Subgroups — #30874360

Order 17152.

808 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569127	C₂	1	1	yes	maximal, characteristic
2	#22569125	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863517	C₂	1	1	yes	maximal, characteristic
4	#22568819	C₂	1	2	yes	maximal, complemented
5	#22561000	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560452	C₂	1	2	yes	maximal, complemented
7	#22560446	C₂	1	2	yes	maximal, complemented
8	#22560382	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560262	C₂	1	2	yes	maximal, complemented
10	#22560094	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	#16864162	C₂²	1	2	yes	—
14	#16864162	C₂²	1	2	yes	—
15	C₂×#14524051	C₂²	1	2	yes	—
16	#16863519	—	2	2	no	—
17	#16863517	—	2	2	no	—
18	#16863517	C₄	1	4	yes	complemented
19	#16863517	C₄	1	4	yes	complemented
20	#16863514	—	2	2	no	—
21	#16863512	—	2	2	no	—
22	#16863445	C₂²	1	2	yes	—
23	#16863443	—	2	4	no	—
24	C₂×#14523979	C₂²	1	1	yes	characteristic
25	C₂×#14523978	C₂²	1	2	yes	—
26	C₂×#14523978	C₂²	1	1	yes	characteristic
27	#108×D₆₇	—	2	4	no	—
28	#16863432	C₂²	1	1	yes	characteristic
29	#16863419	C₂²	1	2	yes	—
30	#16863414	C₂²	1	2	yes	—
31	#16863412	C₂²	1	2	yes	—
32	C₂×#14523969	C₂²	1	2	yes	—
33	#16863351	C₂²	1	1	yes	characteristic
34	C₂×#14523953	C₂²	1	1	yes	characteristic
35	#16863168	C₂²	1	2	yes	—
36	#16863157	C₂²	1	1	yes	characteristic
37	#16863155	C₂²	1	1	yes	characteristic
38	#430×C₆₇	C₂²	1	2	yes	—
39	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
41	#405×C₆₇	C₂²	1	1	yes	characteristic
42	#344×C₆₇	C₂²	1	2	yes	—
43	C₂×#11761628	C₂³	1	2	yes	—
44	#14524065	C₂×C₄	1	2	yes	—
45	#14524065	C₂×C₄	1	2	yes	—
46	#14524065	D₄	1	2	yes	—
47	#14524065	D₄	1	2	yes	—
48	C₂×Dic₂₆₈	Q₈	1	1	yes	characteristic
49	C₂×#11761617	Q₈	1	2	yes	—
50	C₂×D₅₃₆	Q₈	1	1	yes	characteristic
51	#14524051	—	2	2	no	—
52	#14524051	—	2	2	no	—
53	#14524051	—	2	4	no	—
54	#14524051	—	4	2	no	—
55	#14524051	C₂×C₄	1	4	yes	—
56	#14524051	C₂×C₄	1	4	yes	—
57	#14524049	—	2	4	no	—
58	#14524049	—	4	2	no	—
59	C₂×#11761612	C₂³	1	2	yes	—
60	C₂×#11761612	C₂³	1	1	yes	characteristic
61	C₂×#11761611	—	2	2	no	—
62	M₁₆×D₆₇	—	2	4	no	—
63	M₁₆×D₆₇	—	2	4	no	—
64	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₆₇	—	2	2	no	—
66	#14524002	C₂³	1	1	yes	characteristic
67	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
68	#14523979	—	2	4	no	—
69	#14523979	—	2	1	no	—
70	#14523979	—	2	1	no	—
71	#14523978	—	2	2	no	—
72	#14523978	—	2	2	no	—
73	#14523978	D₄	1	4	yes	—
74	#14523976	—	2	1	no	—
75	#14523974	—	2	1	no	—
76	Dic₆₇⋊C₈	—	4	2	no	—
77	#14523971	—	2	4	no	—
78	#14523969	—	2	4	no	—
79	#14523969	C₂×C₄	1	8	yes	—
80	#14523953	—	2	1	no	—
81	#14523953	C₂×C₄	1	2	yes	—
82	#14523953	C₂×C₄	1	2	yes	—
83	C₈×Dic₆₇	—	2	1	no	—
84	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
85	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
86	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	#117×C₆₇	C₂³	1	1	yes	characteristic
89	#108×C₆₇	—	2	4	no	—
