Subgroups — #30826034

Order 17152.

658 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	#22570241	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863526	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic
4	#22559101	C₂	1	4	yes	maximal, complemented
5	#22559047	C₂	1	4	yes	maximal, complemented
6	#22558980	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558639	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558609	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524049	—	2	1	no	—
11	#16863946	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	#16863715	C₄	1	2	yes	—
14	#16863682	C₄	1	2	yes	—
15	#16863597	C₄	1	2	yes	—
16	#16863537	C₄	1	2	yes	—
17	#16863526	—	2	1	no	—
18	#16863526	—	2	1	no	—
19	#16863526	C₂²	1	2	yes	—
20	#16863526	C₂²	1	2	yes	—
21	C₄×#11761617	C₂²	1	4	yes	—
22	#16863475	—	2	1	no	—
23	#16863475	C₂²	1	2	yes	—
24	C₈×D₂₆₈	—	2	1	no	—
25	C₈×D₂₆₈	C₂²	1	2	yes	—
26	#16863420	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×Dic₆₇⋊C₈	—	2	1	no	—
29	#16863362	—	2	1	no	—
30	C₂×#14523953	—	2	1	no	—
31	#16863147	C₂²	1	4	yes	—
32	#16863128	C₂²	1	4	yes	—
33	#16863125	C₂²	1	4	yes	—
34	#431×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#322×C₆₇	C₂²	1	2	yes	—
38	C₈²×C₆₇	C₂²	1	2	yes	—
39	C₂×#11761628	—	2	1	no	—
40	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
41	C₂×#11761617	C₂×C₄	1	4	yes	—
42	#14524049	—	4	4	no	—
43	C₂×#11761612	—	2	1	no	—
44	C₂×#11761611	—	2	2	no	—
45	C₂²×C₆₇⋊C₈	—	2	1	no	—
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×Dic₁₃₄	—	2	1	no	—
53	C₄×Dic₁₃₄	—	2	1	no	—
54	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃₄	C₂³	1	2	yes	—
56	C₄×D₂₆₈	—	2	1	no	—
57	C₄×D₂₆₈	C₂³	1	2	yes	—
58	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×Dic₆₇⋊C₄	—	2	1	no	—
60	C₂×C₄×Dic₆₇	—	2	1	no	—
61	#14523975	C₂×C₄	1	4	yes	—
62	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
63	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	#14523968	C₂×C₄	1	4	yes	—
69	#14523965	C₂×C₄	1	4	yes	—
70	(C₂×D₆₇)⋊C₈	—	2	2	no	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	#14523953	—	2	2	no	—
73	#14523953	—	2	2	no	—
74	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	1	no	—
78	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	2	yes	—
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#11761628	—	4	2	no	—
88	#11761628	—	4	2	no	—
89	C₂×Dic₁₃₄	—	2	1	no	—
90	C₂×Dic₁₃₄	—	2	1	no	—
91	C₂×Dic₁₃₄	—	4	2	no	—
92	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
93	C₂×D₂₆₈	—	2	1	no	—
94	C₂×D₂₆₈	—	2	1	no	—
95	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₆₇	—	2	2	no	—
97	C₂²×Dic₆₇	—	2	1	no	—
98	C₂²×Dic₆₇	—	2	1	no	—
99	C₂×C₄×D₆₇	—	2	2	no	—
100	#11761617	—	4	4	no	—
101	#11761612	—	4	2	no	—
102	#11761611	—	4	4	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	4	2	no	—
106	C₈×D₆₇	—	4	4	no	—
107	C₄⋊Dic₆₇	—	2	1	no	—
108	C₄⋊Dic₆₇	—	2	1	no	—
109	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
111	Dic₆₇⋊C₄	—	2	2	no	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	(C₂×D₆₇)⋊C₄	—	2	2	no	—
114	(C₂×D₆₇)⋊C₄	—	2	2	no	—
115	C₄×Dic₆₇	—	2	2	no	—
116	C₄×Dic₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
123	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₆₇	#135	1	2	yes	—
