Subgroups — #30886713

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569141	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569113	C₂	1	1	yes	maximal, characteristic
3	#22561790	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561785	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561661	C₂	1	2	yes	maximal, complemented
6	#22561516	C₂	1	1	yes	maximal, characteristic
7	#22561497	C₂	1	1	yes	maximal, characteristic
8	#22561380	C₂	1	2	yes	maximal, complemented
9	#22560048	C₂	1	2	yes	maximal, complemented
10	#22560010	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864196	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	2	yes	—
14	#16863949	C₂²	1	1	yes	characteristic
15	#16863524	—	2	2	no	—
16	C₈×Dic₁₃₄	—	2	2	no	—
17	#16863519	—	2	2	no	—
18	#16863514	—	2	2	no	—
19	#16863445	C₂²	1	1	yes	characteristic
20	#16863424	C₂²	1	1	yes	characteristic
21	#16863423	C₂²	1	1	yes	characteristic
22	#16863419	C₂²	1	1	yes	characteristic
23	#16863414	C₂²	1	2	yes	—
24	C₂×#14523969	C₂²	1	2	yes	—
25	#16863390	C₂²	1	2	yes	—
26	#16863378	C₂²	1	2	yes	—
27	#16863369	C₂²	1	2	yes	—
28	#16863351	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
30	#16863168	C₂²	1	2	yes	—
31	#16863166	C₂²	1	2	yes	—
32	#16863160	C₂²	1	2	yes	—
33	#16863155	C₂²	1	2	yes	—
34	#405×C₆₇	C₂²	1	2	yes	—
35	#405×C₆₇	C₂²	1	1	yes	characteristic
36	#338×C₆₇	C₂²	1	1	yes	characteristic
37	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761628	C₂³	1	2	yes	—
40	#14524076	—	2	2	no	—
41	#14524071	—	2	2	no	—
42	#14524065	—	2	2	no	—
43	#14524049	—	2	2	no	—
44	#14524049	—	2	2	no	—
45	#14524049	—	2	2	no	—
46	#14524049	—	2	2	no	—
47	C₂×#11761612	C₂³	1	1	yes	characteristic
48	C₂×#11761612	C₂³	1	2	yes	—
49	C₂×#11761611	—	2	2	no	—
50	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₆₇	—	2	2	no	—
52	#14524042	—	2	1	no	—
53	#14524034	—	2	1	no	—
54	#14524020	—	2	2	no	—
55	C₄×Dic₁₃₄	—	2	2	no	—
56	#14524002	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
58	#14523979	—	2	2	no	—
59	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
60	Dic₆₇⋊C₈	—	2	2	no	—
61	Dic₆₇⋊C₈	—	2	2	no	—
62	#14523971	—	2	4	no	—
63	#14523969	—	2	2	no	—
64	#14523969	—	2	2	no	—
65	#14523966	—	2	2	no	—
66	#14523963	—	2	2	no	—
67	#14523961	—	2	2	no	—
68	#14523959	—	2	2	no	—
69	#14523957	—	2	4	no	—
70	#14523953	—	2	1	no	—
71	C₈×Dic₆₇	—	2	1	no	—
72	C₈×Dic₆₇	—	2	1	no	—
73	C₈×Dic₆₇	—	2	1	no	—
74	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	#117×C₆₇	C₂³	1	2	yes	—
78	#117×C₆₇	C₂³	1	1	yes	characteristic
79	#97×C₆₇	—	2	2	no	—
80	#97×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	D₄	1	2	yes	—
83	#11761628	—	2	2	no	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	#11761628	—	2	2	no	—
87	C₂×Dic₁₃₄	—	2	2	no	—
88	C₂×Dic₁₃₄	#41	1	2	yes	—
89	C₂×D₂₆₈	#41	1	2	yes	—
90	C₂×C₂²⋊D₆₇	—	2	2	no	—
91	C₂²×Dic₆₇	—	2	1	no	—
92	C₂×C₄×D₆₇	—	2	2	no	—
93	Dic₂₆₈	—	2	2	no	—
94	#11761617	—	2	4	no	—
95	D₅₃₆	—	2	2	no	—
96	#11761612	—	2	2	no	—
97	#11761612	—	2	2	no	—
98	#11761612	—	2	2	no	—
99	#11761611	—	4	2	no	—
100	#11761611	—	4	2	no	—
101	C₂×C₆₇⋊C₈	—	2	1	no	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₈×D₆₇	—	4	2	no	—
106	C₈×D₆₇	—	4	2	no	—
107	C₄⋊Dic₆₇	—	2	2	no	—
108	C₄⋊Dic₆₇	—	2	1	no	—
109	Dic₆₇⋊C₄	—	2	2	no	—
110	Dic₆₇⋊C₄	—	2	2	no	—
111	C₂²⋊Dic₆₇	#41	1	2	yes	—
112	C₄×Dic₆₇	—	2	1	no	—
113	C₄×Dic₆₇	—	2	1	no	—
114	C₄×Dic₆₇	#41	1	1	yes	characteristic