90	#108×C₆₇	—	2	1	no	—
91	#108×C₆₇	—	2	1	no	—
92	#108×C₆₇	C₂×C₄	1	2	yes	—
93	#108×C₆₇	C₂×C₄	1	2	yes	—
94	#97×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	1	no	—
96	#11761628	—	2	2	no	—
97	#11761628	—	2	2	no	—
98	#11761628	C₂²×C₄	1	4	yes	—
99	#11761628	C₂×D₄	1	4	yes	—
100	C₂×Dic₁₃₄	C₂×Q₈	1	2	yes	—
101	C₂×D₂₆₈	C₂×Q₈	1	2	yes	—
102	C₂×C₂²⋊D₆₇	—	2	2	no	—
103	C₂²×Dic₆₇	—	2	1	no	—
104	C₂×C₄×D₆₇	—	2	2	no	—
105	Dic₂₆₈	C₄⋊C₄	1	2	yes	—
106	Dic₂₆₈	C₄⋊C₄	1	2	yes	—
107	#11761617	C₄⋊C₄	1	4	yes	—
108	#11761617	C₄⋊C₄	1	4	yes	—
109	D₅₃₆	C₄⋊C₄	1	2	yes	—
110	D₅₃₆	C₄⋊C₄	1	2	yes	—
111	#11761612	—	2	1	no	—
112	#11761612	—	2	2	no	—
113	#11761612	—	2	2	no	—
114	#11761612	C₂×D₄	1	2	yes	—
115	#11761611	—	2	4	no	—
116	#11761611	—	2	4	no	—
117	#11761611	—	4	4	no	—
118	#11761611	—	4	4	no	—
119	C₂×C₆₇⋊C₈	—	2	1	no	—
120	C₂×C₆₇⋊C₈	—	2	2	no	—
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₆₇⋊C₈	#41	1	1	yes	characteristic
124	C₂×C₆₇⋊C₈	#41	1	1	yes	characteristic
125	C₈×D₆₇	—	2	4	no	—
126	C₈×D₆₇	—	2	4	no	—
127	C₈×D₆₇	—	4	4	no	—
128	C₈×D₆₇	—	4	4	no	—
129	C₄⋊Dic₆₇	—	2	1	no	—
130	C₂²⋊Dic₆₇	—	2	1	no	—
131	C₄×Dic₆₇	—	2	1	no	—
132	C₄×Dic₆₇	—	2	1	no	—
133	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
135	M₁₆×C₆₇	—	2	2	no	—
136	M₁₆×C₆₇	—	2	1	no	—
137	M₁₆×C₆₇	—	2	2	no	—
138	M₁₆×C₆₇	—	2	2	no	—
139	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
140	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂²×D₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²⋊D₆₇	—	2	4	no	—
163	C₂²⋊D₆₇	—	2	4	no	—
164	C₂×Dic₆₇	—	2	2	no	—
165	C₂×Dic₆₇	—	2	2	no	—
166	C₂×Dic₆₇	—	4	1	no	—
167	C₂×Dic₆₇	—	8	1	no	—
168	D₂₆₈	—	2	2	no	—
169	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
170	C₄×D₆₇	—	2	4	no	—
171	C₄×D₆₇	—	2	4	no	—
172	Dic₁₃₄	—	2	2	no	—
173	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
174	C₈×C₆₇	—	2	4	no	—
175	C₈×C₆₇	—	2	2	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₆₇⋊C₈	—	2	2	no	—
182	C₆₇⋊C₈	—	4	2	no	—
183	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
184	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	#520	1	1	yes	characteristic
187	C₂²×C₆₇	#547	1	1	yes	characteristic
188	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
189	C₂×D₆₇	—	2	4	no	—
190	C₂×D₆₇	—	8	2	no	—
191	C₄×C₆₇	—	8	1	no	—
192	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₄×C₆₇	D₄²	1	1	yes	characteristic
195	C₄×C₆₇	#554	1	1	yes	characteristic
196	Dic₆₇	—	2	4	no	—
197	Dic₆₇	—	4	2	no	—
198	Dic₆₇	—	8	1	no	—
199	#22695	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	—	2	1	no	—
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
204	D₆₇	—	4	4	no	—
205	#2957	—	67	1	no	—
206	#2957	—	67	1	no	—
207	C₂×#444	—	67	1	no	—
208	#2916	—	67	2	no	—
209	#1953	—	67	1	no	—
210	#1953	D₆₇	1	1	yes	characteristic, complemented
211	#1861	—	67	2	no	—
212	#1861	—	67	2	no	—
213	#1849	—	67	1	no	—
214	#1814	—	67	2	no	—
215	#1773	—	67	1	no	—
216	C₆₇	#22695	1	1	yes	characteristic, complemented, Sylow
