Subgroups — #30825065

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761628	C₂	1	1	yes	maximal, characteristic
2	#22568774	C₂	1	2	yes	maximal
3	#22558983	C₂	1	2	yes	maximal, complemented
4	#22558982	C₂	1	4	yes	maximal, complemented
5	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558966	C₂	1	2	yes	maximal, complemented
7	#335×D₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524049	—	2	2	no	—
10	C₄×#11761628	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
13	C₈×D₂₆₈	C₄	1	1	yes	characteristic
14	C₈×D₂₆₈	C₄	1	1	yes	characteristic
15	C₈×C₂²⋊D₆₇	C₄	1	2	yes	—
16	C₈×C₂²⋊D₆₇	C₄	1	2	yes	—
17	#16863449	C₂²	1	1	yes	characteristic
18	#16863447	C₂²	1	2	yes	—
19	#16863422	—	2	2	no	—
20	#16863421	—	2	4	no	—
21	#16863420	C₂²	1	2	yes	—
22	#16863418	C₂²	1	4	yes	—
23	C₄⋊C₈×D₆₇	C₂²	1	4	yes	—
24	C₄×C₈×D₆₇	C₂²	1	2	yes	—
25	#16863149	C₂²	1	2	yes	—
26	#16863149	C₂²	1	2	yes	—
27	#16863148	C₂²	1	4	yes	—
28	#16863148	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#335×C₆₇	C₂²	1	2	yes	—
32	#335×C₆₇	C₂²	1	1	yes	characteristic
33	#335×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761628	D₄	1	1	yes	characteristic
35	#14524049	—	4	8	no	—
36	C₂×#11761612	—	2	2	no	—
37	C₂×#11761611	—	2	4	no	—
38	C₂×C₈×D₆₇	—	2	4	no	—
39	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
43	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
44	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₂²⋊D₆₇	Q₈	1	2	yes	—
47	#14524002	D₄	1	1	yes	characteristic
48	#14524000	D₄	1	2	yes	—
49	C₄²×D₆₇	C₂³	1	2	yes	—
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
52	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
53	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
54	Dic₆₇⋊C₈	—	2	4	no	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
57	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
59	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
60	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
61	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
62	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
63	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
64	C₂²×C₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
70	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#11761628	D₈	1	4	yes	—
76	#11761628	Q₁₆	1	4	yes	—
77	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
83	#11761612	—	4	4	no	—
84	#11761611	—	4	8	no	—
85	C₂×C₆₇⋊C₈	—	2	4	no	—
86	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
88	C₈×D₆₇	—	2	4	no	—
89	C₈×D₆₇	—	4	2	no	—
90	C₈×D₆₇	—	4	8	no	—
91	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
93	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
94	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
95	C₂²⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
96	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
97	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
98	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
99	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
100	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₂×C₈×C₆₇	—	4	4	no	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂²⋊D₆₇	#107	1	8	yes	—
131	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
134	D₂₆₈	#107	1	4	yes	—
135	C₄×D₆₇	—	2	2	no	—
136	C₄×D₆₇	—	2	2	no	—
137	C₄×D₆₇	—	4	2	no	—
138	C₄×D₆₇	C₂×D₈	1	4	yes	—
139	C₄×D₆₇	C₂×Q₁₆	1	4	yes	—
140	Dic₁₃₄	#107	1	4	yes	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	—	4	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₈×C₆₇	#131	1	2	yes	complemented
147	C₆₇⋊C₈	—	2	2	no	—
148	C₆₇⋊C₈	—	2	2	no	—
149	C₆₇⋊C₈	—	4	4	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	C₂×#107	1	2	yes	—
153	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂×D₆₇	—	2	2	no	—
155	C₂×D₆₇	—	2	2	no	—
156	C₂×D₆₇	C₂×#107	1	4	yes	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	C₂×#107	1	2	yes	—
161	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
162	C₄×C₆₇	#568	1	1	yes	characteristic
163	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
165	Dic₆₇	—	2	2	no	—
166	Dic₆₇	—	2	2	no	—
167	Dic₆₇	C₂×#107	1	4	yes	—
168	#17828	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	—	2	2	no	—
170	C₂×C₆₇	#2906	1	1	yes	characteristic
171	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
172	C₂×C₆₇	#2916	1	1	yes	characteristic
173	D₆₇	—	2	4	no	—
174	C₈×#41	—	67	1	no	—
175	#2906	—	67	2	no	—
176	#1571	—	67	2	no	—
177	#1571	—	67	4	no	—
178	#1567	—	67	1	no	—
179	#1567	—	67	2	no	—
180	C₂×#335	—	67	2	no	—
