Subgroups — #30869336

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570532	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic
4	#22561186	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561037	C₂	1	1	yes	maximal, characteristic
6	#22560858	C₂	1	4	yes	maximal, complemented
7	#22560487	C₂	1	4	yes	maximal, complemented
8	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1744	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	#16863751	C₄	1	4	yes	complemented
14	#16863747	C₄	1	1	yes	characteristic, complemented
15	#16863747	C₄	1	1	yes	characteristic, complemented
16	#16863743	C₄	1	1	yes	characteristic, complemented
17	#16863743	C₄	1	1	yes	characteristic, complemented
18	#16863526	—	2	1	no	—
19	#16863526	—	2	1	no	—
20	C₈×Dic₁₃₄	—	2	1	no	—
21	C₈×Dic₁₃₄	—	2	1	no	—
22	#16863475	—	2	1	no	—
23	#16863475	—	2	1	no	—
24	C₈×D₂₆₈	—	2	1	no	—
25	C₈×D₂₆₈	—	2	1	no	—
26	C₂×#14523979	C₂²	1	4	yes	—
27	#16863422	C₂²	1	2	yes	—
28	#16863420	C₂²	1	1	yes	characteristic
29	#16863420	C₂²	1	1	yes	characteristic
30	C₂×#14523971	C₂²	1	4	yes	—
31	#16863393	C₂²	1	4	yes	—
32	#16863362	C₂²	1	2	yes	—
33	#16863351	C₂²	1	2	yes	—
34	#16863156	C₂²	1	4	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₄³×C₆₇	C₂²	1	1	yes	characteristic
38	#336×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	2	yes	—
40	C₂×#11761628	C₂³	1	2	yes	—
41	#14524049	—	4	4	no	—
42	C₂×#11761612	C₂³	1	2	yes	—
43	C₂×#11761612	C₂³	1	2	yes	—
44	C₂×#11761611	—	2	2	no	—
45	C₂×C₈×D₆₇	—	2	2	no	—
46	#14524043	C₂×C₄	1	1	yes	characteristic
47	#14524043	C₂×C₄	1	1	yes	characteristic
48	#14524043	C₂×C₄	1	1	yes	characteristic
49	#14524040	C₂×C₄	1	1	yes	characteristic
50	#14524035	C₂×C₄	1	2	yes	—
51	#14524031	C₂×C₄	1	2	yes	—
52	#14524022	—	2	1	no	—
53	#14524022	—	2	1	no	—
54	#14524017	—	2	1	no	—
55	#14524017	—	2	1	no	—
56	C₄×Dic₁₃₄	—	2	1	no	—
57	C₄×Dic₁₃₄	—	2	1	no	—
58	C₄×D₂₆₈	—	2	1	no	—
59	C₄×D₂₆₈	—	2	1	no	—
60	#14524002	C₂³	1	2	yes	—
61	#14523979	—	4	4	no	—
62	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
63	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
65	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	#14523971	—	4	4	no	—
69	#14523968	C₂×C₄	1	4	yes	—
70	#14523968	C₂×C₄	1	4	yes	—
71	#14523965	C₂×C₄	1	4	yes	—
72	#14523965	C₂×C₄	1	4	yes	—
73	(C₂×D₆₇)⋊C₈	—	2	2	no	—
74	(C₂×D₆₇)⋊C₈	—	2	2	no	—
75	#14523953	—	2	2	no	—
76	C₈×Dic₆₇	—	2	2	no	—
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	—	2	1	no	—
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
87	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
88	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
90	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
91	#11761628	—	2	4	no	—
92	#11761628	—	2	4	no	—
93	C₂×Dic₁₃₄	—	2	1	no	—
94	C₂×Dic₁₃₄	—	2	1	no	—
95	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
96	C₂×D₂₆₈	—	2	1	no	—
97	C₂×D₂₆₈	—	2	1	no	—
98	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
99	C₂×C₂²⋊D₆₇	—	2	2	no	—
100	C₂×C₄×D₆₇	—	2	2	no	—
101	#11761612	—	2	4	no	—
102	#11761612	—	2	4	no	—
103	#11761611	—	4	4	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₈×D₆₇	—	4	4	no	—
107	C₄⋊Dic₆₇	—	2	1	no	—
108	C₄⋊Dic₆₇	—	2	1	no	—
109	C₄⋊Dic₆₇	—	2	1	no	—
110	C₄⋊Dic₆₇	—	2	1	no	—
111	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
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	C₄×Dic₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₆₇	—	2	4	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	4	2	no	—
137	C₄²×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	C₂×D₄	1	1	yes	characteristic
