Subgroups — #30827557

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569295	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22559118	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559062	C₂	1	2	yes	maximal, complemented
5	#22559057	C₂	1	2	yes	maximal, complemented
6	#22559029	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559027	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558969	C₂	1	1	yes	maximal, characteristic
9	#22558654	C₂	1	2	yes	maximal
10	#22558495	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₆₇	C₂²	1	2	yes	—
14	#16863527	C₂²	1	2	yes	—
15	#16863525	—	2	2	no	—
16	#16863524	—	2	2	no	—
17	C₈×Dic₁₃₄	C₂²	1	2	yes	—
18	#16863520	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic
21	#16863490	C₄	1	1	yes	characteristic
22	C₄×#11761618	C₂²	1	2	yes	—
23	#16863488	C₄	1	2	yes	—
24	#16863452	C₂²	1	2	yes	—
25	#16863451	—	2	2	no	—
26	#16863448	—	2	2	no	—
27	#16863447	C₂²	1	1	yes	characteristic
28	#107×D₆₇	C₄	1	1	yes	characteristic
29	#103×D₆₇	C₄	1	2	yes	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	2	yes	—
33	#16863147	C₂²	1	1	yes	characteristic
34	#16863128	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863117	C₂²	1	1	yes	characteristic
39	#16863117	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#334×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
47	C₂×#11761618	C₂×C₄	1	2	yes	—
48	Q₁₆×D₆₇	—	2	4	no	—
49	C₂×#11761611	—	2	1	no	—
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
52	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₃₄	—	2	2	no	—
54	C₄×Dic₁₃₄	C₂³	1	2	yes	—
55	#14524008	—	2	2	no	—
56	#14524008	C₂×C₄	1	2	yes	—
57	C₄⋊C₄×D₆₇	—	2	2	no	—
58	C₄⋊C₄×D₆₇	C₂×C₄	1	2	yes	—
59	C₄²×D₆₇	—	2	2	no	—
60	C₄²×D₆₇	C₂³	1	1	yes	characteristic
61	#14523977	C₂×C₄	1	1	yes	characteristic
62	#14523976	C₂×C₄	1	2	yes	—
63	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₆₇⋊C₈	—	2	2	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	D₄	1	2	yes	—
68	#14523968	C₂×C₄	1	2	yes	—
69	#14523967	C₂×C₄	1	2	yes	—
70	#14523966	C₂×C₄	1	2	yes	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
73	#14523954	—	2	2	no	—
74	#14523953	—	2	2	no	—
75	#14523953	—	2	1	no	—
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₆₇	—	2	2	no	—
80	C₈×Dic₆₇	—	2	2	no	—
81	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
85	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	—	2	2	no	—
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#103×C₆₇	C₂×C₄	1	2	yes	—
89	#97×C₆₇	—	2	2	no	—
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×Dic₁₃₄	—	2	2	no	—
94	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
95	Q₈×D₆₇	—	2	4	no	—
96	Q₈×D₆₇	—	4	2	no	—
97	C₂×C₄×D₆₇	—	2	2	no	—
98	C₂×C₄×D₆₇	—	2	2	no	—
99	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
102	Dic₂₆₈	M₁₆	1	4	yes	complemented
103	#11761618	—	2	4	no	—
104	#11761611	—	4	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	1	no	—
107	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
110	C₈×D₆₇	—	2	2	no	—
111	C₈×D₆₇	—	4	1	no	—
112	C₄⋊Dic₆₇	—	2	2	no	—
113	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
114	Dic₆₇⋊C₄	—	2	2	no	—
115	Dic₆₇⋊C₄	—	2	2	no	—
116	Dic₆₇⋊C₄	—	2	2	no	—
117	C₄×Dic₆₇	—	2	2	no	—
118	C₄×Dic₆₇	—	2	2	no	—
