Subgroups — #30859199

Order 17152.

982 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
3	#22562832	C₂	1	2	yes	maximal
4	#22560953	C₂	1	4	yes	maximal, complemented
5	#22560943	C₂	1	4	yes	maximal, complemented
6	#22559930	C₂	1	2	yes	maximal
7	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	C₂×#14523977	—	2	4	no	—
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523977	C₂²	1	2	yes	—
14	#16863351	—	2	4	no	—
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	#16863295	C₂²	1	8	yes	—
18	#16863271	—	2	4	no	—
19	#16863266	—	2	2	no	—
20	#16863262	—	2	2	no	—
21	#16863158	C₂²	1	8	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	#427×C₆₇	C₄	1	8	yes	complemented
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	C₂³×Dic₆₇	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	Q₈	1	4	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	#14524002	—	2	2	no	—
33	#14524002	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	8	no	—
35	C₂×C₄⋊Dic₆₇	—	2	4	no	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	4	no	—
38	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
41	C₂×C₂²⋊Dic₆₇	D₄	1	2	yes	—
42	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
43	#14523977	—	2	4	no	—
44	#14523977	—	2	4	no	—
45	#14523953	—	2	4	no	—
46	C₈×Dic₆₇	—	2	4	no	—
47	C₆₇⋊#95	—	2	8	no	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	D₄	1	4	yes	—
51	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	#117×C₆₇	—	2	4	no	—
55	#117×C₆₇	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	#107×C₆₇	C₂×C₄	1	8	yes	complemented
60	#106×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₂²×Dic₆₇	—	2	4	no	—
62	C₂²×Dic₆₇	—	4	4	no	—
63	C₂²×Dic₆₇	—	8	4	no	—
64	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
65	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
66	#11761612	—	2	8	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₆₇	—	2	2	no	—
71	C₄⋊Dic₆₇	—	2	2	no	—
72	C₄⋊Dic₆₇	—	2	4	no	—
73	C₄⋊Dic₆₇	—	2	4	no	—
74	C₄⋊Dic₆₇	—	4	4	no	—
75	C₄⋊Dic₆₇	—	8	8	no	—
76	C₂²⋊Dic₆₇	—	2	4	no	—
77	C₄×Dic₆₇	—	2	2	no	—
78	C₄×Dic₆₇	—	2	2	no	—
79	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
98	C₂²⋊C₄×C₆₇	—	2	8	no	—
99	C₄²×C₆₇	—	2	8	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	C₄×D₄	1	2	yes	—
103	C₂³×C₆₇	#121	1	2	yes	—
104	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	8	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₆₇	#121	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₆₇	#125	1	4	yes	—
115	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₆₇	—	2	2	no	—
123	C₂×Dic₆₇	—	2	2	no	—
124	C₂×Dic₆₇	—	2	4	no	—
125	C₂×Dic₆₇	—	2	4	no	—
126	C₂×Dic₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	4	8	no	—
128	C₂×Dic₆₇	—	8	8	no	—
129	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₆₇⋊C₈	—	2	4	no	—
132	C₂²×C₆₇	—	2	4	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	#389	1	4	yes	—
137	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂²×C₆₇	#576	1	2	yes	—
141	C₂²×C₆₇	#578	1	2	yes	—
142	C₄×C₆₇	—	4	8	no	—
143	C₄×C₆₇	#389	1	4	yes	—
144	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
148	Dic₆₇	—	4	2	no	—
149	Dic₆₇	—	4	2	no	—
150	Dic₆₇	—	8	4	no	—
151	#21156	—	67	1	no	maximal, Sylow
152	C₂×C₆₇	—	2	4	no	—
153	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
154	C₂×C₆₇	#2951	1	2	yes	—
155	C₂×C₆₇	#3058	1	1	yes	characteristic
156	C₂×C₆₇	#3059	1	1	yes	characteristic
157	C₂×C₆₇	#3164	1	2	yes	—
158	C₂×#427	D₆₇	1	1	yes	characteristic
159	C₂²×#107	—	67	1	no	—
160	C₂×#405	—	67	1	no	—
161	#2317	—	67	2	no	—
162	#1944	—	67	4	no	—
163	#1942	—	67	4	no	—
164	#1736	—	67	2	no	—
165	C₆₇	#21156	1	1	yes	characteristic, complemented, Sylow
166	C₂²×M₁₆	—	67	1	no	—
167	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#117	—	67	1	no	—
169	C₂×#117	C₂×D₆₇	1	2	yes	—
170	C₂²×C₄⋊C₄	—	67	2	no	—
171	#427	Dic₆₇	1	8	yes	complemented
172	C₂×#107	—	67	2	no	—
173	C₂×#107	—	67	2	no	—
174	C₂×#107	—	134	4	no	—
175	C₂×#107	C₂×D₆₇	1	2	yes	—
176	C₂×#106	C₂×D₆₇	1	2	yes	—
177	#405	—	134	4	no	—
178	C₂×#97	—	67	2	no	—
179	C₂×C₄×C₈	—	67	2	no	—
180	#389	—	67	8	no	—
181	#385	—	134	4	no	—
182	#384	—	134	2	no	—
183	#382	—	134	2	no	—
184	#340	—	67	8	no	—
185	C₂³×C₄	—	134	2	no	—
186	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
187	C₂×M₁₆	—	67	4	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	—	134	2	no	—
190	C₂²×C₈	—	67	2	no	—
191	C₂²×C₈	Dic₁₃₄	1	4	yes	—
