Subgroups — #30855478

Order 17152.

1324 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	#22568799	C₂	1	2	yes	maximal, complemented
2	C₂×#16863370	C₂	1	2	yes	maximal
3	#22560255	C₂	1	8	yes	maximal, complemented
4	C₂×#16863153	C₂	1	2	yes	maximal, complemented
5	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524002	C₂²	1	4	yes	—
7	#16863438	—	2	8	no	—
8	#16863434	C₂²	1	8	yes	complemented
9	#16863370	—	2	4	no	—
10	#16863370	C₂²	1	8	yes	—
11	C₂×#14523958	—	2	4	no	—
12	#16863153	C₂²	1	8	yes	complemented
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	#568×C₆₇	—	2	8	no	—
15	C₂×#131×C₆₇	C₄	1	4	yes	complemented
16	C₂×#131×C₆₇	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₆₇	C₄	1	4	yes	complemented
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	#14524002	—	2	4	no	—
21	#14524002	—	2	4	no	—
22	#14524002	—	2	4	no	—
23	#14524002	C₂³	1	8	yes	—
24	C₂×C₄⋊Dic₆₇	—	2	4	no	—
25	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
26	C₂×C₄×Dic₆₇	—	2	4	no	—
27	#14523977	—	2	8	no	—
28	#14523975	—	2	8	no	—
29	#14523958	—	2	16	no	—
30	C₂×#41×C₆₇	—	2	4	no	—
31	C₂×#41×C₆₇	—	2	4	no	—
32	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×#41×C₆₇	D₄	1	2	yes	—
34	C₂×#41×C₆₇	Q₈	1	2	yes	—
35	C₂×#41×C₆₇	C₂³	1	2	yes	—
36	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	D₄	1	2	yes	—
38	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	#131×C₆₇	—	2	16	no	—
42	#131×C₆₇	—	2	16	no	—
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	—	2	16	no	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
49	C₂×M₁₆×C₆₇	D₄	1	8	yes	—
50	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×Dic₆₇	—	4	4	no	—
55	C₄⋊Dic₆₇	—	2	8	no	—
56	C₄⋊Dic₆₇	—	4	4	no	—
57	C₂²⋊Dic₆₇	—	2	8	no	—
58	C₂²⋊Dic₆₇	—	4	4	no	—
59	C₄×Dic₆₇	—	2	8	no	—
60	C₄×Dic₆₇	—	4	4	no	—
61	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
62	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
63	#41×C₆₇	—	2	4	no	—
64	#41×C₆₇	—	2	4	no	—
65	#41×C₆₇	—	2	8	no	—
66	#41×C₆₇	—	2	8	no	—
67	#41×C₆₇	—	2	4	no	—
68	#41×C₆₇	—	2	4	no	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
72	C₂×D₄×C₆₇	—	2	8	no	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
77	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	4	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	16	no	—
94	M₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
95	M₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	2	16	no	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
101	C₂³×C₆₇	—	2	4	no	—
102	C₂³×C₆₇	—	8	1	no	—
103	C₂³×C₆₇	—	8	2	no	—
104	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₆₇	#121	1	4	yes	—
108	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	4	no	—
110	Q₈×C₆₇	—	2	4	no	—
111	D₄×C₆₇	—	2	8	no	—
112	D₄×C₆₇	—	2	8	no	—
113	D₄×C₆₇	—	2	4	no	—
114	D₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	2	8	no	—
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	2	yes	—
125	C₂×C₄×C₆₇	#117	1	2	yes	—
126	C₂×C₄×C₆₇	#121	1	4	yes	—
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
133	C₂×Dic₆₇	—	4	8	no	—
134	C₂×Dic₆₇	—	8	8	no	—
135	C₈×C₆₇	—	2	4	no	—
136	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
137	C₂²×C₆₇	—	2	4	no	—
138	C₂²×C₆₇	—	4	4	no	—
139	C₂²×C₆₇	—	4	1	no	—
140	C₂²×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	—	4	1	no	—
142	C₂²×C₆₇	—	4	4	no	—
143	C₂²×C₆₇	—	8	4	no	—
144	C₂²×C₆₇	#380	1	2	yes	—
145	C₂²×C₆₇	#380	1	2	yes	—
146	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₆₇	C₂×#121	1	2	yes	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	#380	1	2	yes	—
150	C₄×C₆₇	#380	1	2	yes	—
151	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
154	Dic₆₇	—	8	8	no	—
155	#20671	—	67	1	no	maximal, Sylow
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	—	4	4	no	—
161	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
162	C₂×#568	D₆₇	1	1	yes	characteristic
163	#2911	—	67	2	no	—
164	C₂×#410	—	67	2	no	—
165	#1812	—	67	8	no	—
166	C₂×H(3,ℤ/4)	—	67	2	no	—
167	C₆₇	#20671	1	1	yes	characteristic, complemented, Sylow
168	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
169	#568	—	2	8	no	—
170	C₂×#131	Dic₆₇	1	4	yes	complemented
171	C₂×#131	C₂×D₆₇	1	4	yes	—
172	C₂²×M₁₆	Dic₆₇	1	4	yes	complemented
173	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
174	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
175	C₂×#117	—	67	4	no	—
176	#428	—	134	8	no	—
177	#427	—	67	8	no	—
178	#410	—	67	8	no	—
179	#410	—	134	4	no	—
180	C₂×#99	—	134	4	no	—
181	H(3,ℤ/4)	—	67	8	no	—
182	C₂×#41	—	2	4	no	—
183	C₂×#41	—	2	4	no	—
184	C₂×#41	Dic₁₃₄	1	2	yes	—
185	C₂×#41	D₂₆₈	1	2	yes	—
186	C₂×#41	C₂×Dic₆₇	1	2	yes	—
187	C₂×#41	C₂²×D₆₇	1	2	yes	—
188	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
