Subgroups — #30859406

Order 17152.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863445	C₂	1	2	yes	maximal
2	C₂×#16863445	C₂	1	1	yes	maximal, characteristic
3	#22568627	C₂	1	1	yes	maximal, characteristic
4	#22561000	C₂	1	8	yes	maximal, complemented
5	C₂×#16863168	C₂	1	2	yes	maximal
6	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	1	yes	characteristic
10	#16863445	—	2	4	no	—
11	#16863445	—	2	2	no	—
12	#16863445	—	2	2	no	—
13	#16863445	C₂²	1	8	yes	—
14	C₂×#14523979	—	2	4	no	—
15	C₂×#14523979	C₂²	1	4	yes	—
16	#16863354	—	2	2	no	—
17	#16863351	C₂²	1	4	yes	—
18	M₁₆×Dic₆₇	—	2	2	no	—
19	#16863168	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	#430×C₆₇	C₄	1	8	yes	complemented
23	C₂×#108×C₆₇	C₂²	1	4	yes	—
24	C₂×#11761612	—	2	1	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	D₄	1	4	yes	—
32	C₂×#11761612	C₂³	1	4	yes	—
33	C₂×#11761612	C₂³	1	2	yes	—
34	C₂²×C₆₇⋊C₈	—	2	1	no	—
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	#14524002	—	2	1	no	—
37	#14524002	—	2	1	no	—
38	#14524002	Q₈	1	2	yes	—
39	#14524002	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	—	2	1	no	—
41	C₂×C₂²⋊Dic₆₇	Q₈	1	2	yes	—
42	C₂×C₄×Dic₆₇	—	2	1	no	—
43	#14523979	—	2	16	no	—
44	#14523979	—	2	8	no	—
45	#14523953	—	2	4	no	—
46	C₈×Dic₆₇	—	2	4	no	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
55	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
56	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
57	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	#108×C₆₇	C₂×C₄	1	16	yes	complemented
61	C₂²×Dic₆₇	—	2	2	no	—
62	#11761612	—	2	4	no	—
63	#11761612	—	2	2	no	—
64	#11761612	—	2	4	no	—
65	#11761612	—	2	4	no	—
66	#11761612	—	4	2	no	—
67	#11761612	—	4	2	no	—
68	#11761612	C₂×D₄	1	4	yes	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	4	2	no	—
73	C₂×C₆₇⋊C₈	—	4	2	no	—
74	C₄⋊Dic₆₇	—	2	2	no	—
75	C₄⋊Dic₆₇	—	2	2	no	—
76	C₂²⋊Dic₆₇	—	2	2	no	—
77	C₂²⋊Dic₆₇	C₂×Q₈	1	4	yes	—
78	C₄×Dic₆₇	—	2	2	no	—
79	C₄×Dic₆₇	—	2	2	no	—
80	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₆₇	—	2	4	no	—
92	M₁₆×C₆₇	—	2	4	no	—
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
95	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
96	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	2	4	no	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
102	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
103	C₂³×C₆₇	—	8	1	no	—
104	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
105	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	#121	1	2	yes	—
115	C₂×C₄×C₆₇	#121	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#125	1	2	yes	—
119	C₂×C₄×C₆₇	#125	1	2	yes	—
120	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₆₇	—	2	4	no	—
128	C₂×Dic₆₇	—	8	2	no	—
129	C₈×C₆₇	—	2	8	no	—
130	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
131	C₆₇⋊C₈	—	2	4	no	—
132	C₆₇⋊C₈	—	4	4	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	—	4	2	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	#389	1	2	yes	—
139	C₂²×C₆₇	#389	1	2	yes	—
140	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₆₇	#389	1	2	yes	—
144	C₄×C₆₇	#389	1	2	yes	—
145	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
149	Dic₆₇	—	4	4	no	—
150	#21184	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
157	C₂×#430	—	67	1	no	—
158	C₂×#430	—	67	2	no	—
159	C₂×#430	D₆₇	1	1	yes	characteristic
160	#2873	—	67	1	no	—
161	#1953	—	67	8	no	—
162	C₂×#344	—	67	2	no	—
163	C₆₇	#21184	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	67	1	no	—
165	C₂²×M₁₆	—	67	2	no	—
166	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
167	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#117	—	67	1	no	—
169	#430	—	67	8	no	—
170	#430	—	134	4	no	—
171	#430	—	134	2	no	—
172	#430	—	134	2	no	—
173	#430	Dic₆₇	1	8	yes	complemented
174	C₂×#108	—	67	4	no	—
175	C₂×#108	—	134	4	no	—
176	C₂×#108	C₂×D₆₇	1	4	yes	—
177	#405	—	67	4	no	—
178	#405	—	134	2	no	—
179	C₄×M₁₆	—	134	2	no	—
180	#344	—	67	8	no	—
181	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	67	4	no	—
186	C₂×M₁₆	—	67	2	no	—
