Subgroups — #30749000

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560229	C₂	1	2	yes	maximal
2	C₄×#14523977	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558111	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	1	no	—
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523970	—	2	2	no	—
9	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	#16863159	—	2	1	no	—
11	#16863155	—	2	2	no	—
12	#16863149	—	2	1	no	—
13	#16863149	—	2	1	no	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
17	C₈×D₄×C₆₇	—	2	1	no	—
18	C₈×D₄×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
20	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#431×C₆₇	—	2	1	no	—
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄×Dic₆₇	—	2	2	no	—
28	#14523977	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	#14523970	—	4	4	no	—
32	C₆₇⋊#95	—	2	2	no	—
33	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₆₇	—	2	1	no	—
41	C₄×D₄×C₆₇	—	2	1	no	—
42	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₆₇	Q₈	1	1	yes	characteristic
44	#117×C₆₇	—	2	1	no	—
45	#117×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	—	2	1	no	—
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₆₇	—	2	1	no	—
53	C₂²⋊C₈×C₆₇	—	2	1	no	—
54	C₂²⋊C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₂²×Dic₆₇	—	4	2	no	—
63	#11761612	—	4	2	no	—
64	#11761612	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₄⋊Dic₆₇	—	2	4	no	—
67	C₄⋊Dic₆₇	—	4	2	no	—
68	C₄×Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	#41×C₆₇	—	2	1	no	—
71	#41×C₆₇	—	2	1	no	—
72	#41×C₆₇	D₈	1	2	yes	—
73	#41×C₆₇	Q₁₆	1	2	yes	—
74	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂×D₄×C₆₇	—	2	1	no	—
76	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	—	2	1	no	—
79	C₂²×C₄×C₆₇	—	2	1	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	1	no	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	4	1	no	—
87	C₂×C₈×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	4	2	no	—
89	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂²⋊C₄×C₆₇	—	2	1	no	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₄²×C₆₇	—	2	1	no	—
103	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₆₇	#41	1	1	yes	characteristic
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	#121	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	1	no	—
110	Q₈×C₆₇	#107	1	2	yes	complemented
111	D₄×C₆₇	—	2	2	no	—
112	D₄×C₆₇	—	2	2	no	—
113	D₄×C₆₇	—	2	1	no	—
114	D₄×C₆₇	#107	1	2	yes	complemented
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
136	C₂×Dic₆₇	—	4	4	no	—
137	C₂×Dic₆₇	—	8	2	no	—
138	C₂×Dic₆₇	—	8	2	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	4	1	no	—
143	C₈×C₆₇	—	4	1	no	—
144	C₈×C₆₇	#104	1	2	yes	complemented
145	C₆₇⋊C₈	—	8	2	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	4	1	no	—
150	C₂²×C₆₇	—	4	1	no	—
151	C₂²×C₆₇	#380	1	1	yes	characteristic
152	C₂²×C₆₇	#466	1	1	yes	characteristic
153	C₂²×C₆₇	#467	1	1	yes	characteristic
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	4	1	no	—
159	C₄×C₆₇	—	4	1	no	—
160	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
161	C₄×C₆₇	#421	1	1	yes	characteristic
162	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
163	C₄×C₆₇	#427	1	1	yes	characteristic
164	C₄×C₆₇	#466	1	1	yes	characteristic
165	C₄×C₆₇	#467	1	1	yes	characteristic
166	Dic₆₇	—	8	4	no	—
167	#8667	—	67	1	no	maximal, Sylow
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	2	1	no	—
170	C₂×C₆₇	—	4	2	no	—
171	C₂×C₆₇	#1806	1	1	yes	characteristic
172	C₂×C₆₇	#1808	1	1	yes	characteristic
173	C₂×C₆₇	#1814	1	1	yes	characteristic
174	C₈×#41	D₆₇	1	1	yes	characteristic
175	#1806	—	67	2	no	—
176	C₄×#107	—	67	1	no	—
177	#1567	—	67	1	no	—
178	#1303	—	67	2	no	—
179	C₆₇	#8667	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
181	C₈×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
182	C₈×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
183	C₈×D₄	—	2	1	no	—
184	C₈×D₄	—	2	1	no	—
185	C₈×D₄	Dic₆₇	1	1	yes	characteristic, complemented
186	C₈×D₄	Dic₆₇	1	1	yes	characteristic, complemented
187	#431	—	2	1	no	—
188	#431	C₂×D₆₇	1	1	yes	characteristic
