Subgroups — #30864707

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570732	C₂	1	1	yes	maximal, characteristic
2	#22569148	C₂	1	2	yes	maximal
3	#22561184	C₂	1	2	yes	maximal
4	#22560962	C₂	1	1	yes	maximal, characteristic
5	#22560954	C₂	1	2	yes	maximal
6	#22560382	C₂	1	1	yes	maximal, characteristic
7	#22560092	C₂	1	2	yes	maximal
8	#22560091	C₂	1	1	yes	maximal, characteristic
9	#22560090	C₂	1	1	yes	maximal, characteristic
10	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1946	C₂	1	1	yes	maximal, characteristic
12	#16863949	C₂²	1	2	yes	—
13	#16863804	C₄	1	4	yes	complemented
14	#16863803	C₄	1	4	yes	complemented
15	#16863525	—	2	4	no	—
16	#16863523	—	2	4	no	—
17	C₂×#14523979	C₂²	1	1	yes	characteristic
18	C₂×#14523978	C₂²	1	2	yes	—
19	#16863434	C₂²	1	1	yes	characteristic
20	#16863432	C₂²	1	2	yes	—
21	#16863422	C₂²	1	2	yes	—
22	#16863419	C₂²	1	2	yes	—
23	#16863354	C₂²	1	2	yes	—
24	#16863352	C₂²	1	2	yes	—
25	C₂×#14523954	C₂²	1	1	yes	characteristic
26	C₂×#14523953	C₂²	1	1	yes	characteristic
27	C₂×#14523953	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
30	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
31	#16863157	C₂²	1	2	yes	—
32	#16863156	C₂²	1	2	yes	—
33	#16863155	C₂²	1	2	yes	—
34	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
35	#427×C₆₇	C₂²	1	1	yes	characteristic
36	#424×C₆₇	C₂²	1	2	yes	—
37	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
38	#339×C₆₇	C₂²	1	2	yes	—
39	C₂×#11761612	C₂³	1	2	yes	—
40	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
42	#14524042	C₂×C₄	1	4	yes	—
43	#14524034	C₂×C₄	1	4	yes	—
44	#14524020	—	2	4	no	—
45	C₄×Dic₁₃₄	—	2	4	no	—
46	#14524002	C₂³	1	2	yes	—
47	C₂×C₄×Dic₆₇	—	2	2	no	—
48	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
50	#14523979	—	2	2	no	—
51	#14523978	—	2	4	no	—
52	#14523977	C₂×C₄	1	2	yes	—
53	#14523976	C₂×C₄	1	4	yes	—
54	#14523975	C₂×C₄	1	2	yes	—
55	#14523974	C₂×C₄	1	4	yes	—
56	Dic₆₇⋊C₈	—	2	4	no	—
57	Dic₆₇⋊C₈	—	2	4	no	—
58	#14523954	—	2	2	no	—
59	#14523954	Q₈	1	4	yes	complemented
60	#14523953	—	2	2	no	—
61	#14523953	C₂×C₄	1	2	yes	—
62	#14523953	C₂×C₄	1	2	yes	—
63	#14523953	D₄	1	2	yes	—
64	#14523953	D₄	1	2	yes	—
65	C₄×C₆₇⋊C₈	—	2	2	no	—
66	C₈×Dic₆₇	—	2	2	no	—
67	C₈×Dic₆₇	—	2	2	no	—
68	C₈×Dic₆₇	Q₈	1	4	yes	complemented
69	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	#117×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#108×C₆₇	—	2	2	no	—
75	#107×C₆₇	C₂×C₄	1	2	yes	—
76	#106×C₆₇	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	#97×C₆₇	—	2	2	no	—
79	C₂×Dic₁₃₄	—	2	4	no	—
80	C₂²×Dic₆₇	—	2	1	no	—
81	C₂²×Dic₆₇	#41	1	2	yes	—
82	#11761612	—	2	2	no	—
83	#11761612	—	2	2	no	—
84	C₂×C₆₇⋊C₈	—	2	2	no	—
85	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
91	C₂×C₆₇⋊C₈	#41	1	2	yes	—
92	C₄⋊Dic₆₇	—	2	4	no	—
93	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
94	Dic₆₇⋊C₄	—	2	4	no	—
95	Dic₆₇⋊C₄	—	2	4	no	—
96	C₂²⋊Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₄×Dic₆₇	—	2	1	no	—