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₆₇	—	4	4	no	—
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	4	2	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	D₂₆₈	—	2	4	no	—
151	D₂₆₈	—	4	2	no	—
152	D₂₆₈	—	4	2	no	—
153	C₄×D₆₇	—	4	2	no	—
154	C₄×D₆₇	—	4	2	no	—
155	Dic₁₃₄	—	2	2	no	—
156	Dic₁₃₄	—	2	2	no	—
157	Dic₁₃₄	—	4	2	no	—
158	Dic₁₃₄	—	4	2	no	—
159	C₈×C₆₇	—	2	4	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
164	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
165	C₆₇⋊C₈	—	4	2	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	C₄×SD₁₆	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	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
178	C₄×C₆₇	#568	1	2	yes	—
179	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
180	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
181	Dic₆₇	—	4	2	no	—
182	Dic₆₇	—	4	2	no	—
183	Dic₆₇	—	4	2	no	—
184	#17929	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	—	2	2	no	—
186	C₂×C₆₇	#2930	1	1	yes	characteristic
187	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
188	C₂×C₆₇	#2955	1	1	yes	characteristic
189	D₆₇	—	8	2	no	—
190	#3103	—	67	1	no	—
191	C₂×#446	—	67	1	no	—
192	#2931	—	67	1	no	—
193	#1590	—	67	4	no	—
194	#1582	—	67	4	no	—
195	#1570	—	67	1	no	—
196	#1487	—	67	1	no	—
197	#1480	—	67	1	no	—
198	#1450	D₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#17929	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	134	1	no	—
201	#532	—	67	1	no	—
202	C₂×C₄×Q₈	—	67	1	no	—
203	#481	—	67	2	no	—
204	#475	—	67	2	no	—
205	#460	—	67	2	no	—
206	#448	—	67	2	no	—
207	#446	—	67	2	no	—
208	#446	—	67	2	no	—
209	#446	—	134	1	no	—
210	#446	—	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	C₂×D₆₇	1	1	yes	characteristic
217	#431	C₂×D₆₇	1	1	yes	characteristic
218	#423	—	67	1	no	—
219	C₂×C₄⋊C₈	—	67	1	no	—
220	C₂×C₄⋊C₈	—	134	1	no	—
221	#408	—	134	1	no	—
222	C₂×#97	—	134	1	no	—
223	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
224	#334	—	67	4	no	—
225	Q₈⋊C₈	—	67	4	no	—
226	D₄⋊C₈	—	67	4	no	—
227	#322	C₂×D₆₇	1	2	yes	—
228	C₈²	C₂×D₆₇	1	2	yes	—
229	C₂×#41	—	134	1	no	—
230	C₂²×Q₈	—	67	1	no	—
231	C₂×SD₁₆	—	67	4	no	—
232	#131	—	268	4	no	—
233	C₂×M₁₆	—	134	1	no	—
234	C₂×M₁₆	—	134	2	no	—
235	C₂²×C₈	—	134	1	no	—
236	C₂²×C₈	—	134	2	no	—
237	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
238	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
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	—	67	2	no	—
245	C₄×Q₈	—	67	2	no	—
246	C₄×Q₈	—	134	1	no	—
247	C₄×Q₈	—	134	1	no	—
248	C₄×D₄	—	67	2	no	—
249	C₄×D₄	—	134	1	no	—
250	C₂×C₄⋊C₄	—	67	1	no	—
251	C₂×C₄⋊C₄	—	134	1	no	—
252	C₂×C₄²	—	134	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	—	67	2	no	—
257	C₄⋊C₈	—	67	2	no	—
258	C₄⋊C₈	—	134	2	no	—
259	C₄⋊C₈	—	134	2	no	—
260	C₄⋊C₈	—	134	1	no	—
261	C₄⋊C₈	—	134	1	no	—
262	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
263	#103	—	67	4	no	—
264	#102	—	67	4	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	134	2	no	—
267	C₂²⋊C₈	—	134	2	no	—
268	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
269	#97	—	134	2	no	—
270	#97	—	134	2	no	—
271	C₄×C₈	D₂₆₈	1	1	yes	characteristic
272	C₄×C₈	D₂₆₈	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₆₇	1	2	yes	—
274	C₄×C₈	C₂²×D₆₇	1	2	yes	—