115	C₄×Dic₆₇	#41	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	#41	1	2	yes	—
125	C₂×C₈×C₆₇	#41	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₂²⋊C₄×C₆₇	—	2	1	no	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	C₂×D₄	1	2	yes	—
132	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
140	C₂²⋊D₆₇	—	4	2	no	—
141	C₂²⋊D₆₇	—	4	2	no	—
142	C₂×Dic₆₇	—	2	1	no	—
143	C₂×Dic₆₇	—	2	1	no	—
144	C₂×Dic₆₇	—	2	2	no	—
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	4	2	no	—
147	D₂₆₈	—	2	2	no	—
148	D₂₆₈	—	2	2	no	—
149	C₄×D₆₇	—	4	2	no	—
150	C₄×D₆₇	—	4	2	no	—
151	Dic₁₃₄	—	2	2	no	—
152	Dic₁₃₄	—	2	2	no	—
153	Dic₁₃₄	—	4	4	no	—
154	C₈×C₆₇	—	2	2	no	—
155	C₈×C₆₇	—	4	2	no	—
156	C₈×C₆₇	#124	1	2	yes	—
157	C₈×C₆₇	#124	1	1	yes	characteristic
158	C₈×C₆₇	#124	1	1	yes	characteristic
159	C₆₇⋊C₈	—	4	1	no	—
160	C₆₇⋊C₈	—	4	1	no	—
161	C₆₇⋊C₈	—	4	2	no	—
162	C₂²×C₆₇	—	4	1	no	—
163	C₂²×C₆₇	#525	1	1	yes	characteristic
164	C₂²×C₆₇	#529	1	2	yes	—
165	C₂×D₆₇	—	4	2	no	—
166	C₂×D₆₇	—	8	2	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	#525	1	1	yes	characteristic
170	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
171	C₄×C₆₇	#529	1	2	yes	—
172	Dic₆₇	—	2	2	no	—
173	Dic₆₇	—	4	2	no	—
174	Dic₆₇	—	4	1	no	—
175	Dic₆₇	—	4	2	no	—
176	#23833	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	—	2	2	no	—
179	C₂×C₆₇	#2398	1	1	yes	characteristic
180	D₆₇	—	8	2	no	—
181	#2960	—	67	1	no	—
182	#2955	—	67	1	no	—
183	#2103	—	67	1	no	—
184	#2102	—	67	1	no	—
185	#2080	—	67	2	no	—
186	#2053	—	67	1	no	—
187	#2050	—	67	1	no	—
188	#2031	—	67	2	no	—
189	#1764	—	67	2	no	—
190	#1757	—	67	2	no	—
191	#1747	D₆₇	1	1	yes	characteristic, complemented
192	C₆₇	#23833	1	1	yes	characteristic, complemented, Sylow
193	#575	—	67	1	no	—
194	C₂×#131	—	67	2	no	—
195	#533	—	67	1	no	—
196	#446	—	134	2	no	—
197	C₈×Q₈	—	134	2	no	—
198	#444	—	134	2	no	—
199	#443	—	134	2	no	—
200	#430	—	67	1	no	—
201	#424	—	67	1	no	—
202	#424	—	67	1	no	—
203	#423	—	67	1	no	—
204	#421	—	67	2	no	—
205	C₂×#104	—	67	2	no	—
206	#416	—	67	2	no	—
207	#413	—	67	2	no	—
208	#410	—	67	2	no	—
209	#405	—	67	1	no	—
210	#405	C₂×D₆₇	1	2	yes	—
211	#405	C₂×D₆₇	1	1	yes	characteristic
212	C₂×C₄×C₈	—	67	1	no	—
213	#344	—	67	2	no	—
214	#344	—	67	2	no	—
215	#341	—	67	2	no	—
216	#339	—	67	2	no	—
217	#338	C₂×D₆₇	1	1	yes	characteristic
218	H(3,ℤ/4)	C₂×D₆₇	1	1	yes	characteristic
219	H(3,ℤ/4)	C₂×D₆₇	1	2	yes	—
220	C₂×#41	—	67	2	no	—
221	#137	—	134	2	no	—
222	AGL(1,ℤ/8)	—	134	2	no	—
223	#135	—	134	2	no	—
224	#131	—	134	2	no	—
225	#131	—	134	2	no	—
226	#131	—	134	2	no	—
227	#131	—	134	2	no	—
228	C₂×M₁₆	—	67	2	no	—
229	C₂×M₁₆	—	67	1	no	—
230	C₂×M₁₆	—	134	2	no	—
231	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
232	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	—	67	1	no	—
234	C₂²×C₈	—	134	2	no	—
235	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
236	C₄⋊Q₈	—	134	1	no	—
237	#125	—	134	1	no	—
238	C₂²⋊Q₈	—	134	2	no	—
239	C₄×Q₈	—	134	2	no	—
240	#117	—	67	1	no	—
241	#117	C₂²×D₆₇	1	2	yes	—
242	#117	C₂²×D₆₇	1	1	yes	characteristic
243	C₂×C₄²	—	67	1	no	—
244	#108	—	134	2	no	—
245	C₄⋊C₈	—	134	2	no	—
246	C₄⋊C₈	—	134	2	no	—
247	C₄⋊C₈	—	134	2	no	—
248	#104	—	134	2	no	—
249	#104	—	134	2	no	—
250	#104	—	134	4	no	—
251	#103	—	134	2	no	—
252	#102	—	134	2	no	—
253	#101	—	134	2	no	—
254	#100	—	134	2	no	—
255	#99	—	134	4	no	—
256	#97	—	2	2	no	—
257	#97	—	134	1	no	—