189	C₂²×D₄	Dic₁₃₄	1	1	yes	characteristic
190	C₂²×D₄	D₂₆₈	1	2	yes	—
191	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
192	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
193	#131	—	2	16	no	—
194	#131	—	2	16	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	16	no	—
197	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
198	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
199	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
200	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	complemented
201	C₂×M₁₆	C₂²⋊D₆₇	1	8	yes	—
202	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
203	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
204	C₂²×C₈	C₂×Dic₆₇	1	8	yes	complemented
205	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
206	#117	—	67	8	no	—
207	#117	—	134	4	no	—
208	#117	—	134	4	no	—
209	#117	—	134	4	no	—
210	C₂×C₄⋊C₄	—	134	4	no	—
211	C₂×C₂²⋊C₄	—	134	4	no	—
212	C₂×C₄²	—	134	4	no	—
213	#107	—	134	8	no	—
214	#106	—	134	8	no	—
215	#99	—	134	16	no	—
216	C₂⁴	C₄⋊Dic₆₇	1	2	yes	—
217	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
218	#41	—	2	4	no	—
219	#41	—	2	4	no	—
220	#41	—	2	8	no	—
221	#41	—	2	8	no	—
222	#41	—	2	4	no	—
223	#41	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
226	C₂×Q₈	C₂×Dic₁₃₄	1	2	yes	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
230	C₂×D₄	C₄⋊Dic₆₇	1	4	yes	—
231	C₂×D₄	C₂×D₂₆₈	1	4	yes	—
232	C₂×D₄	C₂×Dic₁₃₄	1	2	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	268	4	no	—
236	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
237	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
238	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
239	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
242	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
244	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
245	C₂²×C₄	#11761628	1	4	yes	—
246	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
247	M₁₆	—	2	4	no	—
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	16	no	—
250	M₁₆	C₂²⋊Dic₆₇	1	8	yes	—
251	M₁₆	C₂²⋊Dic₆₇	1	8	yes	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	16	no	—
254	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
255	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
256	C₂×C₈	C₂×C₂²⋊D₆₇	1	8	yes	—
257	C₄⋊C₄	—	268	4	no	—
258	C₄⋊C₄	—	134	8	no	—
259	C₂²⋊C₄	—	268	4	no	—
260	C₂²⋊C₄	—	134	8	no	—
261	C₄²	—	268	4	no	—
262	C₄²	—	134	8	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	8	1	no	—
265	C₂³	—	8	2	no	—
266	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
267	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
268	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
269	C₂³	#14524017	1	4	yes	—
270	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	2	4	no	—
273	D₄	—	2	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	268	8	no	—
282	C₂×C₄	—	536	8	no	—
283	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
286	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
288	C₂×C₄	#14524002	1	2	yes	—
289	C₂×C₄	#14524002	1	2	yes	—
290	C₂×C₄	#14524017	1	4	yes	—
291	C₂×C₄	#14524022	1	4	yes	—
292	C₂×C₄	#14524022	1	4	yes	—
293	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
295	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761628	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	C₂×C₂²⋊Dic₆₇	1	8	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	4	no	—
306	C₂²	#16863256	1	2	yes	—
307	C₂²	#16863256	1	2	yes	—
308	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
309	C₂²	C₂×#14524017	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	536	8	no	—
312	C₄	#16863256	1	2	yes	—
313	C₄	#16863256	1	2	yes	—
314	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
315	C₄	C₂×#14524002	1	1	yes	characteristic
316	C₄	C₂×#14524022	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	4	no	—
322	C₂	C₂×#16863256	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#568×C₆₇	C₂
socle	C₂×C₆₇	C₂×#380
prime radical	C₂²×#41×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#568	D₆₇
hypocenter	C₆₇	#20671
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
1. #22568799	C₂	semidirect
3. #22560255	C₂	semidirect
4. C₂×#16863153	C₂	semidirect
8. #16863434	C₂²	semidirect
12. #16863153	C₂²	semidirect
15. C₂×#131×C₆₇	C₄	semidirect
17. C₂²×M₁₆×C₆₇	C₄	semidirect
48. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
52. C₂²×C₈×C₆₇	C₂×C₄	semidirect
167. C₆₇	#20671	semidirect
170. C₂×#131	Dic₆₇	semidirect
172. C₂²×M₁₆	Dic₆₇	semidirect
200. C₂×M₁₆	C₂×Dic₆₇	semidirect
204. C₂²×C₈	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568799	2
C₂×#16863370	2
#22560255	2
C₂×#16863153	2
C₂×#568×C₆₇	2
#20671	67