141	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂²×D₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	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	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
155	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
156	C₂²⋊D₆₇	—	4	4	no	—
157	C₂×Dic₆₇	—	2	2	no	—
158	C₂×Dic₆₇	—	2	2	no	—
159	C₂×Dic₆₇	—	2	2	no	—
160	D₂₆₈	—	2	4	no	—
161	D₂₆₈	—	4	2	no	—
162	D₂₆₈	—	4	2	no	—
163	C₄×D₆₇	—	4	2	no	—
164	C₄×D₆₇	—	4	2	no	—
165	Dic₁₃₄	—	2	4	no	—
166	Dic₁₃₄	—	4	2	no	—
167	Dic₁₃₄	—	4	2	no	—
168	C₈×C₆₇	—	2	2	no	—
169	C₈×C₆₇	—	2	2	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	2	1	no	—
173	C₈×C₆₇	—	2	1	no	—
174	C₆₇⋊C₈	—	4	2	no	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂²×C₆₇	#542	1	2	yes	—
180	C₂×D₆₇	—	4	2	no	—
181	C₂×D₆₇	—	4	2	no	—
182	C₂×D₆₇	—	4	2	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	4	4	no	—
186	C₄×C₆₇	#443	1	2	yes	—
187	C₄×C₆₇	#444	1	2	yes	—
188	C₄×C₆₇	#520	1	2	yes	—
189	C₄×C₆₇	#535	1	1	yes	characteristic
190	C₄×C₆₇	#536	1	1	yes	characteristic
191	Dic₆₇	—	4	2	no	—
192	Dic₆₇	—	4	2	no	—
193	Dic₆₇	—	4	2	no	—
194	#22234	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	—	2	2	no	—
196	C₂×C₆₇	#2326	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
199	D₆₇	—	8	2	no	—
200	#3131	—	67	1	no	—
201	#2960	—	67	1	no	—
202	#2931	—	67	1	no	—
203	#1994	—	67	1	no	—
204	#1962	—	67	1	no	—
205	#1929	—	67	4	no	—
206	#1866	—	67	4	no	—
207	#1848	—	67	1	no	—
208	#1744	D₆₇	1	1	yes	characteristic, complemented
209	C₆₇	#22234	1	1	yes	characteristic, complemented, Sylow
210	C₂×#131	—	67	2	no	—
211	#533	—	67	1	no	—
212	#532	—	67	1	no	—
213	#488	—	67	4	no	—
214	#487	—	67	1	no	—
215	#487	—	67	1	no	—
216	#486	—	67	1	no	—
217	#486	—	67	1	no	—
218	#446	—	134	1	no	—
219	#446	—	134	1	no	—
220	C₈×Q₈	—	134	1	no	—
221	C₈×Q₈	—	134	1	no	—
222	#436	—	134	1	no	—
223	#436	—	134	1	no	—
224	C₈×D₄	—	134	1	no	—
225	C₈×D₄	—	134	1	no	—
226	C₂×#108	—	67	4	no	—
227	#424	—	67	2	no	—
228	#423	—	67	1	no	—
229	#423	—	67	1	no	—
230	C₂×#104	—	67	4	no	—
231	#416	—	67	4	no	—
232	#408	—	67	2	no	—
233	#405	—	67	2	no	—
234	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
236	C₄³	C₂×D₆₇	1	1	yes	characteristic
237	#339	—	67	4	no	—
238	#336	C₂×D₆₇	1	2	yes	—
239	#336	C₂×D₆₇	1	2	yes	—
240	C₂×#41	—	67	2	no	—
241	#131	—	268	4	no	—
242	C₂×M₁₆	—	67	2	no	—
243	C₂×M₁₆	—	67	2	no	—
244	C₂×M₁₆	—	134	2	no	—
245	C₂²×C₈	—	134	2	no	—
246	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
247	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
249	C₄⋊Q₈	—	67	1	no	—
250	C₄⋊Q₈	—	67	1	no	—
251	C₄⋊Q₈	—	67	1	no	—
252	#127	—	67	1	no	—
253	#125	—	67	2	no	—
254	#124	—	67	2	no	—
255	C₂²⋊Q₈	—	134	1	no	—
256	C₂²⋊Q₈	—	134	1	no	—
257	#121	—	134	1	no	—
258	#121	—	134	1	no	—
259	C₄×Q₈	—	134	1	no	—
260	C₄×Q₈	—	134	1	no	—
261	C₄×D₄	—	134	1	no	—
262	C₄×D₄	—	134	1	no	—
263	#117	—	67	2	no	—
264	C₂×C₄²	—	2	1	no	—
265	C₂×C₄²	—	2	1	no	—
266	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
267	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
268	#108	—	268	4	no	—
269	C₄⋊C₈	—	134	2	no	—
270	C₄⋊C₈	—	134	2	no	—
271	C₄⋊C₈	—	134	1	no	—
272	C₄⋊C₈	—	134	1	no	—
273	C₄⋊C₈	—	134	1	no	—
274	C₄⋊C₈	—	134	1	no	—
275	#104	—	268	4	no	—
276	#103	—	67	4	no	—
277	#103	—	67	4	no	—
278	#102	—	67	4	no	—
279	#102	—	67	4	no	—
280	C₂²⋊C₈	—	134	2	no	—
281	C₂²⋊C₈	—	134	2	no	—
282	#97	—	134	2	no	—
283	C₄×C₈	—	134	2	no	—
284	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
285	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
286	C₄×C₈	C₄×D₆₇	1	2	yes	—
287	C₄×C₈	D₂₆₈	1	1	yes	characteristic
288	C₄×C₈	D₂₆₈	1	1	yes	characteristic
289	C₄×C₈	D₂₆₈	1	1	yes	characteristic
290	C₄×C₈	D₂₆₈	1	1	yes	characteristic
