Subgroups — #30825685

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569322	C₂	1	2	yes	maximal, complemented
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22559113	C₂	1	4	yes	maximal, complemented
4	#22559107	C₂	1	2	yes	maximal, complemented
5	#22559095	C₂	1	2	yes	maximal, complemented
6	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558597	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₄	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	2	yes	—
11	#16863526	C₂²	1	4	yes	—
12	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
13	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
14	C₄×#11761617	C₄	1	2	yes	—
15	C₄×#11761617	C₂²	1	2	yes	—
16	C₄×D₅₃₆	C₄	1	1	yes	characteristic
17	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
18	#16863475	C₂²	1	4	yes	—
19	#16863464	—	2	4	no	—
20	#16863452	—	2	2	no	—
21	#16863434	C₄	1	1	yes	characteristic
22	#16863418	—	2	2	no	—
23	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
24	#16863393	C₄	1	2	yes	—
25	#16863149	C₂²	1	2	yes	—
26	#16863149	C₂²	1	2	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#16863125	C₂²	1	4	yes	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	#431×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#322×C₆₇	C₂²	1	2	yes	—
33	C₈²×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761628	C₂×C₄	1	2	yes	—
35	#14524065	—	2	4	no	—
36	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#11761617	C₂×C₄	1	2	yes	—
38	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#11761611	—	2	4	no	—
40	C₂²×C₆₇⋊C₈	—	2	2	no	—
41	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₃₄	C₂³	1	2	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
44	C₄×D₂₆₈	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₆₇	—	2	2	no	—
46	C₄×C₂²⋊D₆₇	—	2	2	no	—
47	#14524002	C₂×C₄	1	2	yes	—
48	#14524000	—	2	2	no	—
49	C₄²×D₆₇	—	2	2	no	—
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	#14523977	C₂×C₄	1	1	yes	characteristic
52	#14523975	C₂×C₄	1	2	yes	—
53	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	#14523968	C₂×C₄	1	2	yes	—
57	#14523968	C₂×C₄	1	2	yes	—
58	#14523965	C₂×C₄	1	2	yes	—
59	#14523965	C₂×C₄	1	2	yes	—
60	(C₂×D₆₇)⋊C₈	—	2	4	no	—
61	#14523953	—	2	4	no	—
62	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
67	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₆₇	—	2	1	no	—
69	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
70	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂³	1	2	yes	—
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	2	yes	—
76	#11761628	—	2	4	no	—
77	#11761628	—	2	4	no	—
78	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
79	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₆₇	—	2	2	no	—
81	C₂×C₄×D₆₇	—	2	2	no	—
82	C₂×C₄×D₆₇	—	2	2	no	—
83	C₂×C₄×D₆₇	—	2	2	no	—
84	Dic₂₆₈	—	2	2	no	—
85	#11761617	—	2	4	no	—
86	D₅₃₆	—	2	2	no	—
87	#11761611	—	8	4	no	—
88	C₂×C₆₇⋊C₈	—	2	4	no	—
89	C₂×C₆₇⋊C₈	—	8	2	no	—
90	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
91	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
92	Dic₆₇⋊C₄	—	2	2	no	—
93	Dic₆₇⋊C₄	—	2	2	no	—
94	C₂²⋊Dic₆₇	—	2	2	no	—
95	(C₂×D₆₇)⋊C₄	—	2	2	no	—
96	(C₂×D₆₇)⋊C₄	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₄×Dic₆₇	—	2	2	no	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	M₁₆	1	4	yes	complemented