99	C₄×Dic₆₇	—	4	4	no	—
100	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
101	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
102	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₆₇	—	2	2	no	—
106	M₁₆×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
114	C₂×C₈×C₆₇	#41	1	2	yes	—
115	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
116	C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	1	no	—
119	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
120	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₆₇	—	2	4	no	—
131	C₂×Dic₆₇	—	2	2	no	—
132	C₂×Dic₆₇	—	2	4	no	—
133	C₂×Dic₆₇	—	4	2	no	—
134	C₂×Dic₆₇	C₄×D₄	1	4	yes	—
135	Dic₁₃₄	—	8	4	no	—
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	—	4	2	no	—
138	C₈×C₆₇	C₄×Q₈	1	2	yes	—
139	C₈×C₆₇	C₄×Q₈	1	2	yes	—
140	C₆₇⋊C₈	—	4	2	no	—
141	C₆₇⋊C₈	C₄×Q₈	1	4	yes	—
142	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
143	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₆₇	#533	1	1	yes	characteristic
147	C₂²×C₆₇	#533	1	1	yes	characteristic
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	4	4	no	—
150	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₄×C₆₇	#532	1	1	yes	characteristic
154	Dic₆₇	—	2	4	no	—
155	Dic₆₇	—	2	4	no	—
156	Dic₆₇	—	4	4	no	—
157	#21736	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
161	C₂×C₆₇	#2956	1	2	yes	—
162	#3162	—	67	1	no	—
163	#2961	—	67	2	no	—
164	#1994	—	67	2	no	—
165	#1946	—	67	1	no	—
166	#1946	D₆₇	1	1	yes	characteristic
167	#1945	—	67	2	no	—
168	#1849	—	67	1	no	—
169	#1773	—	67	2	no	—
170	#1772	—	67	1	no	—
171	#1772	—	67	1	no	—
172	C₄×#97	—	67	1	no	—
173	C₆₇	#21736	1	1	yes	characteristic, complemented, Sylow
174	#533	—	67	2	no	—
175	#501	—	67	4	no	—
176	#501	—	67	4	no	—
177	#446	—	134	4	no	—
178	#446	—	134	4	no	—
179	C₂×#108	—	67	2	no	—
180	C₂×#108	—	67	1	no	—
181	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
182	#427	—	67	1	no	—
183	#427	—	67	2	no	—
184	#427	C₂×D₆₇	1	1	yes	characteristic
185	#424	—	67	2	no	—
186	#424	C₂×D₆₇	1	2	yes	—
187	#423	—	67	2	no	—
188	#405	—	67	2	no	—
189	#405	—	67	2	no	—
190	C₂×#97	—	67	1	no	—
191	C₂×#97	—	67	1	no	—
192	C₂×#97	—	67	1	no	—
193	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄×C₈	—	67	1	no	—
195	C₂×C₄×C₈	—	67	1	no	—
196	C₄³	—	67	1	no	—
197	#339	—	67	2	no	—
198	#339	—	67	2	no	—
199	#339	—	67	2	no	—
200	#339	C₂×D₆₇	1	2	yes	—
201	C₂×M₁₆	—	67	2	no	—
202	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
203	C₂²×C₈	—	67	1	no	—
204	C₂²×C₈	—	67	1	no	—
205	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	C₄⋊Q₈	—	67	4	no	—
208	#125	—	67	4	no	—
209	C₂²⋊Q₈	—	134	4	no	—
210	C₄×Q₈	—	134	4	no	—
211	#117	—	67	2	no	—
212	#117	C₂²×D₆₇	1	2	yes	—
213	C₂×C₄²	—	67	1	no	—
214	C₂×C₄²	—	67	1	no	—
215	C₂×C₄²	—	134	2	no	—
216	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
217	#108	—	2	2	no	—
218	#108	—	134	4	no	—
219	#108	—	134	2	no	—
220	#107	—	67	2	no	—
221	#107	—	67	4	no	—