189	C₂×#107	—	134	1	no	—
190	#423	—	67	2	no	—
191	C₂×#104	—	134	2	no	—
192	C₂×C₄×C₈	—	2	1	no	—
193	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₄×C₄⋊C₄	—	67	2	no	—
195	#340	—	134	1	no	—
196	#339	—	134	2	no	—
197	#335	—	134	1	no	—
198	#335	—	134	1	no	—
199	C₂×#41	D₂₆₈	1	1	yes	characteristic
200	C₂×M₁₆	—	134	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
205	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
206	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	Dic₁₃₄	1	1	yes	characteristic
210	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	#117	D₂₆₈	1	1	yes	characteristic
213	C₂×C₄⋊C₄	—	134	2	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	134	2	no	—
216	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
217	#107	—	134	4	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	134	2	no	—
221	C₄⋊C₈	—	134	2	no	—
222	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
223	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
224	#104	—	268	4	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
232	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
235	#95	—	134	2	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	D₅₃₆	1	2	yes	—
239	#41	Dic₂₆₈	1	2	yes	—
240	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	268	2	no	—
247	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
249	C₂²×C₄	#11761628	1	1	yes	characteristic
250	M₁₆	—	268	2	no	—
251	M₁₆	—	268	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	268	2	no	—
259	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	268	2	no	—
266	C₄⋊C₄	—	134	4	no	—
267	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
268	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	268	2	no	—
276	C₄²	—	268	2	no	—
277	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
279	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
280	C₄²	#11761628	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	#14524017	1	1	yes	characteristic
283	Q₈	—	2	1	no	—
284	Q₈	#14523977	1	2	yes	complemented
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	1	no	—
288	D₄	#14523977	1	2	yes	complemented
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	268	4	no	—
301	C₂×C₄	—	536	2	no	—
302	C₂×C₄	—	536	2	no	—
303	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
305	C₂×C₄	#14524002	1	1	yes	characteristic
306	C₂×C₄	#14524017	1	1	yes	characteristic
307	C₂×C₄	#14524022	1	1	yes	characteristic
308	C₂×C₄	#14524022	1	1	yes	characteristic
309	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
310	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
311	C₂×C₄	#14524065	1	1	yes	characteristic
312	C₂×C₄	#14524065	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	536	2	no	—
319	C₈	#14523970	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#16863256	1	1	yes	characteristic
326	C₂²	#16863632	1	1	yes	characteristic
327	C₂²	#16863637	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	536	4	no	—
335	C₄	C₂×#14523970	1	1	yes	characteristic
336	C₄	#16863413	1	1	yes	characteristic
337	C₄	C₂×#14523977	1	1	yes	characteristic
338	C₄	#16863434	1	1	yes	characteristic
339	C₄	#16863632	1	1	yes	characteristic
340	C₄	#16863637	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#22560229	1	1	yes	characteristic
345	C₂	#22560238	1	1	yes	characteristic
346	C₂	#22560263	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#8667
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14523977	C₂	semidirect
3. #22558968	C₂	semidirect
4. #22558111	C₂	semidirect
15. C₈×Q₈×C₆₇	C₄	semidirect
16. C₈×Q₈×C₆₇	C₄	semidirect
19. C₈×D₄×C₆₇	C₄	semidirect
20. C₈×D₄×C₆₇	C₄	semidirect
110. Q₈×C₆₇	#107	semidirect
114. D₄×C₆₇	#107	semidirect
144. C₈×C₆₇	#104	semidirect
179. C₆₇	#8667	semidirect
181. C₈×Q₈	Dic₆₇	semidirect
182. C₈×Q₈	Dic₆₇	semidirect
185. C₈×D₄	Dic₆₇	semidirect
186. C₈×D₄	Dic₆₇	semidirect
284. Q₈	#14523977	semidirect
288. D₄	#14523977	semidirect
319. C₈	#14523970	semidirect

Maximal subgroups

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

subgroup	index
#22560229	2
C₄×#14523977	2
#22558968	2
#22558111	2
C₈×#41×C₆₇	2
#8667	67