Subgroups — #30839624

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570002	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559657	C₂	1	4	yes	maximal, complemented
4	#22559605	C₂	1	2	yes	maximal, complemented
5	#22559571	C₂	1	2	yes	maximal, complemented
6	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558899	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₆₇	—	2	1	no	—
10	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
11	#16863693	—	2	4	no	—
12	#16863683	C₂²	1	4	yes	—
13	#16863673	C₂²	1	4	yes	—
14	C₈×D₂₆₈	—	2	1	no	—
15	C₈×D₂₆₈	—	2	1	no	—
16	C₈×D₂₆₈	C₂²	1	2	yes	—
17	C₈×D₂₆₈	C₂²	1	2	yes	—
18	#16863439	—	2	2	no	—
19	#16863420	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#16863400	—	2	2	no	—
22	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
23	#16863149	C₂²	1	2	yes	complemented
24	#16863149	C₂²	1	2	yes	complemented
25	#16863145	C₂²	1	4	yes	complemented
26	#16863142	C₂²	1	4	yes	complemented
27	#533×C₆₇	C₂²	1	2	yes	—
28	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#487×C₆₇	C₂²	1	2	yes	complemented
31	#467×C₆₇	—	2	1	no	—
32	#467×C₆₇	—	2	1	no	—
33	C₄×Q₁₆×C₆₇	—	2	1	no	—
34	C₄×Q₁₆×C₆₇	—	2	1	no	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₆₇	—	2	1	no	—
38	C₂×#11761612	—	2	2	no	—
39	C₂²×C₆₇⋊C₈	—	2	1	no	—
40	C₂×C₈×D₆₇	—	2	2	no	—
41	C₂×C₈×D₆₇	—	2	2	no	—
42	C₂×C₈×D₆₇	—	4	4	no	—
43	C₄×D₂₆₈	—	2	1	no	—
44	C₄×D₂₆₈	—	2	1	no	—
45	C₄×D₂₆₈	D₄	1	2	yes	—
46	C₄×D₂₆₈	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
49	#14523976	—	2	4	no	—
50	#14523974	—	2	4	no	—
51	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
55	#14523963	—	2	4	no	—
56	#14523963	—	2	4	no	—
57	(C₂×D₆₇)⋊C₈	—	2	2	no	—
58	(C₂×D₆₇)⋊C₈	—	2	2	no	—
59	C₂×Q₁₆×C₆₇	—	2	1	no	—
60	C₂×Q₁₆×C₆₇	—	2	1	no	—
61	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
65	#125×C₆₇	—	2	2	no	—
66	C₂²⋊Q₈×C₆₇	—	2	2	no	—
67	C₂²⋊Q₈×C₆₇	—	2	2	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×Q₈×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	—	2	1	no	—
74	#103×C₆₇	—	2	2	no	—
75	#103×C₆₇	—	2	2	no	—
76	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂³×D₆₇	—	2	1	no	—
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×D₂₆₈	—	2	2	no	—
83	C₂×D₂₆₈	Q₁₆	1	4	yes	complemented
84	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
85	C₂²×Dic₆₇	—	2	1	no	—
86	C₂×C₄×D₆₇	—	2	2	no	—
87	C₂×C₄×D₆₇	—	2	2	no	—
88	C₂×C₄×D₆₇	—	4	2	no	—
89	C₂×C₄×D₆₇	—	4	2	no	—
90	#11761612	—	4	4	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₂×C₆₇⋊C₈	—	2	4	no	—
93	C₂×C₆₇⋊C₈	—	4	2	no	—
94	C₈×D₆₇	—	4	4	no	—
95	C₈×D₆₇	—	4	4	no	—
96	C₄⋊Dic₆₇	—	2	1	no	—
97	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
98	(C₂×D₆₇)⋊C₄	—	2	2	no	—
99	(C₂×D₆₇)⋊C₄	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
105	Q₁₆×C₆₇	—	4	2	no	—
106	Q₁₆×C₆₇	—	4	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₂²⋊C₄×C₆₇	—	4	2	no	—
120	C₄²×C₆₇	—	4	2	no	—
121	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₆₇	#41	1	1	yes	characteristic
124	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂³×C₆₇	#121	1	1	yes	characteristic
126	C₂²×D₆₇	—	2	2	no	—
127	C₂²×D₆₇	—	2	4	no	—
128	C₂²×D₆₇	—	4	2	no	—
129	C₂²×D₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	2	no	—
