Subgroups — #30755704

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561187	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561038	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558664	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558660	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
8	#16863420	C₂²	1	1	yes	characteristic
9	#16863420	C₂²	1	1	yes	characteristic
10	C₂×#14523970	—	2	2	no	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₆₇	—	2	2	no	—
13	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
14	#16863155	—	2	2	no	—
15	#16863147	—	2	2	no	—
16	Q₈⋊(C₆₇⋊C₈)	—	2	2	no	—
17	D₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	#533×C₆₇	C₂²	1	1	yes	characteristic
19	#532×C₆₇	C₂²	1	1	yes	characteristic
20	#498×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#492×C₆₇	C₄	1	1	yes	characteristic, complemented
22	#487×C₆₇	C₄	1	1	yes	characteristic, complemented
23	#486×C₆₇	C₄	1	1	yes	characteristic, complemented
24	#465×C₆₇	—	2	2	no	—
25	C₄×SD₁₆×C₆₇	—	2	2	no	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761612	—	2	1	no	—
28	C₂×#11761612	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₄×Dic₆₇	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	#14523970	—	4	4	no	—
38	C₄×C₆₇⋊C₈	—	2	1	no	—
39	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
40	C₈×Dic₆₇	—	2	8	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
48	#125×C₆₇	—	2	1	no	—
49	#124×C₆₇	—	2	1	no	—
50	C₂²⋊Q₈×C₆₇	—	2	1	no	—
51	C₂²⋊Q₈×C₆₇	—	2	1	no	—
52	#121×C₆₇	—	2	1	no	—
53	#121×C₆₇	—	2	1	no	—
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×Q₈×C₆₇	—	2	1	no	—
56	C₄×D₄×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	#117×C₆₇	—	2	1	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#106×C₆₇	—	2	2	no	—
61	#103×C₆₇	—	2	2	no	—
62	#102×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	D₄	1	2	yes	—
66	C₂²×Dic₆₇	—	2	2	no	—
67	#11761612	—	4	2	no	—
68	#11761612	—	4	2	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	4	1	no	—
72	C₂×C₆₇⋊C₈	—	4	1	no	—
73	C₄×Dic₆₇	—	2	4	no	—
74	C₄×Dic₆₇	—	2	4	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₄×Dic₆₇	SD₁₆	1	4	yes	complemented
77	#41×C₆₇	—	4	2	no	—
78	#41×C₆₇	—	4	2	no	—
79	C₂×Q₈×C₆₇	—	2	1	no	—
80	C₂×Q₈×C₆₇	—	2	1	no	—
81	C₂×Q₈×C₆₇	—	2	1	no	—
82	C₂×D₄×C₆₇	—	2	1	no	—
83	C₂×D₄×C₆₇	—	2	1	no	—
84	C₂×D₄×C₆₇	—	2	1	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
90	SD₁₆×C₆₇	—	4	4	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	4	1	no	—
102	C₂²⋊C₄×C₆₇	—	4	1	no	—
103	C₂²⋊C₄×C₆₇	—	4	1	no	—
104	C₂²⋊C₄×C₆₇	—	4	1	no	—
105	C₄²×C₆₇	—	4	1	no	—
106	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	#41	1	1	yes	characteristic
110	C₂³×C₆₇	—	4	1	no	—
111	C₂³×C₆₇	#121	1	1	yes	characteristic
112	Q₈×C₆₇	—	4	2	no	—
113	Q₈×C₆₇	—	4	2	no	—
114	Q₈×C₆₇	—	4	2	no	—
115	D₄×C₆₇	—	4	2	no	—
116	D₄×C₆₇	—	4	2	no	—
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	8	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	8	1	no	—
125	C₂×C₄×C₆₇	—	8	1	no	—
126	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
131	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₆₇	#135	1	2	yes	—
134	C₂×Dic₆₇	—	2	4	no	—
135	C₂×Dic₆₇	—	2	4	no	—
136	C₂×Dic₆₇	—	2	4	no	—
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	#104	1	4	yes	complemented
139	C₆₇⋊C₈	—	2	4	no	—
140	C₆₇⋊C₈	—	8	1	no	—
141	C₆₇⋊C₈	—	8	1	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	8	1	no	—
145	C₂²×C₆₇	—	8	1	no	—
146	C₂²×C₆₇	—	8	1	no	—
147	C₂²×C₆₇	#390	1	1	yes	characteristic
148	C₂²×C₆₇	#465	1	2	yes	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	8	1	no	—
152	C₄×C₆₇	—	8	1	no	—
153	C₄×C₆₇	—	8	1	no	—
154	C₄×C₆₇	C₂×#104	1	2	yes	—
155	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
156	C₄×C₆₇	#443	1	2	yes	—
157	C₄×C₆₇	#492	1	1	yes	characteristic
158	C₄×C₆₇	#495	1	1	yes	characteristic
159	Dic₆₇	—	2	8	no	—
160	#9843	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	2	no	—
162	C₂×C₆₇	—	16	1	no	—
163	C₂×C₆₇	#1962	1	1	yes	characteristic
164	C₂×C₆₇	#1968	1	1	yes	characteristic
165	C₂×C₆₇	#1974	1	1	yes	characteristic
166	#3167	D₆₇	1	1	yes	characteristic
167	#1994	—	67	1	no	—
168	#1962	—	67	1	no	—
169	C₄²×C₈	—	67	1	no	—
170	#1566	—	67	1	no	—
171	#1493	—	67	1	no	—
172	#1492	—	67	1	no	—
173	C₆₇	#9843	1	1	yes	characteristic, complemented, Sylow
174	#533	C₂×D₆₇	1	1	yes	characteristic