181	C₂×#335	D₆₇	1	1	yes	characteristic, complemented
182	C₆₇	#17828	1	1	yes	characteristic, complemented, Sylow
183	C₂×#131	—	134	2	no	—
184	C₄×#41	—	67	1	no	—
185	C₈×Q₈	—	67	1	no	—
186	C₈×Q₈	—	67	1	no	—
187	C₈×D₄	—	67	2	no	—
188	C₈×D₄	—	67	2	no	—
189	C₈×D₄	—	67	1	no	—
190	C₈×D₄	—	67	1	no	—
191	#431	—	67	1	no	—
192	#431	—	67	2	no	—
193	#424	—	134	4	no	—
194	#424	—	134	2	no	—
195	#423	—	67	2	no	—
196	#423	—	67	4	no	—
197	C₂×C₄⋊C₈	—	67	4	no	—
198	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
199	C₂×C₄×C₈	—	67	2	no	—
200	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
201	#335	—	67	4	no	—
202	#335	—	67	4	no	—
203	#335	—	67	2	no	—
204	#335	—	67	2	no	—
205	#335	C₂×D₆₇	1	1	yes	characteristic
206	#335	C₂×D₆₇	1	1	yes	characteristic
207	#335	C₂×D₆₇	1	2	yes	—
208	C₂×#41	—	67	1	no	—
209	#131	—	268	8	no	—
210	C₂×M₁₆	—	134	4	no	—
211	C₂×M₁₆	—	134	2	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	67	2	no	—
214	C₂²×C₈	—	67	2	no	—
215	C₂²×C₈	—	134	4	no	—
216	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
218	C₄×Q₈	—	67	1	no	—
219	C₄×Q₈	—	67	1	no	—
220	C₄×D₄	—	67	2	no	—
221	C₄×D₄	—	67	2	no	—
222	C₄×D₄	—	67	1	no	—
223	C₄×D₄	—	67	1	no	—
224	#117	—	67	1	no	—
225	#117	—	67	2	no	—
226	C₂×C₄²	—	67	2	no	—
227	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	67	2	no	—
230	C₄⋊C₈	—	67	2	no	—
231	C₄⋊C₈	—	67	1	no	—
232	C₄⋊C₈	—	67	1	no	—
233	C₄⋊C₈	—	67	4	no	—
234	C₄⋊C₈	—	134	4	no	—
235	C₄⋊C₈	—	134	4	no	—
236	C₄⋊C₈	—	134	2	no	—
237	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
238	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
239	C₂²⋊C₈	—	67	2	no	—
240	C₂²⋊C₈	—	67	2	no	—
241	C₂²⋊C₈	—	67	2	no	—
242	C₄×C₈	—	67	2	no	—
243	C₄×C₈	—	67	2	no	—
244	C₄×C₈	—	67	2	no	—
245	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
246	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
249	#41	—	67	4	no	—
250	#41	—	67	4	no	—
251	C₂×Q₈	—	67	1	no	—
252	C₂×D₄	—	67	2	no	—
253	C₂×D₄	—	67	1	no	—
254	C₂²×C₄	—	67	2	no	—
255	C₂²×C₄	—	67	2	no	—
256	C₂²×C₄	—	67	2	no	—
257	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
260	M₁₆	—	268	8	no	—
261	M₁₆	—	268	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	268	2	no	—
268	C₂×C₈	—	268	8	no	—
269	C₂×C₈	—	67	2	no	—
270	C₂×C₈	—	67	2	no	—
271	C₂×C₈	—	134	4	no	—
272	C₂×C₈	—	134	4	no	—
273	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	67	2	no	—
278	C₄⋊C₄	—	67	2	no	—
279	C₄⋊C₄	—	67	1	no	—
280	C₄⋊C₄	—	67	1	no	—
281	C₂²⋊C₄	—	67	2	no	—
282	C₂²⋊C₄	—	67	2	no	—
283	C₂²⋊C₄	—	67	2	no	—
284	C₄²	—	67	2	no	—
285	C₄²	—	67	2	no	—
286	C₄²	—	67	2	no	—
287	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
288	C₄²	D₄×D₆₇	1	1	yes	characteristic
289	C₄²	Q₈×D₆₇	1	1	yes	characteristic
290	C₄²	C₂³×D₆₇	1	1	yes	characteristic
291	C₂³	—	67	2	no	—
292	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
293	Q₈	—	67	4	no	—
294	D₄	—	67	8	no	—
295	D₄	—	67	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	268	2	no	—
300	C₂×C₄	—	67	2	no	—
301	C₂×C₄	—	67	4	no	—
302	C₂×C₄	—	67	4	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	—	134	2	no	—
307	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
309	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
310	C₂×C₄	D₈×D₆₇	1	2	yes	—
311	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
312	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	2	no	—
320	C₈	—	268	4	no	—
321	C₈	—	134	2	no	—
322	C₈	—	134	2	no	—
323	C₈	#14524049	1	2	yes	complemented
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	67	4	no	—
327	C₂²	—	134	2	no	—
328	C₂²	—	134	2	no	—
329	C₂²	#107×D₆₇	1	2	yes	—
330	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	67	4	no	—
335	C₄	—	134	2	no	—
336	C₄	—	134	2	no	—
337	C₄	#107×D₆₇	1	2	yes	—
338	C₄	C₂×#14524049	1	1	yes	characteristic
339	C₄	#16864162	1	1	yes	characteristic
340	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
341	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	134	4	no	—
344	C₂	#22568774	1	1	yes	characteristic
345	C₂	C₂×#107×D₆₇	1	1	yes	characteristic
346	C₂	#22568817	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#17828
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #22558983	C₂	semidirect
4. #22558982	C₂	semidirect
5. #22558968	C₂	semidirect
6. #22558966	C₂	semidirect
7. #335×D₆₇	C₂	semidirect
8. C₂×#335×C₆₇	C₂	semidirect
146. C₈×C₆₇	#131	semidirect
181. C₂×#335	D₆₇	semidirect
182. C₆₇	#17828	semidirect
323. C₈	#14524049	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11761628	2
#22568774	2
#22558983	2
#22558982	2
#22558968	2
#22558966	2
#335×D₆₇	2
C₂×#335×C₆₇	2
#17828	67