131	Q₈×C₆₇	—	4	2	no	—
132	Q₈×C₆₇	—	8	2	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₆₇	#135	1	2	yes	—
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	4	2	no	—
148	D₂₆₈	—	2	2	no	—
149	D₂₆₈	—	2	4	no	—
150	D₂₆₈	C₂×Q₁₆	1	4	yes	—
151	C₄×D₆₇	—	4	4	no	—
152	C₄×D₆₇	—	4	2	no	—
153	C₄×D₆₇	—	4	2	no	—
154	C₈×C₆₇	—	2	1	no	—
155	C₈×C₆₇	—	2	1	no	—
156	C₈×C₆₇	—	2	1	no	—
157	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
158	C₆₇⋊C₈	—	4	4	no	—
159	C₆₇⋊C₈	—	4	2	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	#467	1	2	yes	—
163	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
164	C₂×D₆₇	—	2	4	no	—
165	C₂×D₆₇	—	4	8	no	—
166	C₂×D₆₇	—	4	2	no	—
167	C₂×D₆₇	—	4	2	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	8	2	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	#467	1	2	yes	—
174	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
175	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
176	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₄×C₆₇	#575	1	1	yes	characteristic
178	Dic₆₇	—	4	2	no	—
179	#19256	—	67	1	no	maximal, Sylow
180	C₂×C₆₇	—	2	2	no	—
181	C₂×C₆₇	#3036	1	1	yes	characteristic
182	C₂×C₆₇	C₂×#467	1	1	yes	characteristic
183	C₂×C₆₇	#3067	1	1	yes	characteristic
184	D₆₇	—	4	4	no	—
185	#3170	D₆₇	1	1	yes	characteristic, complemented
186	#3077	—	67	2	no	—
187	C₂×C₈×D₄	—	67	1	no	—
188	#1683	—	67	4	no	—
189	#1675	—	67	2	no	—
190	#1669	—	67	2	no	—
191	#1567	—	67	1	no	—
192	#1550	—	67	2	no	—
193	C₆₇	#19256	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₈	—	134	1	no	—
195	#533	C₂×D₆₇	1	2	yes	—
196	C₂×C₄×D₄	—	67	1	no	—
197	#500	C₂×D₆₇	1	1	yes	characteristic, complemented
198	#494	C₂×D₆₇	1	1	yes	characteristic, complemented
199	#487	C₂×D₆₇	1	2	yes	complemented
200	#478	—	134	4	no	—
201	#476	—	67	4	no	—
202	#474	—	67	4	no	—
203	#467	—	2	1	no	—
204	#467	—	2	1	no	—
205	C₄×Q₁₆	—	2	1	no	—
206	C₄×Q₁₆	—	2	1	no	—
207	C₈×D₄	—	67	2	no	—
208	C₈×D₄	—	67	2	no	—
209	C₈×D₄	—	134	1	no	—
210	C₈×D₄	—	134	1	no	—
211	#428	—	134	2	no	—
212	#423	—	67	2	no	—
213	C₂×C₄⋊C₈	—	67	1	no	—
214	#418	—	134	2	no	—
215	C₂×C₂²⋊C₈	—	134	1	no	—
216	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
217	#335	—	67	2	no	—
218	#335	—	67	2	no	—
219	#333	—	67	4	no	—
220	#332	—	67	4	no	—
221	C₂²×D₄	—	67	1	no	—
222	C₂³×C₄	—	134	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×M₁₆	—	134	2	no	—
226	C₂²×C₈	—	268	4	no	—
227	C₂²×C₈	—	134	1	no	—
228	C₂²×C₈	—	134	2	no	—
229	C₂²×C₈	—	134	2	no	—
230	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
232	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
233	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
234	#125	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×D₄	—	67	2	no	—
240	C₄×D₄	—	67	2	no	—
241	C₄×D₄	—	134	1	no	—
242	C₄×D₄	—	134	1	no	—
243	#117	—	2	2	no	—
244	C₂×C₄⋊C₄	—	67	1	no	—
245	C₂×C₂²⋊C₄	—	134	1	no	—
246	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
247	#107	—	2	1	no	—
248	#107	—	2	1	no	—
249	#107	—	134	4	no	—
250	#106	—	134	4	no	—
251	C₄⋊C₈	—	67	2	no	—
252	C₄⋊C₈	—	67	4	no	—
253	C₄⋊C₈	—	134	2	no	—
254	C₄⋊C₈	—	134	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#102	—	134	4	no	—
258	#102	—	134	4	no	—
259	C₂²⋊C₈	—	134	2	no	—