291	#41	—	134	4	no	—
292	#41	—	134	4	no	—
293	C₂×Q₈	—	67	2	no	—
294	C₂×Q₈	—	134	1	no	—
295	C₂×Q₈	—	134	1	no	—
296	C₂×D₄	—	67	2	no	—
297	C₂×D₄	—	134	2	no	—
298	C₂×D₄	—	134	1	no	—
299	C₂×D₄	—	134	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	134	2	no	—
302	C₂²×C₄	#11761628	1	1	yes	characteristic
303	C₂²×C₄	#11761628	1	1	yes	characteristic
304	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
305	M₁₆	—	268	4	no	—
306	M₁₆	—	134	4	no	—
307	M₁₆	—	134	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	268	4	no	—
313	C₂×C₈	—	134	2	no	—
314	C₂×C₈	—	134	2	no	—
315	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
316	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
317	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
318	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
319	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
320	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
321	C₄⋊C₄	—	67	2	no	—
322	C₄⋊C₄	—	67	2	no	—
323	C₄⋊C₄	—	134	1	no	—
324	C₄⋊C₄	—	134	1	no	—
325	C₄⋊C₄	—	134	2	no	—
326	C₄⋊C₄	—	134	2	no	—
327	C₄⋊C₄	—	134	1	no	—
328	C₄⋊C₄	—	134	1	no	—
329	C₂²⋊C₄	—	134	2	no	—
330	C₂²⋊C₄	—	134	2	no	—
331	C₂²⋊C₄	—	134	2	no	—
332	C₄²	—	2	4	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	4	2	no	—
336	C₄²	—	134	2	no	—
337	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
338	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
339	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
340	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
341	C₂³	—	134	2	no	—
342	C₂³	C₂×#11761628	1	1	yes	characteristic
343	Q₈	—	268	2	no	—
344	Q₈	—	268	2	no	—
345	Q₈	—	134	4	no	—
346	D₄	—	268	4	no	—
347	D₄	—	268	2	no	—
348	D₄	—	268	2	no	—
349	D₄	—	134	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	268	2	no	—
356	C₂×C₄	—	268	2	no	—
357	C₂×C₄	—	134	2	no	—
358	C₂×C₄	—	134	2	no	—
359	C₂×C₄	—	134	2	no	—
360	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
361	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
362	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
363	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
366	C₂×C₄	#14524089	1	2	yes	—
367	C₂×C₄	#14524090	1	2	yes	—
368	C₈	—	2	2	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	268	2	no	—
375	C₈	—	268	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	268	2	no	—
379	C₂²	—	268	2	no	—
380	C₂²	—	268	2	no	—
381	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
382	C₂²	#16864017	1	2	yes	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	4	no	—
386	C₄	—	268	2	no	—
387	C₄	—	268	2	no	—
388	C₄	—	268	2	no	—
389	C₄	#16863514	1	2	yes	—
390	C₄	#16863519	1	2	yes	—
391	C₄	#16863876	1	2	yes	—
392	C₄	#16863965	1	1	yes	characteristic
393	C₄	#16863972	1	1	yes	characteristic
394	C₂	—	2	2	no	—
395	C₂	—	536	2	no	—
396	C₂	#22562920	1	1	yes	characteristic
397	C₂	C₂×#16863514	1	1	yes	characteristic
398	C₂	C₂×#16863519	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×#1744	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#22234
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
2. #22569143	C₂	semidirect
4. #22561186	C₂	semidirect
6. #22560858	C₂	semidirect
7. #22560487	C₂	semidirect
8. #22560381	C₂	semidirect
9. C₆₇×#1744	C₂	semidirect
13. #16863751	C₄	semidirect
14. #16863747	C₄	semidirect
15. #16863747	C₄	semidirect
16. #16863743	C₄	semidirect
17. #16863743	C₄	semidirect
208. #1744	D₆₇	semidirect
209. C₆₇	#22234	semidirect

Maximal subgroups

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

subgroup	index
#22570532	2
#22569143	2
#22568922	2
#22561186	2
#22561037	2
#22560858	2
#22560487	2
#22560381	2
C₆₇×#1744	2
#22234	67