217	C₂×#135	—	67	1	no	—
218	#568	—	67	2	no	—
219	#568	—	67	2	no	—
220	C₂×#131	—	67	2	no	—
221	#444	—	67	4	no	—
222	#444	—	67	4	no	—
223	#444	—	134	2	no	—
224	#444	—	134	2	no	—
225	#443	—	134	2	no	—
226	#443	—	134	2	no	—
227	#430	—	67	2	no	—
228	#430	—	134	4	no	—
229	#430	C₂×D₆₇	1	2	yes	—
230	C₂×#108	—	67	1	no	—
231	C₂×#108	—	67	2	no	—
232	C₂×#108	—	67	1	no	—
233	C₂×#108	—	134	4	no	—
234	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
235	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
236	#427	—	67	1	no	—
237	#423	—	67	2	no	—
238	#421	—	67	2	no	—
239	#421	—	67	2	no	—
240	C₂×#104	—	67	2	no	—
241	#405	—	67	1	no	—
242	#405	C₂×D₆₇	1	1	yes	characteristic
243	C₂×#97	—	67	1	no	—
244	#344	—	67	2	no	—
245	#344	C₂×D₆₇	1	2	yes	—
246	#339	—	67	1	no	—
247	#339	—	67	1	no	—
248	C₂×#41	—	67	2	no	—
249	#135	—	67	2	no	—
250	#135	—	67	2	no	—
251	#135	—	67	2	no	—
252	#135	—	67	2	no	—
253	C₂×Q₁₆	—	67	1	no	—
254	C₂×SD₁₆	—	67	2	no	—
255	C₂×D₈	—	67	1	no	—
256	#131	—	268	2	no	—
257	#131	—	268	2	no	—
258	#131	—	67	4	no	—
259	#131	—	67	4	no	—
260	#131	—	134	4	no	—
261	#131	—	134	4	no	—
262	#131	—	134	2	no	—
263	#131	—	134	2	no	—
264	C₂×M₁₆	—	67	1	no	—
265	C₂×M₁₆	—	67	2	no	—
266	C₂×M₁₆	—	134	2	no	—
267	C₂×M₁₆	—	134	4	no	—
268	C₂×M₁₆	—	134	4	no	—
269	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
270	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
271	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
272	C₂²×C₈	—	67	1	no	—
273	C₂²×C₈	—	134	2	no	—
274	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
275	#117	—	67	1	no	—
276	#117	C₂²×D₆₇	1	1	yes	characteristic
277	C₂×C₄²	—	67	1	no	—
278	#108	—	2	4	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	67	4	no	—
282	#108	—	134	4	no	—
283	#108	—	134	2	no	—
284	#108	—	134	2	no	—
285	#108	—	134	1	no	—
286	#108	—	134	1	no	—
287	#108	C₄×D₆₇	1	2	yes	—
288	#108	C₄×D₆₇	1	2	yes	—
289	#107	—	134	1	no	—
290	#106	—	134	1	no	—
291	C₄⋊C₈	—	268	2	no	—
292	#104	—	67	8	no	—
293	#104	—	134	4	no	—
294	#104	—	134	4	no	—
295	#97	—	2	1	no	—
296	#97	—	67	2	no	—
297	#97	—	67	2	no	—
298	#97	—	134	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	134	1	no	—
301	#41	—	67	4	no	—
302	#41	—	67	4	no	—
303	#41	—	134	2	no	—
304	#41	—	134	2	no	—
305	C₂×Q₈	—	67	2	no	—
306	C₂×D₄	—	67	2	no	—
307	C₂×D₄	—	134	2	no	—
308	C₂²×C₄	—	134	1	no	—
309	C₂²×C₄	—	134	2	no	—
310	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
311	Q₁₆	—	67	2	no	—
312	Q₁₆	—	67	2	no	—
313	SD₁₆	—	67	4	no	—
314	SD₁₆	—	67	4	no	—
315	D₈	—	67	2	no	—
316	D₈	—	67	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	1	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	268	4	no	—
322	M₁₆	—	268	4	no	—
323	M₁₆	—	67	2	no	—
324	M₁₆	—	134	4	no	—
325	M₁₆	—	134	4	no	—
326	M₁₆	—	134	2	no	—
327	M₁₆	—	134	2	no	—
328	M₁₆	—	134	1	no	—
329	M₁₆	C₂×C₄×D₆₇	1	4	yes	—
330	M₁₆	D₄×D₆₇	1	2	yes	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	268	4	no	—
335	C₂×C₈	—	268	4	no	—
336	C₂×C₈	—	67	2	no	—
