Subgroups — #30860261

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569011	C₂	1	1	yes	maximal, characteristic
2	#22568861	C₂	1	2	yes	maximal, complemented
3	#22561100	C₂	1	1	yes	maximal, characteristic
4	#22561095	C₂	1	1	yes	maximal, characteristic
5	#22560960	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560955	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560918	C₂	1	2	yes	maximal, complemented
8	#22560911	C₂	1	2	yes	maximal, complemented
9	C₄×#14523969	C₂	1	2	yes	maximal, complemented
10	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	2	yes	—
14	#16863509	C₄	1	2	yes	complemented
15	#16863504	C₄	1	2	yes	complemented
16	#16863498	C₄	1	4	yes	complemented
17	#16863455	—	2	4	no	—
18	#16863451	—	2	4	no	—
19	#16863434	C₂²	1	1	yes	characteristic
20	#16863424	C₂²	1	2	yes	—
21	#16863424	C₂²	1	2	yes	—
22	C₂×#14523969	C₂²	1	2	yes	—
23	C₂×#14523969	C₂²	1	2	yes	—
24	#16863393	C₂²	1	2	yes	—
25	#16863364	C₂²	1	2	yes	—
26	#16863364	C₂²	1	2	yes	—
27	C₂×#14523954	C₂²	1	1	yes	characteristic
28	C₂×#14523953	C₂²	1	1	yes	characteristic
29	C₂×#14523953	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863157	C₂²	1	2	yes	—
34	#16863157	C₂²	1	2	yes	—
35	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
38	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
39	#341×C₆₇	C₂²	1	2	yes	—
40	C₂×#11761628	C₂³	1	2	yes	—
41	#14524065	—	2	2	no	—
42	#14524065	—	2	2	no	—
43	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
44	C₂×#11761617	C₂×C₄	1	2	yes	—
45	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
49	C₄×D₂₆₈	C₂×C₄	1	4	yes	—
50	C₄×C₂²⋊D₆₇	—	2	4	no	—
51	#14524002	C₂³	1	2	yes	—
52	#14524000	—	2	2	no	—
53	C₂×C₄×Dic₆₇	—	2	2	no	—
54	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
56	C₄²×D₆₇	—	2	2	no	—
57	#14523977	C₂×C₄	1	2	yes	—
58	#14523975	C₂×C₄	1	2	yes	—
59	Dic₆₇⋊C₈	—	2	4	no	—
60	Dic₆₇⋊C₈	—	2	4	no	—
61	#14523969	—	2	4	no	—
62	#14523969	—	2	4	no	—
63	#14523968	C₂×C₄	1	4	yes	—
64	#14523965	C₂×C₄	1	4	yes	—
65	(C₂×D₆₇)⋊C₈	—	2	4	no	—
66	(C₂×D₆₇)⋊C₈	—	2	4	no	—
67	#14523954	D₄	1	4	yes	—
68	#14523953	—	2	2	no	—
69	#14523953	—	2	2	no	—
70	C₈×Dic₆₇	—	2	2	no	—
71	C₈×Dic₆₇	—	2	2	no	—
72	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#108×C₆₇	—	2	2	no	—
78	#108×C₆₇	—	2	2	no	—
79	#97×C₆₇	—	2	2	no	—
80	#97×C₆₇	C₂×C₄	1	2	yes	—
81	#97×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	#11761628	—	2	2	no	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	#11761628	—	2	2	no	—
87	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
88	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₆₇	—	2	2	no	—
90	C₂²×Dic₆₇	—	2	1	no	—
91	C₂²×Dic₆₇	—	2	1	no	—
92	C₂×C₄×D₆₇	—	2	2	no	—
93	C₂×C₄×D₆₇	—	2	4	no	—
94	Dic₂₆₈	—	2	2	no	—
95	#11761617	—	2	4	no	—
96	D₅₃₆	—	2	2	no	—
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	—	2	1	no	—
99	C₂×C₆₇⋊C₈	—	2	1	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
103	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
104	Dic₆₇⋊C₄	—	2	4	no	—
105	C₂²⋊Dic₆₇	—	2	2	no	—
106	(C₂×D₆₇)⋊C₄	—	2	4	no	—
107	C₄×Dic₆₇	—	2	4	no	—
108	C₄×Dic₆₇	—	2	2	no	—
109	C₄×Dic₆₇	—	2	1	no	—
110	C₄×Dic₆₇	—	2	1	no	—
111	C₄×Dic₆₇	—	4	4	no	—
112	C₄×Dic₆₇	#41	1	2	yes	—
113	C₄×Dic₆₇	#41	1	2	yes	—
114	C₂²×C₄×C₆₇	#41	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₆₇	—	2	2	no	—
