Subgroups — #30761592

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863164	C₂	1	2	yes	maximal
2	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
3	#22559930	C₂	1	2	yes	maximal
4	C₂×#16863159	C₂	1	1	yes	maximal, characteristic
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	#16863164	C₄	1	8	yes	complemented
9	#16863161	—	2	2	no	—
10	#16863159	—	2	1	no	—
11	#16863159	—	2	1	no	—
12	#16863158	—	2	2	no	—
13	#16863158	—	2	2	no	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	#405×C₆₇	—	2	2	no	—
18	#405×C₆₇	C₄	1	4	yes	complemented
19	C₂×#97×C₆₇	—	2	1	no	—
20	C₂×C₄×C₈×C₆₇	—	2	1	no	—
21	C₂³×Dic₆₇	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	#117×C₆₇	—	2	1	no	—
48	#117×C₆₇	—	2	1	no	—
49	#117×C₆₇	C₂×C₄	1	2	yes	—
50	#117×C₆₇	Q₈	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄²×C₆₇	—	2	1	no	—
55	#97×C₆₇	—	2	4	no	—
56	C₄×C₈×C₆₇	—	2	4	no	—
57	C₂²×Dic₆₇	—	2	4	no	—
58	C₂²×Dic₆₇	—	4	2	no	—
59	C₂²×Dic₆₇	—	4	2	no	—
60	C₂²×Dic₆₇	—	8	2	no	—
61	C₂²×Dic₆₇	—	8	2	no	—
62	#11761612	—	2	4	no	—
63	#11761612	—	2	4	no	—
64	#11761612	—	4	2	no	—
65	#11761612	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	4	2	no	—
68	C₂×C₆₇⋊C₈	—	4	2	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	4	2	no	—
72	C₄⋊Dic₆₇	—	4	2	no	—
73	C₂⁴×C₆₇	#41	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	M₁₆×C₆₇	—	2	2	no	—
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	C₄²	1	4	yes	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	4	no	—
96	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
98	C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₆₇	—	2	2	no	—
103	C₂²⋊C₄×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	#117	1	2	yes	—
110	C₂³×C₆₇	#117	1	1	yes	characteristic
111	C₂³×C₆₇	#121	1	1	yes	characteristic
112	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂³×C₆₇	#126	1	2	yes	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#117	1	2	yes	—
123	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#123	1	2	yes	—
133	C₂×C₄×C₆₇	#123	1	2	yes	—
134	C₂×C₄×C₆₇	#124	1	2	yes	—
135	C₂×C₄×C₆₇	#125	1	2	yes	—
136	C₂×C₄×C₆₇	#126	1	2	yes	—
137	C₂×Dic₆₇	—	4	4	no	—
138	C₂×Dic₆₇	—	8	4	no	—
139	C₂×Dic₆₇	—	8	2	no	—
140	C₂×Dic₆₇	—	8	2	no	—
141	C₂×Dic₆₇	—	8	4	no	—
142	C₈×C₆₇	—	2	4	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	2	2	no	—
145	C₆₇⋊C₈	—	4	4	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	#376	1	1	yes	characteristic
155	C₂²×C₆₇	#380	1	1	yes	characteristic
156	C₂²×C₆₇	#383	1	1	yes	characteristic
157	C₂²×C₆₇	#387	1	2	yes	—
158	C₂²×C₆₇	#388	1	2	yes	—
159	C₂²×C₆₇	#471	1	2	yes	—
160	C₂²×C₆₇	#472	1	1	yes	characteristic
161	C₂²×C₆₇	#473	1	1	yes	characteristic
162	C₄×C₆₇	—	4	2	no	—
163	C₄×C₆₇	—	4	2	no	—
164	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₆₇	#381	1	1	yes	characteristic
167	C₄×C₆₇	#382	1	1	yes	characteristic
168	C₄×C₆₇	#387	1	2	yes	—
169	C₄×C₆₇	#388	1	2	yes	—
170	Dic₆₇	—	8	4	no	—
171	#10827	—	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	#1434	1	1	yes	characteristic
176	C₂×C₆₇	#1815	1	1	yes	characteristic
177	C₂×C₆₇	#1816	1	1	yes	characteristic
178	C₂×C₆₇	#1930	1	2	yes	—
179	C₂×C₆₇	#1931	1	2	yes	—
180	C₂×#405	D₆₇	1	1	yes	characteristic
181	C₂×#343	—	67	2	no	—
182	C₂×#341	—	67	1	no	—
183	#1736	—	67	2	no	—
184	C₂×#340	—	67	1	no	—
185	C₆₇	#10827	1	1	yes	characteristic, complemented, Sylow
186	C₂²×M₁₆	—	67	2	no	—
187	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
188	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
189	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
190	C₂²×C₄⋊C₄	—	67	2	no	—
191	#405	—	2	2	no	—
192	#405	Dic₆₇	1	4	yes	complemented
193	C₂×#97	—	2	1	no	—
194	C₂×C₄×C₈	—	2	1	no	—
195	#343	—	67	8	no	—
196	#341	—	134	2	no	—
197	#340	—	134	1	no	—
198	#340	—	134	1	no	—
199	#340	—	134	2	no	—
200	#340	—	134	2	no	—
201	C₂³×C₄	—	134	2	no	—
202	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	67	4	no	—
206	C₂×M₁₆	—	134	2	no	—
207	C₂×M₁₆	—	134	2	no	—
208	C₂×M₁₆	—	134	2	no	—
