×Dic₆₇	—	2	2	no	—
155	D₂₆₈	—	2	4	no	—
156	D₂₆₈	—	4	2	no	—
157	D₂₆₈	—	4	2	no	—
158	C₄×D₆₇	—	4	2	no	—
159	C₄×D₆₇	—	4	2	no	—
160	Dic₁₃₄	—	2	4	no	—
161	Dic₁₃₄	—	4	2	no	—
162	Dic₁₃₄	—	4	2	no	—
163	C₈×C₆₇	—	2	4	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	—	2	1	no	—
168	C₆₇⋊C₈	—	4	2	no	—
169	C₆₇⋊C₈	—	4	2	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₆₇	#542	1	2	yes	—
174	C₂×D₆₇	—	4	2	no	—
175	C₂×D₆₇	—	4	2	no	—
176	C₂×D₆₇	—	4	2	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	8	2	no	—
182	C₄×C₆₇	#520	1	2	yes	—
183	C₄×C₆₇	#535	1	1	yes	characteristic
184	C₄×C₆₇	#536	1	1	yes	characteristic
185	Dic₆₇	—	4	2	no	—
186	Dic₆₇	—	4	2	no	—
187	Dic₆₇	—	4	2	no	—
188	#22235	—	67	1	no	maximal, Sylow
189	C₂×C₆₇	—	2	2	no	—
190	C₂×C₆₇	#2326	1	1	yes	characteristic
191	C₂×C₆₇	#2955	1	1	yes	characteristic
192	C₂×C₆₇	#2955	1	1	yes	characteristic
193	D₆₇	—	8	2	no	—
194	#3135	—	67	1	no	—
195	#2960	—	67	1	no	—
196	#2931	—	67	1	no	—
197	#1995	—	67	1	no	—
198	#1963	—	67	1	no	—
199	#1929	—	67	4	no	—
200	#1866	—	67	4	no	—
201	#1849	—	67	1	no	—
202	#1746	D₆₇	1	1	yes	characteristic, complemented
203	C₆₇	#22235	1	1	yes	characteristic, complemented, Sylow
204	C₂×#131	—	67	2	no	—
205	#533	—	67	1	no	—
206	#532	—	67	1	no	—
207	#491	—	67	2	no	—
208	#490	—	67	2	no	—
209	#489	—	67	2	no	—
210	#489	—	67	2	no	—
211	#446	—	134	1	no	—
212	#446	—	134	1	no	—
213	C₈×Q₈	—	134	1	no	—
214	C₈×Q₈	—	134	1	no	—
215	#436	—	134	1	no	—
216	#436	—	134	1	no	—
217	C₈×D₄	—	134	1	no	—
218	C₈×D₄	—	134	1	no	—
219	C₂×#108	—	67	4	no	—
220	#424	—	67	2	no	—
221	#423	—	67	1	no	—
222	#423	—	67	1	no	—
223	C₂×#104	—	67	4	no	—
224	#416	—	67	4	no	—
225	#408	—	67	2	no	—
226	#405	—	67	2	no	—
227	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
229	#376	C₂×D₆₇	1	1	yes	characteristic
230	#339	—	67	4	no	—
231	#336	C₂×D₆₇	1	2	yes	—
232	#336	C₂×D₆₇	1	2	yes	—
233	C₂×#41	—	67	2	no	—
234	#131	—	268	4	no	—
235	C₂×M₁₆	—	67	2	no	—
236	C₂×M₁₆	—	67	2	no	—
237	C₂×M₁₆	—	134	2	no	—
238	C₂²×C₈	—	134	2	no	—
239	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
240	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
242	C₄⋊Q₈	—	67	1	no	—
243	C₄⋊Q₈	—	67	1	no	—
244	C₄⋊Q₈	—	67	1	no	—
245	#127	—	67	1	no	—
246	#125	—	67	2	no	—
247	#124	—	67	2	no	—
248	C₂²⋊Q₈	—	134	1	no	—
249	C₂²⋊Q₈	—	134	1	no	—
250	#121	—	134	1	no	—
251	#121	—	134	1	no	—
252	C₄×Q₈	—	134	1	no	—
253	C₄×Q₈	—	134	1	no	—
254	C₄×D₄	—	134	1	no	—
255	C₄×D₄	—	134	1	no	—
256	#117	—	67	2	no	—
257	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
258	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
259	#108	—	268	4	no	—
260	C₄⋊C₈	—	134	2	no	—
261	C₄⋊C₈	—	134	2	no	—
262	C₄⋊C₈	—	134	1	no	—
263	C₄⋊C₈	—	134	1	no	—
264	C₄⋊C₈	—	134	1	no	—
265	C₄⋊C₈	—	134	1	no	—
266	#104	—	268	4	no	—
267	#103	—	67	4	no	—
268	#103	—	67	4	no	—
269	#102	—	67	4	no	—
270	#102	—	67	4	no	—
271	C₂²⋊C₈	—	134	2	no	—
272	C₂²⋊C₈	—	134	2	no	—
273	#97	—	134	2	no	—
274	#97	C₄×D₆₇	1	2	yes	—
275	#97	C₄×D₆₇	1	2	yes	—
276	C₄×C₈	—	134	2	no	—
277	C₄×C₈	D₂₆₈	1	1	yes	characteristic
278	C₄×C₈	D₂₆₈	1	1	yes	characteristic
279	C₄×C₈	D₂₆₈	1	1	yes	characteristic
280	C₄×C₈	D₂₆₈	1	1	yes	characteristic
281	#95	—	2	1	no	—
282	#95	—	2	1	no	—
283	#41	—	134	4	no	—
284	#41	—	134	4	no	—
285	C₂×Q₈	—	67	2	no	—
286	C₂×Q₈	—	134	1	no	—
287	C₂×Q₈	—	134	1	no	—
288	C₂×D₄	—	67	2	no	—
289	C₂×D₄	—	134	2	no	—