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	#41	1	2	yes	—
128	C₂×C₈×C₆₇	#41	1	2	yes	—
129	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₂²×D₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₆₇	—	4	2	no	—
142	C₂²⋊D₆₇	—	4	2	no	—
143	C₂×Dic₆₇	—	2	4	no	—
144	C₂×Dic₆₇	—	2	2	no	—
145	C₂×Dic₆₇	—	2	4	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	4	2	no	—
149	D₂₆₈	—	2	2	no	—
150	D₂₆₈	—	2	2	no	—
151	C₄×D₆₇	—	4	2	no	—
152	C₄×D₆₇	—	4	2	no	—
153	C₄×D₆₇	—	8	4	no	—
154	Dic₁₃₄	—	2	2	no	—
155	Dic₁₃₄	—	2	2	no	—
156	C₈×C₆₇	—	2	2	no	—
157	C₈×C₆₇	—	4	2	no	—
158	C₈×C₆₇	—	4	2	no	—
159	C₈×C₆₇	#117	1	2	yes	—
160	C₈×C₆₇	#117	1	2	yes	—
161	C₆₇⋊C₈	—	2	2	no	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₆₇	#529	1	1	yes	characteristic
166	C₂²×C₆₇	#529	1	1	yes	characteristic
167	C₂×D₆₇	—	4	4	no	—
168	C₂×D₆₇	—	4	2	no	—
169	C₄×C₆₇	—	2	4	no	—
170	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
171	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
172	C₄×C₆₇	#532	1	1	yes	characteristic
173	C₄×C₆₇	#532	1	1	yes	characteristic
174	Dic₆₇	—	2	4	no	—
175	Dic₆₇	—	4	2	no	—
176	Dic₆₇	—	4	4	no	—
177	Dic₆₇	—	4	2	no	—
178	#21297	—	67	1	no	maximal, Sylow
179	C₂×C₆₇	—	2	1	no	—
180	C₂×C₆₇	—	2	1	no	—
181	C₂×C₆₇	#2293	1	1	yes	characteristic
182	C₂×C₆₇	#2956	1	2	yes	—
183	D₆₇	—	8	2	no	—
184	#2943	—	67	1	no	—
185	#2923	—	67	2	no	—
186	#1975	—	67	1	no	—
187	#1974	—	67	1	no	—
188	#1946	—	67	1	no	—
189	#1945	—	67	1	no	—
190	#1938	—	67	2	no	—
191	#1937	—	67	2	no	—
192	#1778	D₆₇	1	1	yes	characteristic, complemented
193	C₄×#104	—	67	2	no	—
194	C₄×#97	—	67	1	no	—
195	C₆₇	#21297	1	1	yes	characteristic, complemented, Sylow
196	C₂×#135	—	67	1	no	—
197	C₄×#41	—	67	2	no	—
198	#442	—	67	2	no	—
199	#441	—	67	2	no	—
200	#440	—	67	4	no	—
201	#433	—	134	4	no	—
202	#432	—	134	4	no	—
203	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
204	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
205	#427	—	67	1	no	—
206	#424	—	67	2	no	—
207	#424	—	67	2	no	—
208	C₂×#104	—	67	2	no	—
209	C₂×#104	—	67	2	no	—
210	#416	—	67	2	no	—
211	#408	—	67	2	no	—
212	#408	—	67	2	no	—
213	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
214	C₂×#97	—	67	1	no	—
215	C₂×#97	—	67	1	no	—
216	C₂×#97	—	67	1	no	—
217	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
218	C₂×C₄×C₈	—	67	1	no	—
219	C₂×C₄×C₈	—	67	1	no	—
220	C₄³	—	67	1	no	—
221	#341	C₂×D₆₇	1	2	yes	—
222	#339	—	67	2	no	—
223	#339	—	67	2	no	—
224	C₂×#41	—	67	2	no	—
225	#135	—	134	2	no	—
226	#135	—	134	2	no	—
227	C₂×Q₁₆	—	67	1	no	—
228	C₂×SD₁₆	—	67	2	no	—
229	C₂×D₈	—	67	1	no	—
230	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
231	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	—	67	1	no	—
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
236	C₄×Q₈	—	67	4	no	—
237	C₄×D₄	—	67	4	no	—
238	C₄×D₄	—	134	4	no	—
239	#117	—	67	2	no	—
240	#117	—	134	2	no	—
241	C₂×C₄²	—	67	1	no	—
242	C₂×C₄²	—	67	1	no	—
243	C₂×C₄²	—	134	2	no	—
244	C₂×C₄²	—	134	2	no	—
245	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
246	#108	—	2	2	no	—
247	#108	—	2	2	no	—
248	#107	—	67	2	no	—
249	#106	—	67	2	no	—
250	C₄⋊C₈	—	134	4	no	—
251	C₄⋊C₈	—	134	4	no	—
252	#104	—	134	4	no	—
253	#104	—	134	4	no	—
254	#103	—	67	4	no	—
255	#102	—	67	4	no	—
256	C₂²⋊C₈	—	134	4	no	—
257	C₂²⋊C₈	—	134	4	no	—
258	#97	—	2	2	no	—
259	#97	—	67	4	no	—
260	#97	—	134	2	no	—