187	C₂×M₁₆	—	67	4	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	—	134	2	no	—
190	C₂×M₁₆	—	134	2	no	—
191	C₂×M₁₆	—	134	2	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂×M₁₆	—	134	1	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
196	C₂×M₁₆	D₂₆₈	1	2	yes	—
197	C₂×M₁₆	D₂₆₈	1	2	yes	—
198	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
199	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
200	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
201	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	134	2	no	—
204	C₂²×C₈	—	134	1	no	—
205	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
206	#117	—	67	2	no	—
207	#117	—	67	2	no	—
208	#117	—	134	1	no	—
209	#117	—	134	1	no	—
210	C₂×C₄⋊C₄	—	134	1	no	—
211	C₂×C₂²⋊C₄	—	67	2	no	—
212	C₂×C₄²	—	134	1	no	—
213	#108	—	134	16	no	—
214	#108	—	134	8	no	—
215	#108	C₂×Dic₆₇	1	16	yes	complemented
216	#97	—	134	4	no	—
217	C₄×C₈	—	134	4	no	—
218	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
219	C₂²×C₄	—	134	2	no	—
220	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
221	C₂²×C₄	D₄×D₆₇	1	2	yes	—
222	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
224	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
226	C₂²×C₄	#11761627	1	1	yes	characteristic
227	C₂²×C₄	#11761627	1	1	yes	characteristic
228	C₂²×C₄	#11761629	1	2	yes	—
229	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	268	2	no	—
234	M₁₆	—	268	2	no	—
235	M₁₆	—	67	4	no	—
236	M₁₆	—	134	4	no	—
237	M₁₆	—	134	4	no	—
238	M₁₆	—	134	4	no	—
239	M₁₆	—	134	2	no	—
240	M₁₆	C₄⋊Dic₆₇	1	8	yes	—
241	M₁₆	C₄⋊Dic₆₇	1	8	yes	—
242	M₁₆	C₂²×Dic₆₇	1	8	yes	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	268	2	no	—
247	C₂×C₈	—	268	2	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	—	134	4	no	—
250	C₂×C₈	—	134	2	no	—
251	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
252	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
253	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
254	C₄⋊C₄	—	134	2	no	—
255	C₄⋊C₄	—	134	2	no	—
256	C₂²⋊C₄	—	67	4	no	—
257	C₂²⋊C₄	—	134	2	no	—
258	C₄²	—	134	2	no	—
259	C₄²	—	134	2	no	—
260	C₂³	—	8	1	no	—
261	C₂³	Q₈×Dic₆₇	1	2	yes	—
262	C₂³	#14524023	1	1	yes	characteristic
263	C₂³	#14524023	1	1	yes	characteristic
264	C₂³	#14524042	1	2	yes	—
265	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
266	C₂×C₄	—	536	2	no	—
267	C₂×C₄	—	134	4	no	—
268	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
270	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
271	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
272	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
273	C₂×C₄	#14524018	1	2	yes	—
274	C₂×C₄	#14524018	1	2	yes	—
275	C₂×C₄	#14524023	1	1	yes	characteristic
276	C₂×C₄	#14524023	1	1	yes	characteristic
277	C₂×C₄	#14524034	1	2	yes	—
278	C₂×C₄	#14524034	1	2	yes	—
279	C₂×C₄	#14524042	1	2	yes	—
280	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
285	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
286	C₈	—	2	8	no	—
287	C₈	—	268	4	no	—
288	C₈	—	134	4	no	—
289	C₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#16863295	1	2	yes	—
296	C₂²	#16863295	1	2	yes	—
297	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
298	C₂²	C₂×#14524023	1	1	yes	characteristic
299	C₂²	C₂×#14524042	1	1	yes	characteristic
300	C₄	—	268	4	no	—
301	C₄	#16863295	1	2	yes	—
302	C₄	#16863295	1	2	yes	—
303	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
305	C₄	C₂×#14524018	1	1	yes	characteristic
306	C₄	C₂×#14524034	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₂×#16863295	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524018
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#430×C₆₇	C₂
socle	C₂×C₆₇	C₂×#389
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#430	D₆₇
hypocenter	C₆₇	#21184
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #22561000	C₂	semidirect
22. #430×C₆₇	C₄	semidirect
60. #108×C₆₇	C₂×C₄	semidirect
163. C₆₇	#21184	semidirect
173. #430	Dic₆₇	semidirect
215. #108	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863445	2
#22568627	2
#22561000	2
C₂×#16863168	2
C₂×#430×C₆₇	2
#21184	67