290	C₂×D₄	—	134	1	no	—
291	C₂×D₄	—	134	1	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	134	2	no	—
294	C₂²×C₄	#11761628	1	1	yes	characteristic
295	C₂²×C₄	#11761628	1	1	yes	characteristic
296	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
297	M₁₆	—	268	4	no	—
298	M₁₆	—	134	4	no	—
299	M₁₆	—	134	4	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	268	4	no	—
305	C₂×C₈	—	134	2	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
308	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
309	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
310	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
311	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
312	C₄⋊C₄	—	67	2	no	—
313	C₄⋊C₄	—	67	2	no	—
314	C₄⋊C₄	—	134	1	no	—
315	C₄⋊C₄	—	134	1	no	—
316	C₄⋊C₄	—	134	2	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	—	134	1	no	—
319	C₄⋊C₄	—	134	1	no	—
320	C₂²⋊C₄	—	134	2	no	—
321	C₂²⋊C₄	—	134	2	no	—
322	C₂²⋊C₄	—	134	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	134	2	no	—
326	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
328	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
329	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
330	C₂³	—	134	2	no	—
331	C₂³	C₂×#11761628	1	1	yes	characteristic
332	Q₈	—	268	2	no	—
333	Q₈	—	268	2	no	—
334	Q₈	—	134	4	no	—
335	D₄	—	268	2	no	—
336	D₄	—	268	2	no	—
337	D₄	—	268	4	no	—
338	D₄	—	134	4	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	268	2	no	—
345	C₂×C₄	—	268	2	no	—
346	C₂×C₄	—	134	2	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	2	no	—
349	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
350	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
351	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
352	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
355	C₂×C₄	#14524089	1	2	yes	—
356	C₂×C₄	#14524090	1	2	yes	—
357	C₈	—	2	4	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	268	2	no	—
363	C₈	—	268	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	268	2	no	—
367	C₂²	—	268	2	no	—
368	C₂²	—	268	2	no	—
369	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
370	C₂²	#16864017	1	2	yes	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	268	2	no	—
377	C₄	—	268	2	no	—
378	C₄	—	268	2	no	—
379	C₄	#16863876	1	2	yes	—
380	C₄	#16863965	1	1	yes	characteristic
381	C₄	#16863972	1	1	yes	characteristic
382	C₂	—	2	2	no	—
383	C₂	—	536	2	no	—
384	C₂	#22562920	1	1	yes	characteristic
385	C₂	#22569113	1	1	yes	characteristic
386	C₂	#22569113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1746	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1746	D₆₇
hypocenter	C₆₇	#22235
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1746

Complements

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

N	H ≅ G/N	type
2. #22569143	C₂	semidirect
4. #22561190	C₂	semidirect
6. #22560858	C₂	semidirect
7. #22560487	C₂	semidirect
8. #22560384	C₂	semidirect
9. C₆₇×#1746	C₂	semidirect
13. #16863765	C₄	semidirect
14. #16863761	C₄	semidirect
15. #16863756	C₄	semidirect
16. #16863756	C₄	semidirect
202. #1746	D₆₇	semidirect
203. C₆₇	#22235	semidirect

Maximal subgroups

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

subgroup	index
#22570555	2
#22569143	2
#22568922	2
#22561190	2
#22561041	2
#22560858	2
#22560487	2
#22560384	2
C₆₇×#1746	2
#22235	67