Subgroups — #30859355

Order 17152.

906 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863444	C₂	1	2	yes	maximal
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
4	#22560982	C₂	1	8	yes	maximal, complemented
5	#22559935	C₂	1	2	yes	maximal
6	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	1	yes	characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂²×#11761612	C₂²	1	2	yes	—
10	#16863444	—	2	8	no	—
11	C₂×#14523978	C₂²	1	8	yes	—
12	#16863420	C₄	1	2	yes	—
13	#16863420	C₄	1	4	yes	—
14	#16863420	C₄	1	2	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
16	C₄×#11761612	—	2	2	no	—
17	C₄×#11761612	C₂²	1	4	yes	—
18	C₂×#14523954	—	2	1	no	—
19	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
20	#16863160	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#423×C₆₇	—	2	1	no	—
24	#423×C₆₇	—	2	1	no	—
25	#423×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
27	C₂×#11761612	—	2	1	no	—
28	C₂×#11761612	—	2	1	no	—
29	C₂×#11761612	—	2	4	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	—	2	2	no	—
32	C₂×#11761612	—	2	4	no	—
33	C₂×#11761612	C₂×C₄	1	4	yes	—
34	C₂×#11761612	C₂×C₄	1	4	yes	—
35	C₂×#11761612	D₄	1	4	yes	—
36	C₂×#11761612	Q₈	1	4	yes	—
37	C₂×#11761612	C₂³	1	2	yes	—
38	C₂×#11761612	C₂³	1	2	yes	—
39	C₂²×C₆₇⋊C₈	—	2	1	no	—
40	C₂²×C₆₇⋊C₈	—	2	1	no	—
41	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
42	#14523978	—	2	16	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
44	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
45	#14523954	—	2	4	no	—
46	C₄×C₆₇⋊C₈	—	2	4	no	—
47	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	—	2	4	no	—
52	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	D₄	1	2	yes	—
60	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
63	C₄⋊C₈×C₆₇	—	2	4	no	—
64	C₄⋊C₈×C₆₇	—	2	4	no	—
65	#11761612	—	2	8	no	—
66	#11761612	—	2	8	no	—
67	#11761612	—	2	4	no	—
68	#11761612	—	2	4	no	—
69	#11761612	—	2	2	no	—
70	#11761612	—	2	2	no	—
71	#11761612	C₂×D₄	1	4	yes	—
72	#11761612	C₂×Q₈	1	4	yes	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	2	4	no	—
79	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
80	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	—	4	4	no	—
82	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	2	yes	—
96	C₂²×C₄×C₆₇	#41	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₆₇	—	2	4	no	—
99	M₁₆×C₆₇	—	2	4	no	—
100	M₁₆×C₆₇	—	4	4	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	4	4	no	—
103	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
107	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
108	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₆₇	C₂×D₄	1	2	yes	—
112	C₄²×C₆₇	C₂×D₄	1	2	yes	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂³×C₆₇	#125	1	2	yes	—
117	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
118	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₆₇	#121	1	2	yes	—
136	C₂×C₄×C₆₇	#121	1	2	yes	—
137	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#125	1	2	yes	—
140	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₇	—	4	4	no	—
148	C₆₇⋊C₈	—	2	8	no	—
149	C₆₇⋊C₈	—	2	4	no	—
150	C₆₇⋊C₈	—	2	4	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	—	4	2	no	—
156	C₂²×C₆₇	#389	1	2	yes	—
157	C₂²×C₆₇	#389	1	2	yes	—
158	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	C₄×C₆₇	—	2	2	no	—
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	#389	1	2	yes	—
165	C₄×C₆₇	#389	1	2	yes	—
166	C₄×C₆₇	#430	1	4	yes	complemented
167	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
170	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	#21177	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	—	2	1	no	—
173	C₂×C₆₇	—	2	1	no	—
174	C₂×C₆₇	—	2	2	no	—
175	C₂×C₆₇	—	2	2	no	—
176	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#430	1	2	yes	—
178	C₂×#430	—	67	2	no	—
179	C₂×#423	—	67	1	no	—
180	C₂×#423	D₆₇	1	1	yes	characteristic
181	C₂×C₄×M₁₆	—	67	1	no	—
182	#1950	—	67	8	no	—
183	#1738	—	67	2	no	—
184	C₆₇	#21177	1	1	yes	characteristic, complemented, Sylow
185	C₂²×M₁₆	—	67	2	no	—
186	C₂²×M₁₆	—	67	1	no	—
187	C₂²×M₁₆	—	67	1	no	—
188	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
189	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
190	#430	—	134	8	no	—
191	C₂×#108	—	67	8	no	—
192	#423	—	2	1	no	—
193	#423	—	2	1	no	—
194	#423	—	67	2	no	—
195	#423	—	67	4	no	—
196	#423	—	67	2	no	—
197	#423	C₂×D₆₇	1	4	yes	—
198	C₂×C₄⋊C₈	—	2	2	no	—
199	C₂×C₄⋊C₈	—	67	4	no	—
200	C₄×M₁₆	—	67	4	no	—
201	C₄×M₁₆	—	134	2	no	—
202	C₂×#97	—	134	1	no	—
203	C₂×C₄×C₈	—	134	1	no	—
204	#341	—	67	8	no	—
205	C₂³×C₄	Dic₁₃₄	1	2	yes	—