119	C₄×Dic₆₇	—	2	2	no	—
120	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₆₇	#41	1	1	yes	characteristic
123	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
124	Q₁₆×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₆₇	—	2	2	no	—
143	C₂×Dic₆₇	—	2	2	no	—
144	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	2	1	no	—
149	C₄×D₆₇	—	2	1	no	—
150	C₄×D₆₇	—	2	2	no	—
151	C₄×D₆₇	—	4	4	no	—
152	C₄×D₆₇	—	4	4	no	—
153	Dic₁₃₄	—	2	2	no	—
154	Dic₁₃₄	—	4	4	no	—
155	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
156	C₈×C₆₇	—	2	1	no	—
157	C₈×C₆₇	—	2	2	no	—
158	C₈×C₆₇	—	4	1	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
161	C₆₇⋊C₈	—	2	1	no	—
162	C₆₇⋊C₈	—	2	1	no	—
163	C₆₇⋊C₈	—	2	2	no	—
164	C₆₇⋊C₈	—	4	1	no	—
165	C₆₇⋊C₈	—	4	2	no	—
166	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×D₆₇	—	2	1	no	—
168	C₂×D₆₇	—	2	1	no	—
169	C₂×D₆₇	#444	1	2	yes	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
174	C₄×C₆₇	#568	1	1	yes	characteristic
175	C₄×C₆₇	#578	1	2	yes	—
176	Dic₆₇	—	2	4	no	—
177	Dic₆₇	—	2	1	no	—
178	Dic₆₇	—	2	1	no	—
179	Dic₆₇	—	2	1	no	—
180	Dic₆₇	—	4	2	no	—
181	#18062	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
183	C₂×C₆₇	#2950	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
185	D₆₇	—	2	2	no	—
186	#2984	—	67	2	no	—
187	C₂×C₄×Q₁₆	—	67	1	no	—
188	#1593	—	67	1	no	—
189	#1593	D₆₇	1	1	yes	characteristic, complemented
190	#1584	—	67	2	no	—
191	#1584	—	67	2	no	—
192	#1579	—	67	1	no	—
193	#1579	—	67	1	no	—
194	#1568	—	67	1	no	—
195	#1491	—	67	2	no	—
196	#1451	—	67	1	no	—
197	C₆₇	#18062	1	1	yes	characteristic, complemented, Sylow
198	C₂²×Q₁₆	—	67	1	no	—
199	C₂×C₄×Q₈	—	67	2	no	—
200	#446	—	67	2	no	—
201	#446	—	134	2	no	—
202	#446	—	134	2	no	—
203	#446	C₂×D₆₇	1	2	yes	—
204	C₈×Q₈	—	67	2	no	—
205	C₈×Q₈	—	67	2	no	—
206	C₄×Q₁₆	—	67	1	no	—
207	C₄×Q₁₆	—	67	2	no	—
208	C₄×Q₁₆	—	67	1	no	—
209	C₄×Q₁₆	—	67	2	no	—
210	C₄×Q₁₆	—	67	1	no	—
211	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
212	#432	—	67	2	no	—
213	#432	—	134	2	no	—
214	#431	—	67	1	no	—
215	#431	—	134	2	no	—
216	C₂×#107	—	67	1	no	—
217	C₂×#103	—	67	2	no	—
218	C₄×M₁₆	—	67	1	no	—
219	C₂×C₄×C₈	—	67	1	no	—
220	#335	—	67	2	no	—
221	#334	—	67	1	no	—
222	#334	C₂×D₆₇	1	1	yes	characteristic
223	Q₈⋊C₈	—	67	2	no	—
224	Q₈⋊C₈	—	67	2	no	—
225	Q₈⋊C₈	—	67	2	no	—
226	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
227	#322	—	67	1	no	—
228	#322	—	67	1	no	—
229	#322	—	67	1	no	—
230	#322	C₂×D₆₇	1	1	yes	characteristic
231	C₂²×Q₈	—	67	2	no	—
232	C₂×Q₁₆	—	67	2	no	—
233	C₂×Q₁₆	—	67	1	no	—
234	C₂×Q₁₆	—	134	4	no	—
235	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
236	C₂×M₁₆	—	134	1	no	—
237	C₂²×C₈	—	67	1	no	—
238	C₂²×C₈	—	67	1	no	—
239	C₄×Q₈	—	67	2	no	—
240	C₄×Q₈	—	67	2	no	—
241	C₄×Q₈	—	67	2	no	—
242	C₄×Q₈	—	134	2	no	—
243	C₄×Q₈	—	134	2	no	—
244	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	—	67	2	no	—
246	C₂×C₄⋊C₄	—	134	2	no	—
247	C₂×C₄²	—	67	1	no	—
248	C₂×C₄²	—	134	2	no	—
249	#107	—	67	1	no	—
250	#107	—	67	2	no	—
251	#107	C₄×D₆₇	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	67	2	no	—
254	C₄⋊C₈	—	67	2	no	—
255	C₄⋊C₈	—	134	2	no	—
256	C₄⋊C₈	—	134	2	no	—
257	C₄⋊C₈	—	134	2	no	—
258	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
