Subgroups — #30825773

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570240	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568920	C₂	1	1	yes	maximal, characteristic
4	#22559084	C₂	1	4	yes	maximal, complemented
5	#22559024	C₂	1	4	yes	maximal, complemented
6	#22558985	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558609	C₂	1	1	yes	maximal, characteristic
8	#22558502	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1487	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524051	—	2	1	no	—
11	#16863946	C₂²	1	1	yes	characteristic
12	#16863714	C₄	1	2	yes	—
13	#16863681	C₄	1	2	yes	—
14	#16863596	C₄	1	2	yes	—
15	#16863535	C₄	1	2	yes	—
16	Q₈×C₆₇⋊C₈	—	2	1	no	—
17	Q₈×C₆₇⋊C₈	—	2	1	no	—
18	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
19	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
20	C₄×#11761615	C₂²	1	4	yes	—
21	#16863473	—	2	1	no	—
22	#16863473	C₂²	1	2	yes	—
23	#16863461	—	2	1	no	—
24	#16863461	C₂²	1	2	yes	—
25	#16863449	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
27	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	#16863364	—	2	1	no	—
29	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	#16863148	C₂²	1	4	yes	—
32	#16863125	C₂²	1	4	yes	—
33	#16863118	C₂²	1	2	yes	—
34	#16863117	C₂²	1	2	yes	—
35	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	#423×C₆₇	C₂²	1	1	yes	characteristic
38	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
39	C₂×#11761628	—	2	1	no	—
40	C₂×#11761615	C₂×C₄	1	4	yes	—
41	#14524051	—	4	4	no	—
42	C₂×#11761611	—	2	2	no	—
43	C₂²×C₆₇⋊C₈	—	2	1	no	—
44	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	—	2	2	no	—
47	#14524043	C₂×C₄	1	1	yes	characteristic
48	#14524040	C₂×C₄	1	1	yes	characteristic
49	#14524031	—	2	1	no	—
50	#14524017	—	2	1	no	—
51	#14524017	C₂×C₄	1	2	yes	—
52	C₄×D₂₆₈	—	2	1	no	—
53	C₄×D₂₆₈	C₂³	1	2	yes	—
54	#14523974	C₂×C₄	1	4	yes	—
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
57	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
58	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
59	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
60	#14523967	C₂×C₄	1	4	yes	—
61	#14523963	C₂×C₄	1	4	yes	—
62	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
63	(C₂×D₆₇)⋊C₈	—	2	2	no	—
64	(C₂×D₆₇)⋊C₈	—	2	2	no	—
65	C₄×C₆₇⋊C₈	—	2	2	no	—
66	C₄×C₆₇⋊C₈	—	2	2	no	—
67	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
69	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
70	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₆₇	—	2	1	no	—
74	C₄×Q₈×C₆₇	—	2	1	no	—
75	C₄×Q₈×C₆₇	—	2	1	no	—
76	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	C₄⋊C₈×C₆₇	—	2	1	no	—
84	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
85	C₂²⋊C₈×C₆₇	—	2	1	no	—
86	C₄×C₈×C₆₇	C₂³	1	2	yes	—
87	#11761628	—	4	2	no	—
88	#11761628	—	4	2	no	—
89	C₂×Dic₁₃₄	—	2	1	no	—
90	C₂×D₂₆₈	—	2	1	no	—
91	C₂×D₂₆₈	—	2	1	no	—
92	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
93	C₂×C₂²⋊D₆₇	—	2	2	no	—
94	C₂×C₄×D₆₇	—	2	2	no	—
95	#11761615	—	2	8	no	—
96	#11761611	—	4	4	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	4	2	no	—
101	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₈×D₆₇	—	4	4	no	—
107	C₄⋊Dic₆₇	—	2	1	no	—
108	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
109	(C₂×D₆₇)⋊C₄	—	2	2	no	—