261	#97	—	134	2	no	—
262	#97	C₄×D₆₇	1	2	yes	—
263	#97	C₄×D₆₇	1	2	yes	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	134	2	no	—
266	C₄×C₈	—	134	2	no	—
267	#41	—	134	2	no	—
268	#41	—	134	2	no	—
269	#41	—	134	2	no	—
270	#41	—	134	2	no	—
271	C₂×Q₈	—	67	2	no	—
272	C₂×D₄	—	67	2	no	—
273	C₂×D₄	—	134	2	no	—
274	C₂²×C₄	—	134	1	no	—
275	C₂²×C₄	—	134	2	no	—
276	C₂²×C₄	—	134	4	no	—
277	C₂²×C₄	—	134	1	no	—
278	C₂²×C₄	#11761629	1	2	yes	—
279	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
280	Q₁₆	—	134	2	no	—
281	SD₁₆	—	134	4	no	—
282	D₈	—	134	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	67	2	no	—
292	C₂×C₈	—	67	2	no	—
293	C₂×C₈	—	134	1	no	—
294	C₂×C₈	—	134	1	no	—
295	C₂×C₈	—	134	1	no	—
296	C₂×C₈	—	134	1	no	—
297	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
300	C₂×C₈	#11761628	1	2	yes	—
301	C₂×C₈	#11761628	1	2	yes	—
302	C₄⋊C₄	—	67	4	no	—
303	C₄⋊C₄	—	134	4	no	—
304	C₂²⋊C₄	—	134	2	no	—
305	C₂²⋊C₄	—	134	4	no	—
306	C₄²	—	268	4	no	—
307	C₄²	—	67	2	no	—
308	C₄²	—	67	2	no	—
309	C₄²	—	134	4	no	—
310	C₄²	—	134	2	no	—
311	C₄²	—	134	1	no	—
312	C₄²	—	134	1	no	—
313	C₄²	C₂×C₄×D₆₇	1	2	yes	—
314	C₄²	D₄×D₆₇	1	2	yes	—
315	C₂³	—	134	2	no	—
316	C₂³	C₂×#11761629	1	1	yes	characteristic
317	Q₈	—	134	2	no	—
318	Q₈	—	134	2	no	—
319	D₄	—	268	2	no	—
320	D₄	—	268	2	no	—
321	D₄	—	134	2	no	—
322	D₄	—	134	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	268	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	268	2	no	—
327	C₂×C₄	—	536	4	no	—
328	C₂×C₄	—	134	4	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	—	134	2	no	—
331	C₂×C₄	—	134	4	no	—
332	C₂×C₄	—	134	2	no	—
333	C₂×C₄	#14524005	1	4	yes	—
334	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
336	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
337	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
339	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	134	2	no	—
344	C₈	—	134	2	no	—
345	C₈	#14524000	1	2	yes	—
346	C₈	#14524000	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	268	2	no	—
349	C₂²	—	268	4	no	—
350	C₂²	C₂×#14524005	1	1	yes	characteristic
351	C₂²	#16863933	1	1	yes	characteristic
352	C₂²	#16863933	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	268	2	no	—
355	C₄	—	268	2	no	—
356	C₄	—	268	4	no	—
357	C₄	—	134	4	no	—
358	C₄	C₂×#14524000	1	1	yes	characteristic
359	C₄	#16863864	1	1	yes	characteristic
360	C₄	#16863943	1	1	yes	characteristic
361	C₄	#16863943	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	536	2	no	—
365	C₂	#22562670	1	1	yes	characteristic
366	C₂	#22569118	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1778	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1778	D₆₇
hypocenter	C₆₇	#21297
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
2. #22568861	C₂	semidirect
5. #22560960	C₂	semidirect
6. #22560955	C₂	semidirect
7. #22560918	C₂	semidirect
8. #22560911	C₂	semidirect
9. C₄×#14523969	C₂	semidirect
10. #97×Dic₆₇	C₂	semidirect
11. C₆₇×#1778	C₂	semidirect
14. #16863509	C₄	semidirect
15. #16863504	C₄	semidirect
16. #16863498	C₄	semidirect
192. #1778	D₆₇	semidirect
195. C₆₇	#21297	semidirect

Maximal subgroups

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

subgroup	index
#22569011	2
#22568861	2
#22561100	2
#22561095	2
#22560960	2
#22560955	2
#22560918	2
#22560911	2
C₄×#14523969	2
#97×Dic₆₇	2
C₆₇×#1778	2
#21297	67