222	#107	C₄×D₆₇	1	2	yes	—
223	#106	—	67	2	no	—
224	#106	—	67	4	no	—
225	#106	C₄×D₆₇	1	2	yes	—
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	4	no	—
229	#97	—	2	2	no	—
230	#97	—	67	2	no	—
231	#97	—	67	2	no	—
232	#97	—	67	2	no	—
233	#97	—	67	2	no	—
234	#97	—	67	4	no	—
235	#97	—	134	2	no	—
236	#97	—	134	2	no	—
237	C₄×C₈	—	67	4	no	—
238	C₄×C₈	—	134	2	no	—
239	C₄×C₈	—	134	2	no	—
240	C₄×C₈	—	134	2	no	—
241	C₂×Q₈	—	134	4	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	67	2	no	—
244	C₂²×C₄	—	134	1	no	—
245	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	134	2	no	—
249	M₁₆	—	134	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	67	2	no	—
254	C₂×C₈	—	67	1	no	—
255	C₂×C₈	—	67	1	no	—
256	C₂×C₈	—	67	2	no	—
257	C₂×C₈	—	67	2	no	—
258	C₂×C₈	—	67	2	no	—
259	C₂×C₈	—	67	2	no	—
260	C₂×C₈	—	134	2	no	—
261	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
263	C₂×C₈	D₄×D₆₇	1	2	yes	—
264	C₂×C₈	Q₈×D₆₇	1	2	yes	—
265	C₂×C₈	#11761627	1	2	yes	—
266	C₄⋊C₄	—	67	4	no	—
267	C₄⋊C₄	—	134	4	no	—
268	C₄⋊C₄	—	134	4	no	—
269	C₄⋊C₄	—	134	4	no	—
270	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	134	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	268	4	no	—
276	C₄²	—	67	2	no	—
277	C₄²	—	67	2	no	—
278	C₄²	—	67	2	no	—
279	C₄²	—	134	2	no	—
280	C₄²	—	134	1	no	—
281	C₄²	Q₈×D₆₇	1	2	yes	—
282	C₂³	#41×D₆₇	1	1	yes	characteristic
283	Q₈	—	536	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	268	2	no	—
288	C₂×C₄	—	67	4	no	—
289	C₂×C₄	—	134	4	no	—
290	C₂×C₄	—	134	4	no	—
291	C₂×C₄	—	134	2	no	—
292	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
296	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	268	2	no	—
301	C₈	—	67	4	no	—
302	C₈	—	67	2	no	—
303	C₈	—	67	2	no	—
304	C₈	#14524008	1	2	yes	—
305	C₈	#14524008	1	2	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
308	C₂²	#16863948	1	1	yes	characteristic
309	C₂²	#16863952	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	268	4	no	—
313	C₄	—	134	4	no	—
314	C₄	—	134	4	no	—
315	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
316	C₄	C₂×#14524008	1	1	yes	characteristic
317	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
318	C₄	#16863941	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	#22562780	1	1	yes	characteristic
322	C₂	#22569117	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1946	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1946	D₆₇
hypocenter	C₆₇	#21736
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1946

Complements

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

N	H ≅ G/N	type
13. #16863804	C₄	semidirect
14. #16863803	C₄	semidirect
59. #14523954	Q₈	semidirect
68. C₈×Dic₆₇	Q₈	semidirect
173. C₆₇	#21736	semidirect

Maximal subgroups

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

subgroup	index
#22570732	2
#22569148	2
#22561184	2
#22560962	2
#22560954	2
#22560382	2
#22560092	2
#22560091	2
#22560090	2
#97×Dic₆₇	2
C₆₇×#1946	2
#21736	67