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	#41	1	1	yes	characteristic
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₆₇	#131	1	2	yes	—
126	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₆₇	—	4	4	no	—
130	C₂×Dic₆₇	—	2	2	no	—
131	C₂×Dic₆₇	—	2	2	no	—
132	C₂×Dic₆₇	—	2	2	no	—
133	D₂₆₈	—	2	2	no	—
134	D₂₆₈	—	2	2	no	—
135	C₄×D₆₇	—	4	2	no	—
136	C₄×D₆₇	—	4	2	no	—
137	C₄×D₆₇	—	4	4	no	—
138	C₄×D₆₇	—	8	2	no	—
139	Dic₁₃₄	—	2	2	no	—
140	Dic₁₃₄	—	2	2	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
147	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
148	C₆₇⋊C₈	—	4	4	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	#436	1	2	yes	—
152	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×D₆₇	—	4	2	no	—
154	C₂×D₆₇	—	4	2	no	—
155	C₂×D₆₇	—	4	2	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	#436	1	2	yes	—
160	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
162	C₄×C₆₇	#576	1	1	yes	characteristic
163	C₄×C₆₇	#578	1	1	yes	characteristic
164	Dic₆₇	—	4	2	no	—
165	Dic₆₇	—	4	2	no	—
166	Dic₆₇	—	4	2	no	—
167	#17898	—	67	1	no	maximal, Sylow
168	C₂×C₆₇	—	2	2	no	—
169	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
170	C₂×C₆₇	#2951	1	1	yes	characteristic
171	C₂×C₆₇	#2957	1	1	yes	characteristic
172	D₆₇	—	8	2	no	—
173	#2989	—	67	2	no	—
174	C₄×#135	—	67	1	no	—
175	#1592	—	67	4	no	—
176	#1591	—	67	2	no	—
177	#1589	—	67	2	no	—
178	C₂×#335	—	67	1	no	—
179	#1477	—	67	2	no	—
180	#1450	D₆₇	1	1	yes	characteristic, complemented
181	C₆₇	#17898	1	1	yes	characteristic, complemented, Sylow
182	C₂×#135	—	67	1	no	—
183	C₄×#41	—	67	2	no	—
184	#446	—	67	4	no	—
185	C₄×Q₁₆	—	67	1	no	—
186	C₄×Q₁₆	—	67	1	no	—
187	C₄×SD₁₆	—	67	2	no	—
188	C₄×SD₁₆	—	67	2	no	—
189	C₄×D₈	—	67	1	no	—
190	C₄×D₈	—	67	1	no	—
191	#436	—	67	4	no	—
192	#435	—	134	4	no	—
193	#432	—	134	2	no	—
194	#431	C₂×D₆₇	1	1	yes	characteristic
195	#431	C₂×D₆₇	1	1	yes	characteristic
196	#427	—	67	1	no	—
197	#423	—	134	2	no	—
198	C₂×C₄⋊C₈	—	67	2	no	—
199	#416	—	67	2	no	—
200	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
201	#335	—	67	2	no	—
202	#335	—	67	2	no	—
203	Q₈⋊C₈	—	67	4	no	—
204	D₄⋊C₈	—	67	4	no	—
205	#322	C₂×D₆₇	1	2	yes	—
206	C₈²	C₂×D₆₇	1	2	yes	—
207	C₂×#41	—	67	2	no	—
208	#135	—	134	4	no	—
209	C₂×Q₁₆	—	67	1	no	—
210	C₂×SD₁₆	—	67	2	no	—
211	C₂×D₈	—	67	1	no	—
212	C₂×M₁₆	—	134	4	no	—
213	C₂²×C₈	—	134	2	no	—
214	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
216	C₄×Q₈	—	67	2	no	—
217	C₄×Q₈	—	67	2	no	—
218	C₄×D₄	—	67	2	no	—
219	C₄×D₄	—	67	2	no	—
220	C₄×D₄	—	134	2	no	—
221	C₄×D₄	—	134	2	no	—
222	#117	—	67	2	no	—
223	#117	—	134	2	no	—
224	C₂×C₄²	—	134	2	no	—
225	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
226	#107	—	67	1	no	—
227	#107	—	67	1	no	—
228	#106	—	67	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	67	4	no	—
231	C₄⋊C₈	—	134	4	no	—
232	C₄⋊C₈	—	134	2	no	—
233	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
234	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
235	#103	—	67	2	no	—
236	#103	—	67	2	no	—
237	#102	—	67	2	no	—
238	#102	—	67	2	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	C₂²⋊C₈	—	134	4	no	—
241	C₂²⋊C₈	D₂₆₈	1	2	yes	—
242	#97	—	134	4	no	—
243	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