192	C₂²×C₈	D₂₆₈	1	4	yes	—
193	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
194	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
196	#117	—	2	4	no	—
197	#117	—	134	2	no	—
198	#117	—	134	2	no	—
199	#117	C₂×Dic₆₇	1	8	yes	—
200	C₂×C₄⋊C₄	—	67	1	no	—
201	C₂×C₄⋊C₄	—	67	1	no	—
202	C₂×C₄⋊C₄	—	67	4	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₄⋊C₄	—	134	4	no	—
205	C₂×C₄⋊C₄	—	134	4	no	—
206	C₂×C₄⋊C₄	—	134	8	no	—
207	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	67	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	67	2	no	—
212	#107	—	134	4	no	—
213	#107	—	134	4	no	—
214	#107	C₂×Dic₆₇	1	8	yes	complemented
215	#106	C₂×Dic₆₇	1	8	yes	complemented
216	#97	—	134	4	no	—
217	C₄×C₈	—	134	4	no	—
218	#95	—	134	8	no	—
219	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	268	4	no	—
222	C₂²×C₄	—	536	4	no	—
223	C₂²×C₄	—	67	2	no	—
224	C₂²×C₄	—	67	2	no	—
225	C₂²×C₄	—	134	4	no	—
226	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
227	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
229	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
230	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
231	C₂²×C₄	#11761627	1	2	yes	—
232	C₂²×C₄	#11761629	1	2	yes	—
233	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
234	M₁₆	—	134	8	no	—
235	C₂×C₈	—	67	4	no	—
236	C₂×C₈	—	134	2	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	C₄⋊Dic₆₇	1	16	yes	complemented
239	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
240	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
241	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
242	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
243	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
244	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	268	4	no	—
247	C₄⋊C₄	—	536	8	no	—
248	C₄⋊C₄	—	134	4	no	—
249	C₄⋊C₄	—	134	4	no	—
250	C₄⋊C₄	—	134	2	no	—
251	C₄⋊C₄	—	134	2	no	—
252	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	134	4	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	134	2	no	—
257	C₄²	—	134	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	D₄×Dic₆₇	1	2	yes	—
261	C₂³	#14524018	1	2	yes	—
262	C₂³	#14524042	1	2	yes	—
263	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	268	4	no	—
268	C₂×C₄	—	268	8	no	—
269	C₂×C₄	—	536	8	no	—
270	C₂×C₄	—	134	2	no	—
271	C₂×C₄	—	134	2	no	—
272	C₂×C₄	—	134	4	no	—
273	C₂×C₄	—	134	4	no	—
274	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
275	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
276	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
277	C₂×C₄	#14524018	1	2	yes	—
278	C₂×C₄	#14524023	1	4	yes	—
279	C₂×C₄	#14524034	1	4	yes	—
280	C₂×C₄	#14524042	1	2	yes	—
281	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
283	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
285	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
286	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
287	C₈	—	134	4	no	—
288	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
289	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	#16863295	1	4	yes	—
295	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
296	C₂²	C₂×#14524018	1	1	yes	characteristic
297	C₂²	C₂×#14524042	1	1	yes	characteristic
298	C₂²	#16864198	1	2	yes	—
299	C₂²	#16864210	1	2	yes	—
300	C₄	—	4	8	no	—
301	C₄	—	268	2	no	—
302	C₄	—	268	2	no	—
303	C₄	—	536	4	no	—
304	C₄	#16863295	1	4	yes	—
305	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
306	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
307	C₄	C₂×#14524023	1	1	yes	characteristic
308	C₄	C₂×#14524034	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	C₂×#16863295	1	1	yes	characteristic
311	C₂	#22569085	1	2	yes	—
312	C₂	#22569858	1	1	yes	characteristic
313	C₂	#22569865	1	1	yes	characteristic
314	C₂	#22570749	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#21156
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #22560953	C₂	semidirect
5. #22560943	C₂	semidirect
24. #427×C₆₇	C₄	semidirect
59. #107×C₆₇	C₂×C₄	semidirect
60. #106×C₆₇	C₂×C₄	semidirect
89. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
165. C₆₇	#21156	semidirect
171. #427	Dic₆₇	semidirect
214. #107	C₂×Dic₆₇	semidirect
215. #106	C₂×Dic₆₇	semidirect
238. C₂×C₈	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523977	2
C₂×#16863351	2
#22562832	2
#22560953	2
#22560943	2
#22559930	2
C₂×#427×C₆₇	2
#21156	67