260	C₂²⋊C₈	—	134	2	no	—
261	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
265	C₂⁴	—	134	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×D₄	—	67	4	no	—
269	C₂×D₄	—	67	2	no	—
270	C₂×D₄	—	134	1	no	—
271	C₂×D₄	—	134	2	no	—
272	C₂²×C₄	—	268	2	no	—
273	C₂²×C₄	—	268	2	no	—
274	C₂²×C₄	—	134	1	no	—
275	C₂²×C₄	—	134	2	no	—
276	C₂²×C₄	—	134	2	no	—
277	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
279	C₂²×C₄	#11761627	1	1	yes	characteristic
280	Q₁₆	—	4	2	no	—
281	Q₁₆	—	4	2	no	—
282	M₁₆	—	268	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	268	4	no	—
286	C₂×C₈	—	268	2	no	—
287	C₂×C₈	—	268	4	no	—
288	C₂×C₈	—	134	4	no	—
289	C₂×C₈	—	134	2	no	—
290	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	67	2	no	—
301	C₄⋊C₄	—	134	1	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	134	2	no	—
304	C₂²⋊C₄	—	134	2	no	—
305	C₄²	—	4	2	no	—
306	C₄²	D₄×D₆₇	1	1	yes	characteristic
307	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
308	C₄²	#11761627	1	1	yes	characteristic
309	C₄²	C₂³×D₆₇	1	1	yes	characteristic
310	C₂³	—	268	2	no	—
311	C₂³	—	268	2	no	—
312	C₂³	—	134	2	no	—
313	C₂³	—	134	4	no	—
314	C₂³	#14524014	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	8	2	no	—
318	D₄	—	67	4	no	—
319	D₄	—	134	4	no	—
320	D₄	—	134	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	268	4	no	—
327	C₂×C₄	—	268	2	no	—
328	C₂×C₄	—	268	2	no	—
329	C₂×C₄	—	268	2	no	—
330	C₂×C₄	—	134	2	no	—
331	C₂×C₄	#14524014	1	1	yes	characteristic
332	C₂×C₄	#14524014	1	1	yes	characteristic
333	C₂×C₄	#14524014	1	1	yes	characteristic
334	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
335	C₂×C₄	#14524066	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	268	4	no	—
343	C₈	—	268	2	no	—
344	C₈	#14524068	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	268	8	no	—
348	C₂²	—	268	2	no	—
349	C₂²	—	268	2	no	—
350	C₂²	—	134	4	no	—
351	C₂²	#16863634	1	2	yes	—
352	C₂²	C₂×#14524014	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	—	268	2	no	—
359	C₄	#16863634	1	2	yes	—
360	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
361	C₄	C₂×#14524066	1	1	yes	characteristic
362	C₄	C₂×#14524068	1	1	yes	characteristic
363	C₄	#16864194	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	268	4	no	—
366	C₂	#22569669	1	1	yes	characteristic
367	C₂	C₂×#16863634	1	1	yes	characteristic
368	C₂	#22569925	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂²×D₂₆₈	D₄
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#19256
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #22559657	C₂	semidirect
4. #22559605	C₂	semidirect
5. #22559571	C₂	semidirect
6. #22558968	C₂	semidirect
7. #22558899	C₂	semidirect
8. C₆₇×#3170	C₂	semidirect
23. #16863149	C₂²	semidirect
24. #16863149	C₂²	semidirect
25. #16863145	C₂²	semidirect
26. #16863142	C₂²	semidirect
28. #500×C₆₇	C₂²	semidirect
29. #494×C₆₇	C₂²	semidirect
30. #487×C₆₇	C₂²	semidirect
83. C₂×D₂₆₈	Q₁₆	semidirect
185. #3170	D₆₇	semidirect
193. C₆₇	#19256	semidirect
197. #500	C₂×D₆₇	semidirect
198. #494	C₂×D₆₇	semidirect
199. #487	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22570002	2
C₂×C₈×D₂₆₈	2
#22559657	2
#22559605	2
#22559571	2
#22558968	2
#22558899	2
C₆₇×#3170	2
#19256	67