175	#532	C₂×D₆₇	1	1	yes	characteristic
176	#498	Dic₆₇	1	1	yes	characteristic, complemented
177	#492	Dic₆₇	1	1	yes	characteristic, complemented
178	#487	Dic₆₇	1	1	yes	characteristic, complemented
179	#486	Dic₆₇	1	1	yes	characteristic, complemented
180	#465	—	2	2	no	—
181	C₄×SD₁₆	—	2	2	no	—
182	#423	—	67	1	no	—
183	#423	—	67	1	no	—
184	C₂×#104	—	134	2	no	—
185	C₂×C₄×C₈	—	67	1	no	—
186	C₂×C₄×C₈	—	134	2	no	—
187	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
188	C₄³	—	67	1	no	—
189	#339	—	134	2	no	—
190	#334	—	134	2	no	—
191	Q₈⋊C₈	—	134	2	no	—
192	D₄⋊C₈	—	134	2	no	—
193	C₂×#41	—	2	1	no	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×M₁₆	—	134	1	no	—
196	C₂×M₁₆	—	134	1	no	—
197	C₂²×C₈	—	134	1	no	—
198	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
199	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
201	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
202	#127	C₂×Dic₆₇	1	1	yes	characteristic
203	#125	—	2	1	no	—
204	#124	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	C₂²⋊Q₈	—	2	1	no	—
207	#121	—	2	1	no	—
208	#121	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	#117	—	2	1	no	—
214	C₂×C₄²	—	67	2	no	—
215	C₂×C₄²	—	134	1	no	—
216	C₂×C₄²	—	134	1	no	—
217	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
218	#106	—	2	2	no	—
219	C₄⋊C₈	—	134	1	no	—
220	C₄⋊C₈	—	134	1	no	—
221	C₄⋊C₈	—	134	1	no	—
222	C₄⋊C₈	—	134	1	no	—
223	#104	—	268	4	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	C₄×C₈	—	67	2	no	—
227	C₄×C₈	—	134	8	no	—
228	C₄×C₈	—	134	1	no	—
229	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
230	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	134	2	no	—
243	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
245	C₂²×C₄	#11761627	1	1	yes	characteristic
246	SD₁₆	—	4	4	no	—
247	M₁₆	—	268	2	no	—
248	M₁₆	—	268	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	268	1	no	—
251	C₂×C₈	—	268	1	no	—
252	C₂×C₈	—	134	4	no	—
253	C₂×C₈	—	134	2	no	—
254	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
255	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	67	4	no	—
269	C₄²	—	134	4	no	—
270	C₄²	—	134	4	no	—
271	C₄²	—	134	2	no	—
272	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
273	C₄²	D₄×D₆₇	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₄²	#11761627	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	#14524014	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	134	4	no	—
293	C₂×C₄	—	134	4	no	—
294	C₂×C₄	—	134	4	no	—
295	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
296	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
297	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
298	C₂×C₄	#14524014	1	1	yes	characteristic
299	C₂×C₄	#14524030	1	1	yes	characteristic
300	C₂×C₄	#14524039	1	1	yes	characteristic
301	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
302	C₂×C₄	#14524064	1	2	yes	—
303	C₈	—	2	2	no	—
304	C₈	—	536	1	no	—
305	C₈	—	536	1	no	—
306	C₈	—	134	4	no	—
307	C₈	#14523970	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	#16863299	1	1	yes	characteristic
314	C₂²	#16863622	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	134	8	no	—
321	C₄	C₂×#14523970	1	2	yes	—
322	C₄	SD₁₆×Dic₆₇	1	2	yes	—
323	C₄	#16863513	1	2	yes	—
324	C₄	#16863767	1	1	yes	characteristic
325	C₄	#16863777	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	#22561038	1	1	yes	characteristic
329	C₂	#22561065	1	1	yes	characteristic
330	C₂	#22561098	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#3167	C₂
socle	C₂²×C₆₇	#390
prime radical	#532×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#3167	D₆₇
hypocenter	C₆₇	#9843
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

Complements

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

N	H ≅ G/N	type
1. #22561187	C₂	semidirect
3. C₄×C₈×Dic₆₇	C₂	semidirect
4. #22558965	C₂	semidirect
5. #22558664	C₂	semidirect
20. #498×C₆₇	C₄	semidirect
21. #492×C₆₇	C₄	semidirect
22. #487×C₆₇	C₄	semidirect
23. #486×C₆₇	C₄	semidirect
76. C₄×Dic₆₇	SD₁₆	semidirect
138. C₈×C₆₇	#104	semidirect
173. C₆₇	#9843	semidirect
176. #498	Dic₆₇	semidirect
177. #492	Dic₆₇	semidirect
178. #487	Dic₆₇	semidirect
179. #486	Dic₆₇	semidirect
307. C₈	#14523970	semidirect

Maximal subgroups

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

subgroup	index
#22561187	2
#22561038	2
C₄×C₈×Dic₆₇	2
#22558965	2
#22558664	2
#22558660	2
C₆₇×#3167	2
#9843	67