110	(C₂×D₆₇)⋊C₄	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	4	no	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
119	M₁₆×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₆₇	—	2	2	no	—
134	Q₈×C₆₇	—	2	4	no	—
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	#131	1	4	yes	complemented
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	4	1	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	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₆₇	#137	1	2	yes	—
148	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₆₇	—	4	4	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	D₂₆₈	—	2	4	no	—
154	D₂₆₈	—	4	2	no	—
155	D₂₆₈	—	4	2	no	—
156	C₄×D₆₇	—	4	2	no	—
157	C₄×D₆₇	—	4	2	no	—
158	Dic₁₃₄	—	4	2	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₆₇⋊C₈	—	2	4	no	—
162	C₆₇⋊C₈	—	2	1	no	—
163	C₆₇⋊C₈	—	2	1	no	—
164	C₆₇⋊C₈	—	2	1	no	—
165	C₆₇⋊C₈	—	2	1	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	#440	1	2	yes	—
170	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₆₇	—	4	2	no	—
172	C₂×D₆₇	—	4	2	no	—
173	C₂×D₆₇	—	4	2	no	—
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	#440	1	2	yes	—
180	C₄×C₆₇	C₂×#131	1	2	yes	—
181	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
183	Dic₆₇	—	4	2	no	—
184	#17908	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	—	2	2	no	—
186	C₂×C₆₇	#2929	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
188	C₂×C₆₇	#2955	1	1	yes	characteristic
189	D₆₇	—	8	2	no	—
190	#3103	—	67	1	no	—
191	C₂×C₈×Q₈	—	67	1	no	—
192	#2931	—	67	1	no	—
193	#1588	—	67	4	no	—
194	SD₁₆⋊C₈	—	67	4	no	—
195	#1571	—	67	1	no	—
196	#1487	D₆₇	1	1	yes	characteristic, complemented
197	#1480	—	67	1	no	—
198	#1452	—	67	1	no	—
199	C₆₇	#17908	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	134	1	no	—
201	#532	—	67	1	no	—
202	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
203	#481	—	67	2	no	—
204	#475	—	67	2	no	—
205	#460	—	67	2	no	—
206	#448	—	67	2	no	—
207	C₈×Q₈	—	67	2	no	—
208	C₈×Q₈	—	67	2	no	—
209	C₈×Q₈	—	134	1	no	—
210	C₈×Q₈	—	134	1	no	—
211	C₄×SD₁₆	—	67	4	no	—
212	#436	—	67	2	no	—
213	#436	—	134	1	no	—
214	C₈×D₄	—	67	2	no	—
215	C₈×D₄	—	134	1	no	—
216	#431	—	67	1	no	—
217	#431	C₂×D₆₇	1	1	yes	characteristic
218	#423	C₂×D₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	—	67	1	no	—
220	C₂×C₄⋊C₈	—	134	1	no	—
221	#408	—	134	1	no	—
222	C₂×C₄×C₈	—	67	1	no	—
223	C₂×C₄×C₈	—	134	1	no	—
224	#335	—	67	4	no	—
225	Q₈⋊C₈	C₂×D₆₇	1	4	yes	—
226	D₄⋊C₈	—	67	4	no	—
227	#322	—	67	2	no	—
228	#322	—	67	2	no	—
229	C₂×#41	—	134	1	no	—
230	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
231	C₂×SD₁₆	—	67	4	no	—
232	#131	—	268	4	no	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	134	2	no	—
235	C₂²×C₈	—	67	1	no	—
236	C₂²×C₈	—	67	1	no	—
237	C₂²×C₈	—	134	2	no	—
238	C₂²×C₈	—	134	1	no	—
239	C₄⋊Q₈	—	67	1	no	—
240	#127	—	67	1	no	—
241	#124	—	134	1	no	—
242	#121	—	67	2	no	—
243	#121	—	134	1	no	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₄×D₆₇	1	2	yes	—
247	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
248	C₄×D₄	—	67	2	no	—
249	C₄×D₄	—	134	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
254	#106	—	67	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	67	2	no	—
258	C₄⋊C₈	—	67	2	no	—
259	C₄⋊C₈	—	134	2	no	—
260	C₄⋊C₈	—	134	2	no	—
261	C₄⋊C₈	—	134	1	no	—
262	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