244	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₆₇	1	2	yes	—
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₆₇	1	2	yes	—
249	#41	—	134	4	no	—
250	#41	—	134	4	no	—
251	C₂×Q₈	—	67	2	no	—
252	C₂×D₄	—	67	2	no	—
253	C₂×D₄	—	134	2	no	—
254	C₂²×C₄	—	134	2	no	—
255	C₂²×C₄	—	134	2	no	—
256	C₂²×C₄	—	134	2	no	—
257	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
259	C₂²×C₄	#11761628	1	1	yes	characteristic
260	Q₁₆	—	134	2	no	—
261	SD₁₆	—	134	4	no	—
262	D₈	—	134	2	no	—
263	M₁₆	—	536	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	536	2	no	—
267	C₂×C₈	—	134	4	no	—
268	C₂×C₈	#11761611	1	4	yes	complemented
269	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
273	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
274	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
275	C₄⋊C₄	—	67	2	no	—
276	C₄⋊C₄	—	67	2	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₄⋊C₄	—	134	2	no	—
279	C₂²⋊C₄	—	134	2	no	—
280	C₂²⋊C₄	—	134	2	no	—
281	C₂²⋊C₄	—	134	2	no	—
282	C₄²	—	134	2	no	—
283	C₄²	—	134	2	no	—
284	C₄²	—	134	2	no	—
285	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
286	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
287	C₄²	#11761628	1	1	yes	characteristic
288	C₄²	C₂³×D₆₇	1	1	yes	characteristic
289	C₂³	—	134	2	no	—
290	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
291	Q₈	—	134	2	no	—
292	Q₈	—	134	2	no	—
293	D₄	—	268	4	no	—
294	D₄	—	134	2	no	—
295	D₄	—	134	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	268	2	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	4	no	—
302	C₂×C₄	—	536	2	no	—
303	C₂×C₄	—	134	2	no	—
304	C₂×C₄	—	134	2	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
307	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
308	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
309	C₂×C₄	C₂×#11761611	1	2	yes	—
310	C₂×C₄	#14524049	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
313	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	268	4	no	—
320	C₈	C₂×#11761611	1	2	yes	—
321	C₈	C₂×#11761611	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	268	2	no	—
325	C₂²	—	268	2	no	—
326	C₂²	—	268	2	no	—
327	C₂²	#16863475	1	2	yes	—
328	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	268	2	no	—
333	C₄	—	268	2	no	—
334	C₄	—	268	2	no	—
335	C₄	#16863475	1	2	yes	—
336	C₄	C₂²×#11761611	1	1	yes	characteristic
337	C₄	C₂×#14524049	1	1	yes	characteristic
338	C₄	#16864201	1	1	yes	characteristic
339	C₄	#16864213	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	536	2	no	—
342	C₂	C₂×#16863475	1	1	yes	characteristic
343	C₂	#22569086	1	1	yes	characteristic
344	C₂	#22569127	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	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#17898
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #22569322	C₂	semidirect
3. #22559113	C₂	semidirect
4. #22559107	C₂	semidirect
5. #22559095	C₂	semidirect
6. C₂×#16863149	C₂	semidirect
7. #22558597	C₂	semidirect
8. C₆₇×#1450	C₂	semidirect
105. C₂×C₈×C₆₇	M₁₆	semidirect
180. #1450	D₆₇	semidirect
181. C₆₇	#17898	semidirect
268. C₂×C₈	#11761611	semidirect

Maximal subgroups

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

subgroup	index
#22569322	2
C₄×#14524065	2
#22559113	2
#22559107	2
#22559095	2
C₂×#16863149	2
#22558597	2
C₆₇×#1450	2
#17898	67