259	#103	—	67	2	no	—
260	#103	—	67	2	no	—
261	#103	—	67	2	no	—
262	#103	C₄×D₆₇	1	2	yes	—
263	C₂²⋊C₈	—	67	2	no	—
264	C₂²⋊C₈	—	134	2	no	—
265	#97	—	2	2	no	—
266	#97	—	134	2	no	—
267	#97	—	134	1	no	—
268	#97	—	134	2	no	—
269	C₄×C₈	—	67	1	no	—
270	C₄×C₈	—	67	1	no	—
271	C₄×C₈	—	67	1	no	—
272	C₄×C₈	—	67	1	no	—
273	C₄×C₈	—	67	1	no	—
274	C₄×C₈	—	134	2	no	—
275	C₄×C₈	—	134	2	no	—
276	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
279	C₂×Q₈	—	268	2	no	—
280	C₂×Q₈	—	67	2	no	—
281	C₂×Q₈	—	134	2	no	—
282	C₂×Q₈	—	134	4	no	—
283	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
284	C₂²×C₄	—	67	1	no	—
285	C₂²×C₄	—	67	1	no	—
286	C₂²×C₄	—	67	1	no	—
287	C₂²×C₄	—	134	2	no	—
288	C₂²×C₄	—	134	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	67	4	no	—
291	Q₁₆	—	134	4	no	—
292	M₁₆	—	268	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	268	1	no	—
296	C₂×C₈	—	67	1	no	—
297	C₂×C₈	—	67	1	no	—
298	C₂×C₈	—	67	2	no	—
299	C₂×C₈	—	134	2	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
304	C₂×C₈	D₄×D₆₇	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	67	2	no	—
307	C₄⋊C₄	—	134	2	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₄⋊C₄	—	134	2	no	—
310	C₄⋊C₄	—	134	2	no	—
311	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
312	C₄²	—	2	2	no	—
313	C₄²	—	67	1	no	—
314	C₄²	—	67	1	no	—
315	C₄²	—	67	1	no	—
316	C₄²	—	134	2	no	—
317	C₄²	—	134	2	no	—
318	C₄²	—	134	2	no	—
319	C₄²	C₂³×D₆₇	1	1	yes	characteristic
320	C₂³	—	67	1	no	—
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	268	4	no	—
324	Q₈	—	67	4	no	—
325	Q₈	—	134	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	268	4	no	—
329	C₂×C₄	—	268	4	no	—
330	C₂×C₄	—	67	1	no	—
331	C₂×C₄	—	67	1	no	—
332	C₂×C₄	—	67	1	no	—
333	C₂×C₄	—	134	2	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	—	134	2	no	—
336	C₂×C₄	—	134	1	no	—
337	C₂×C₄	—	134	1	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	2	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	268	1	no	—
347	C₈	—	268	2	no	—
348	C₈	—	134	1	no	—
349	C₈	—	134	1	no	—
350	C₈	—	134	2	no	—
351	C₈	M₁₆×D₆₇	1	2	yes	—
352	C₂²	—	67	2	no	—
353	C₂²	—	134	1	no	—
354	C₂²	—	134	1	no	—
355	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	268	2	no	—
360	C₄	—	134	1	no	—
361	C₄	—	134	1	no	—
362	C₄	—	134	4	no	—
363	C₄	—	134	1	no	—
364	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
365	C₄	#16864161	1	1	yes	characteristic
366	C₄	#16864212	1	2	yes	—
367	C₂	—	134	2	no	—
368	C₂	#22568945	1	1	yes	characteristic
369	C₂	#22569074	1	1	yes	characteristic
370	C₂	#444×D₆₇	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×#1593	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1593	D₆₇
hypocenter	C₆₇	#18062
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #22559118	C₂	semidirect
4. #22559062	C₂	semidirect
5. #22559057	C₂	semidirect
6. #22559029	C₂	semidirect
7. #22559027	C₂	semidirect
11. C₆₇×#1593	C₂	semidirect
102. Dic₂₆₈	M₁₆	semidirect
189. #1593	D₆₇	semidirect
197. C₆₇	#18062	semidirect

Maximal subgroups

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

subgroup	index
#22569295	2
C₄×Q₁₆×D₆₇	2
#22559118	2
#22559062	2
#22559057	2
#22559029	2
#22559027	2
#22558969	2
#22558654	2
#22558495	2
C₆₇×#1593	2
#18062	67