263	#103	—	67	4	no	—
264	#102	—	67	4	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	67	2	no	—
267	C₂²⋊C₈	—	134	2	no	—
268	C₂²⋊C₈	—	134	2	no	—
269	C₄×C₈	—	67	1	no	—
270	C₄×C₈	—	67	1	no	—
271	C₄×C₈	—	67	2	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	2	yes	—
277	#41	—	268	2	no	—
278	#41	—	268	2	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	134	1	no	—
282	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
283	C₂×D₄	—	67	2	no	—
284	C₂×D₄	—	134	2	no	—
285	C₂×D₄	—	134	1	no	—
286	C₂×D₄	—	134	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	134	2	no	—
290	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
292	C₂²×C₄	#11761627	1	1	yes	characteristic
293	SD₁₆	—	134	8	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	268	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	268	4	no	—
299	C₂×C₈	—	268	2	no	—
300	C₂×C₈	—	67	4	no	—
301	C₂×C₈	—	67	1	no	—
302	C₂×C₈	—	67	1	no	—
303	C₂×C₈	—	67	1	no	—
304	C₂×C₈	—	67	1	no	—
305	C₂×C₈	—	134	2	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	—	134	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	67	2	no	—
312	C₄⋊C₄	—	134	1	no	—
313	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
314	C₂²⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
319	C₄²	D₄×D₆₇	1	1	yes	characteristic
320	C₄²	#11761627	1	1	yes	characteristic
321	C₄²	C₂³×D₆₇	1	1	yes	characteristic
322	C₂³	—	134	2	no	—
323	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	268	2	no	—
327	Q₈	#14524051	1	4	yes	complemented
328	D₄	—	268	2	no	—
329	D₄	—	268	4	no	—
330	D₄	—	268	2	no	—
331	D₄	—	134	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	268	2	no	—
339	C₂×C₄	—	268	2	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
342	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
343	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
344	C₂×C₄	#14524070	1	2	yes	—
345	C₂×C₄	#14524075	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	268	2	no	—
352	C₈	—	134	4	no	—
353	C₈	—	134	1	no	—
354	C₈	—	134	1	no	—
355	C₈	—	134	1	no	—
356	C₈	—	134	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	268	2	no	—
360	C₂²	—	268	2	no	—
361	C₂²	—	268	2	no	—
362	C₂²	#16863499	1	2	yes	—
363	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	4	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	268	2	no	—
370	C₄	#16863499	1	2	yes	—
371	C₄	C₂×#14524051	1	2	yes	—
372	C₄	C₂×#14524070	1	1	yes	characteristic
373	C₄	C₂×#14524075	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	536	2	no	—
376	C₂	#22568899	1	1	yes	characteristic
377	C₂	C₂×#16863499	1	1	yes	characteristic
378	C₂	#22569108	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×#1487	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1487	D₆₇
hypocenter	C₆₇	#17908
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1487

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
4. #22559084	C₂	semidirect
5. #22559024	C₂	semidirect
6. #22558985	C₂	semidirect
8. #22558502	C₂	semidirect
9. C₆₇×#1487	C₂	semidirect
136. Q₈×C₆₇	#131	semidirect
196. #1487	D₆₇	semidirect
199. C₆₇	#17908	semidirect
327. Q₈	#14524051	semidirect

Maximal subgroups

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

subgroup	index
#22570240	2
C₂×Q₈×C₆₇⋊C₈	2
#22568920	2
#22559084	2
#22559024	2
#22558985	2
#22558609	2
#22558502	2
C₆₇×#1487	2
#17908	67