209	C₂×M₁₆	—	134	2	no	—
210	C₂×M₁₆	—	134	2	no	—
211	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
212	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	134	2	no	—
217	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
218	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
221	#117	—	2	1	no	—
222	#117	—	2	1	no	—
223	#117	Dic₁₃₄	1	2	yes	—
224	#117	C₂×Dic₆₇	1	2	yes	—
225	C₂×C₄⋊C₄	—	67	4	no	—
226	C₂×C₄⋊C₄	—	134	2	no	—
227	C₂×C₄⋊C₄	—	134	2	no	—
228	C₂×C₄⋊C₄	—	134	2	no	—
229	C₂×C₄⋊C₄	—	134	2	no	—
230	C₂×C₄⋊C₄	—	134	2	no	—
231	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
232	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	#97	—	2	4	no	—
236	C₄×C₈	—	2	4	no	—
237	C₂⁴	#11761628	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	268	2	no	—
241	C₂²×C₄	—	268	2	no	—
242	C₂²×C₄	—	536	2	no	—
243	C₂²×C₄	—	536	2	no	—
244	C₂²×C₄	—	134	4	no	—
245	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
251	C₂²×C₄	#11761628	1	1	yes	characteristic
252	C₂²×C₄	#11761628	1	2	yes	—
253	C₂²×C₄	#11761628	1	2	yes	—
254	C₂²×C₄	#11761628	1	2	yes	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	268	2	no	—
258	M₁₆	—	268	2	no	—
259	M₁₆	—	134	4	no	—
260	M₁₆	—	134	4	no	—
261	M₁₆	C₄×Dic₆₇	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	268	2	no	—
270	C₂×C₈	—	268	2	no	—
271	C₂×C₈	—	134	4	no	—
272	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
273	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	268	2	no	—
277	C₄⋊C₄	—	268	2	no	—
278	C₄⋊C₄	—	134	4	no	—
279	C₄⋊C₄	—	134	4	no	—
280	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
281	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#14524000	1	2	yes	—
290	C₂³	#14524002	1	1	yes	characteristic
291	C₂³	#14524017	1	1	yes	characteristic
292	C₂³	#14524022	1	1	yes	characteristic
293	C₂³	#14524037	1	2	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	268	4	no	—
300	C₂×C₄	—	536	4	no	—
301	C₂×C₄	—	536	2	no	—
302	C₂×C₄	—	536	2	no	—
303	C₂×C₄	—	536	4	no	—
304	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
307	C₂×C₄	#14524000	1	2	yes	—
308	C₂×C₄	#14524002	1	1	yes	characteristic
309	C₂×C₄	#14524002	1	1	yes	characteristic
310	C₂×C₄	#14524002	1	1	yes	characteristic
311	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
312	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
313	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
314	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
315	C₂×C₄	#14524017	1	1	yes	characteristic
316	C₂×C₄	#14524022	1	1	yes	characteristic
317	C₂×C₄	#14524026	1	2	yes	—
318	C₂×C₄	#14524026	1	2	yes	—
319	C₂×C₄	#14524031	1	2	yes	—
320	C₂×C₄	#14524035	1	2	yes	—
321	C₂×C₄	#14524037	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	268	4	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	#16863242	1	1	yes	characteristic
335	C₂²	#16863256	1	1	yes	characteristic
336	C₂²	#16863264	1	1	yes	characteristic
337	C₂²	#16863280	1	2	yes	—
338	C₂²	#16863287	1	2	yes	—
339	C₂²	#16863659	1	2	yes	—
340	C₂²	#16863666	1	1	yes	characteristic
341	C₂²	#16863671	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	536	4	no	—
345	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
346	C₄	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
347	C₄	#16863259	1	1	yes	characteristic
348	C₄	#16863262	1	1	yes	characteristic
349	C₄	#16863280	1	2	yes	—
350	C₄	#16863287	1	2	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	#22558436	1	1	yes	characteristic
355	C₂	#22560268	1	1	yes	characteristic
356	C₂	#22560273	1	1	yes	characteristic
357	C₂	#22560864	1	2	yes	—
358	C₂	#22560871	1	2	yes	—

	subgroup	G/H
center	C₂³	#14524017
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#10827
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #16863164	C₄	semidirect
18. #405×C₆₇	C₄	semidirect
185. C₆₇	#10827	semidirect
192. #405	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863164	2
C₂×#16863161	2
#22559930	2
C₂×#16863159	2
C₂×#405×C₆₇	2
#10827	67