337	C₂×C₈	—	67	2	no	—
338	C₂×C₈	—	67	1	no	—
339	C₂×C₈	—	67	1	no	—
340	C₂×C₈	—	134	4	no	—
341	C₂×C₈	—	134	4	no	—
342	C₂×C₈	—	134	2	no	—
343	C₂×C₈	—	134	1	no	—
344	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
346	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
347	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
348	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
349	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	134	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	134	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	67	2	no	—
356	C₄²	—	134	1	no	—
357	C₄²	—	134	1	no	—
358	C₂³	—	134	2	no	—
359	C₂³	#14524091	1	1	yes	characteristic
360	Q₈	—	67	4	no	—
361	Q₈	—	134	2	no	—
362	D₄	—	67	4	no	—
363	D₄	—	134	4	no	—
364	D₄	—	134	4	no	—
365	D₄	—	134	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	268	1	no	—
368	C₂×C₄	—	536	1	no	—
369	C₂×C₄	—	134	2	no	—
370	C₂×C₄	—	134	2	no	—
371	C₂×C₄	—	134	4	no	—
372	C₂×C₄	—	134	4	no	—
373	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
376	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
377	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
378	C₂×C₄	#14524089	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	4	no	—
381	C₈	—	4	2	no	—
382	C₈	—	268	2	no	—
383	C₈	—	67	2	no	—
384	C₈	—	67	2	no	—
385	C₈	—	134	2	no	—
386	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
387	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
388	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
389	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
390	C₂²	—	4	1	no	—
391	C₂²	—	536	2	no	—
392	C₂²	—	134	4	no	—
393	C₂²	#16863878	1	1	yes	characteristic
394	C₂²	#16864048	1	1	yes	characteristic
395	C₂²	#16864061	1	1	yes	characteristic
396	C₄	—	8	1	no	—
397	C₄	—	268	2	no	—
398	C₄	—	536	1	no	—
399	C₄	—	134	4	no	—
400	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
401	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
402	C₄	#16864030	1	1	yes	characteristic
403	C₄	#16864102	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	268	4	no	—
408	C₂	#22563110	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16864030
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1953	C₂
socle	C₂×C₆₇	D₄×C₄⋊C₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1953	D₆₇
hypocenter	C₆₇	#22695
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
4. #22568819	C₂	semidirect
5. #22561000	C₂	semidirect
6. #22560452	C₂	semidirect
7. #22560446	C₂	semidirect
8. #22560382	C₂	semidirect
9. #22560262	C₂	semidirect
10. #22560094	C₂	semidirect
11. C₆₇×#1953	C₂	semidirect
18. #16863517	C₄	semidirect
19. #16863517	C₄	semidirect
210. #1953	D₆₇	semidirect
216. C₆₇	#22695	semidirect

Maximal subgroups

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

subgroup	index
#22569127	2
#22569125	2
C₂×#16863517	2
#22568819	2
#22561000	2
#22560452	2
#22560446	2
#22560382	2
#22560262	2
#22560094	2
C₆₇×#1953	2
#22695	67