206	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	—	2	4	no	—
210	C₂×M₁₆	—	67	4	no	—
211	C₂×M₁₆	—	67	4	no	—
212	C₂×M₁₆	—	67	4	no	—
213	C₂×M₁₆	—	67	4	no	—
214	C₂×M₁₆	—	67	2	no	—
215	C₂×M₁₆	—	67	2	no	—
216	C₂×M₁₆	—	134	4	no	—
217	C₂×M₁₆	—	134	1	no	—
218	C₂×M₁₆	—	134	1	no	—
219	C₂×M₁₆	—	134	2	no	—
220	C₂×M₁₆	—	134	2	no	—
221	C₂×M₁₆	—	134	4	no	—
222	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	—	67	4	no	—
225	C₂²×C₈	—	134	1	no	—
226	C₂²×C₈	—	134	1	no	—
227	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
228	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
229	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₄²	C₄×D₆₇	1	2	yes	—
231	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
235	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
236	#108	—	134	16	no	—
237	C₄⋊C₈	—	2	4	no	—
238	C₄⋊C₈	—	2	4	no	—
239	C₄⋊C₈	—	67	8	no	—
240	C₄⋊C₈	—	67	8	no	—
241	#97	—	134	4	no	—
242	C₄×C₈	—	134	4	no	—
243	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
246	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
247	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
248	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₆₇	1	2	yes	—
252	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
254	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
255	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
256	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
257	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
258	C₂²×C₄	#11761627	1	2	yes	—
259	C₂²×C₄	#11761629	1	2	yes	—
260	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	4	no	—
263	M₁₆	—	4	4	no	—
264	M₁₆	—	67	4	no	—
265	M₁₆	—	67	4	no	—
266	M₁₆	—	134	8	no	—
267	M₁₆	—	134	8	no	—
268	M₁₆	—	134	4	no	—
269	M₁₆	—	134	4	no	—
270	M₁₆	—	134	2	no	—
271	M₁₆	—	134	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	—	67	8	no	—
275	C₂×C₈	—	134	4	no	—
276	C₂×C₈	—	134	4	no	—
277	C₂×C₈	—	134	4	no	—
278	C₂×C₈	—	134	2	no	—
279	C₂×C₈	—	134	2	no	—
280	C₂×C₈	—	134	4	no	—
281	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
282	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
283	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
284	C₄²	Dic₆₇⋊C₄	1	2	yes	—
285	C₄²	Dic₆₇⋊C₄	1	2	yes	—
286	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₆₇	1	2	yes	—
288	C₄²	C₂×C₄×D₆₇	1	2	yes	—
289	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
291	C₂³	—	4	2	no	—
292	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
293	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
294	C₂³	#14524034	1	2	yes	—
295	C₂³	#14524042	1	2	yes	—
296	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
304	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
305	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
306	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
308	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
309	C₂×C₄	#14524005	1	2	yes	—
310	C₂×C₄	#14524005	1	2	yes	—
311	C₂×C₄	#14524008	1	2	yes	—
312	C₂×C₄	#14524008	1	2	yes	—
313	C₂×C₄	#14524014	1	2	yes	—
314	C₂×C₄	#14524014	1	2	yes	—
315	C₂×C₄	#14524019	1	2	yes	—
316	C₂×C₄	#14524019	1	2	yes	—
317	C₂×C₄	#14524034	1	2	yes	—
318	C₂×C₄	#14524042	1	2	yes	—
319	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
322	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
323	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
324	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
325	C₈	—	4	4	no	—
326	C₈	—	134	8	no	—
327	C₈	—	134	4	no	—
328	C₈	—	134	4	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	#16863292	1	2	yes	—
335	C₂²	#16863292	1	2	yes	—
336	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
337	C₂²	C₂×#14524034	1	1	yes	characteristic
338	C₂²	C₂×#14524042	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	#16863292	1	2	yes	—
343	C₄	#16863292	1	2	yes	—
344	C₄	#16863444	1	4	yes	complemented
345	C₄	C₂×#14524005	1	1	yes	characteristic
346	C₄	C₂×#14524008	1	1	yes	characteristic
347	C₄	C₂×#14524014	1	1	yes	characteristic
348	C₄	C₂×#14524019	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	2	2	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	C₂×#16863292	1	1	yes	characteristic
354	C₂	C₂×#16863444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	#21177
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #22560982	C₂	semidirect
166. C₄×C₆₇	#430	semidirect
184. C₆₇	#21177	semidirect
344. C₄	#16863444	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863444	2
C₂×#16863420	2
C₂×C₄×#11761612	2
#22560982	2
#22559935	2
C₂×#423×C₆₇	2
#21177	67