275	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
277	#41	—	268	2	no	—
278	#41	—	268	2	no	—
279	C₂×Q₈	—	268	2	no	—
280	C₂×Q₈	—	67	2	no	—
281	C₂×Q₈	—	134	1	no	—
282	C₂×Q₈	—	134	1	no	—
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	—	134	1	no	—
288	C₂²×C₄	—	134	1	no	—
289	C₂²×C₄	—	134	2	no	—
290	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
292	C₂²×C₄	#11761628	1	1	yes	characteristic
293	SD₁₆	—	268	4	no	—
294	M₁₆	—	268	2	no	—
295	M₁₆	—	268	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	268	2	no	—
299	C₂×C₈	—	268	4	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	#11761617	1	4	yes	complemented
303	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
304	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
306	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
307	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
308	C₄⋊C₄	—	67	2	no	—
309	C₄⋊C₄	—	67	2	no	—
310	C₄⋊C₄	—	134	1	no	—
311	C₄⋊C₄	—	134	2	no	—
312	C₄⋊C₄	—	134	1	no	—
313	C₄⋊C₄	—	134	2	no	—
314	C₂²⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₄²	—	134	2	no	—
317	C₄²	—	134	2	no	—
318	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
319	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
320	C₄²	#11761628	1	1	yes	characteristic
321	C₄²	C₂³×D₆₇	1	1	yes	characteristic
322	C₂³	—	134	2	no	—
323	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
324	Q₈	—	268	2	no	—
325	Q₈	—	268	2	no	—
326	Q₈	—	134	2	no	—
327	Q₈	—	134	2	no	—
328	D₄	—	268	2	no	—
329	D₄	—	268	4	no	—
330	D₄	—	268	2	no	—
331	D₄	—	134	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	268	2	no	—
336	C₂×C₄	—	268	2	no	—
337	C₂×C₄	—	268	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	—	134	2	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
342	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
343	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761617	1	2	yes	—
345	C₂×C₄	#14524065	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
348	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	268	2	no	—
354	C₈	—	268	2	no	—
355	C₈	C₂×#11761617	1	2	yes	—
356	C₈	C₂×#11761617	1	2	yes	—
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	C₄×#11761617	1	2	yes	—
363	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	268	2	no	—
368	C₄	—	268	2	no	—
369	C₄	—	268	2	no	—
370	C₄	C₄×#11761617	1	2	yes	—
371	C₄	#16864161	1	2	yes	—
372	C₄	C₂²×#11761617	1	1	yes	characteristic
373	C₄	C₂×#14524065	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	536	2	no	—
376	C₂	#22568914	1	1	yes	characteristic
377	C₂	C₂×C₄×#11761617	1	1	yes	characteristic
378	C₂	#22569113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1450	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#17929
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₂×#16863526	C₂	semidirect
4. #22559101	C₂	semidirect
5. #22559047	C₂	semidirect
6. #22558980	C₂	semidirect
7. #22558639	C₂	semidirect
9. C₆₇×#1450	C₂	semidirect
122. C₂×C₈×C₆₇	SD₁₆	semidirect
198. #1450	D₆₇	semidirect
199. C₆₇	#17929	semidirect
302. C₂×C₈	#11761617	semidirect

Maximal subgroups

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

subgroup	index
#22570241	2
C₂×#16863526	2
#22568922	2
#22559101	2
#22559047	2
#22558980	2
#22558639	2
#22558609	2
C₆₇×#1450	2
#17929	67