258	#97	D₂₆₈	1	2	yes	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	134	1	no	—
261	C₄×C₈	—	134	1	no	—
262	C₄×C₈	—	134	1	no	—
263	C₄×C₈	D₂₆₈	1	2	yes	—
264	#41	—	134	2	no	—
265	#41	—	134	2	no	—
266	#41	—	134	2	no	—
267	#41	—	134	2	no	—
268	C₂×Q₈	—	67	2	no	—
269	C₂×Q₈	—	134	2	no	—
270	C₂×D₄	—	67	2	no	—
271	C₂×D₄	—	134	2	no	—
272	C₂²×C₄	—	134	1	no	—
273	C₂²×C₄	—	134	2	no	—
274	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
275	Q₁₆	—	134	2	no	—
276	SD₁₆	—	134	4	no	—
277	D₈	—	134	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	268	2	no	—
282	M₁₆	—	268	2	no	—
283	M₁₆	—	134	2	no	—
284	M₁₆	—	134	2	no	—
285	M₁₆	—	134	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	268	2	no	—
288	C₂×C₈	—	268	2	no	—
289	C₂×C₈	—	134	1	no	—
290	C₂×C₈	—	134	1	no	—
291	C₂×C₈	—	134	2	no	—
292	C₂×C₈	—	134	2	no	—
293	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
294	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
295	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
296	C₂×C₈	#11761628	1	2	yes	—
297	C₂×C₈	#11761628	1	2	yes	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	134	2	no	—
301	C₄⋊C₄	—	134	2	no	—
302	C₄⋊C₄	—	134	2	no	—
303	C₄⋊C₄	—	134	1	no	—
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	67	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	67	1	no	—
309	C₄²	—	67	1	no	—
310	C₄²	—	134	1	no	—
311	C₄²	—	134	1	no	—
312	C₄²	C₂×D₂₆₈	1	2	yes	—
313	C₂³	—	134	2	no	—
314	C₂³	#41×D₆₇	1	1	yes	characteristic
315	Q₈	—	268	4	no	—
316	Q₈	—	134	2	no	—
317	Q₈	—	134	2	no	—
318	D₄	—	268	2	no	—
319	D₄	—	268	2	no	—
320	D₄	—	134	2	no	—
321	D₄	—	134	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	268	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	268	2	no	—
327	C₂×C₄	—	134	1	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	—	134	1	no	—
331	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
332	C₂×C₄	#41×D₆₇	1	2	yes	—
333	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×#11761628	1	2	yes	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	268	1	no	—
338	C₈	—	268	1	no	—
339	C₈	—	268	2	no	—
340	C₈	#14524031	1	2	yes	—
341	C₈	#14524031	1	1	yes	characteristic
342	C₈	#14524031	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	268	2	no	—
345	C₂²	—	536	2	no	—
346	C₂²	#16863908	1	1	yes	characteristic
347	C₂²	#16863924	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	268	1	no	—
351	C₄	—	268	2	no	—
352	C₄	—	268	2	no	—
353	C₄	—	134	2	no	—
354	C₄	#16863908	1	1	yes	characteristic
355	C₄	C₂×#14524031	1	1	yes	characteristic
356	C₄	#16863924	1	2	yes	—
357	C₂	—	2	1	no	—
358	C₂	—	2	2	no	—
359	C₂	—	536	2	no	—
360	C₂	#22563495	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524031
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1747	C₂
socle	C₂×C₆₇	#2398
prime radical	#16864196	C₂²
hypercenter	#1747	D₆₇
hypocenter	C₆₇	#23833
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
1. #22569141	C₂	semidirect
3. #22561790	C₂	semidirect
4. #22561785	C₂	semidirect
5. #22561661	C₂	semidirect
8. #22561380	C₂	semidirect
9. #22560048	C₂	semidirect
10. #22560010	C₂	semidirect
11. C₆₇×#1747	C₂	semidirect
191. #1747	D₆₇	semidirect
192. C₆₇	#23833	semidirect

Maximal subgroups

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

subgroup	index
#22569141	2
#22569113	2
#22561790	2
#22561785	2
#22561661	2
#22561516	2
#22561497	2
#22561380	2
#22560048	2
#22560010	2
C₆₇×#1747	2
#23833	67