Subgroups — #30874261

Order 17152.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863490	C₂	1	2	yes	maximal
3	#22563121	C₂	1	2	yes	maximal
4	#22560940	C₂	1	2	yes	maximal
5	#22560460	C₂	1	4	yes	maximal
6	#22560211	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
11	C₂×#14524008	C₂²	1	4	yes	—
12	C₄×Dic₂₆₈	C₄	1	8	yes	complemented
13	#16863490	—	2	8	no	—
14	C₂×#14523977	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#14523968	C₂²	1	2	yes	—
17	C₂×#14523966	C₂²	1	4	yes	—
18	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
19	#16863292	C₂²	1	4	yes	—
20	#16863291	—	2	4	no	—
21	#16863265	—	2	2	no	—
22	C₄⋊C₄×Dic₆₇	—	2	2	no	—
23	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
24	#16863158	C₂²	1	4	yes	—
25	#16863150	C₂²	1	2	yes	—
26	C₂×#107×C₆₇	C₂²	1	2	yes	—
27	C₂×#107×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#384×C₆₇	C₂²	1	2	yes	—
30	C₂²×Dic₁₃₄	C₂³	1	2	yes	—
31	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
32	C₂×Dic₂₆₈	D₄	1	4	yes	—
33	C₂×Dic₂₆₈	Q₈	1	4	yes	—
34	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₄×Dic₁₃₄	—	2	4	no	—
36	C₄×Dic₁₃₄	C₂×C₄	1	8	yes	—
37	#14524008	—	2	8	no	—
38	#14524008	—	4	4	no	—
39	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
42	C₂×Dic₆₇⋊C₄	—	2	4	no	—
43	C₂×Dic₆₇⋊C₄	—	2	4	no	—
44	C₂×Dic₆₇⋊C₄	—	2	2	no	—
45	C₂×C₄×Dic₆₇	—	2	4	no	—
46	C₂×C₄×Dic₆₇	—	2	2	no	—
47	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
48	#14523977	C₂×C₄	1	4	yes	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	#14523968	C₂×C₄	1	8	yes	—
51	#14523966	—	2	8	no	—
52	C₈×Dic₆₇	—	2	4	no	—
53	C₆₇⋊#95	—	2	4	no	—
54	C₆₇⋊#95	D₄	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	—	2	4	no	—
62	#107×C₆₇	—	4	2	no	—
63	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×Dic₁₃₄	—	2	2	no	—
65	C₂×Dic₁₃₄	—	2	2	no	—
66	C₂×Dic₁₃₄	—	2	2	no	—
67	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₃₄	C₂×D₄	1	4	yes	—
69	C₂×Dic₁₃₄	C₂×Q₈	1	4	yes	—
70	C₂²×Dic₆₇	—	2	2	no	—
71	C₂²×Dic₆₇	—	2	2	no	—
72	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
73	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
74	Dic₂₆₈	C₄⋊C₄	1	16	yes	complemented
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	—	2	2	no	—
80	C₄⋊Dic₆₇	Q₁₆	1	4	yes	complemented
81	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
82	Dic₆₇⋊C₄	—	2	8	no	—
83	Dic₆₇⋊C₄	—	4	8	no	—
84	Dic₆₇⋊C₄	—	8	4	no	—
85	C₄×Dic₆₇	—	2	8	no	—
86	C₄×Dic₆₇	—	2	2	no	—
87	C₄×Dic₆₇	—	2	2	no	—
88	C₄×Dic₆₇	—	4	8	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	—	4	4	no	—
102	C₄⋊C₄×C₆₇	—	8	2	no	—
103	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₆₇	#41	1	2	yes	—
105	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₆₇	#135	1	2	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×Dic₆₇	—	2	4	no	—
122	C₂×Dic₆₇	—	2	2	no	—
123	C₂×Dic₆₇	—	2	4	no	—
124	C₂×Dic₆₇	—	2	4	no	—
125	C₂×Dic₆₇	—	2	2	no	—
126	C₂×Dic₆₇	—	2	4	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	C₂×Q₁₆	1	4	yes	—
133	Dic₁₃₄	—	2	4	no	—
134	Dic₁₃₄	—	2	4	no	—
135	Dic₁₃₄	—	2	4	no	—
136	Dic₁₃₄	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
139	C₆₇⋊C₈	—	4	2	no	—
140	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₆₇	D₄²	1	1	yes	characteristic
142	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
143	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
145	C₂²×C₆₇	#576	1	1	yes	characteristic
146	C₂²×C₆₇	#578	1	1	yes	characteristic
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	8	4	no	—
149	C₄×C₆₇	C₄×Q₁₆	1	4	yes	complemented
150	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₆₇	#520	1	1	yes	characteristic
152	C₄×C₆₇	#547	1	1	yes	characteristic
153	C₄×C₆₇	#554	1	1	yes	characteristic
154	Dic₆₇	—	2	8	no	—
155	Dic₆₇	—	2	4	no	—
156	Dic₆₇	—	4	4	no	—
157	Dic₆₇	—	4	2	no	—
158	C₄⋊C₄×Q₁₆	—	67	1	no	maximal, Sylow
159	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
161	C₂×C₆₇	#2951	1	1	yes	characteristic
162	C₂×C₆₇	D₄×Q₁₆	1	1	yes	characteristic
163	C₂×C₆₇	#3287	1	1	yes	characteristic
164	C₂×C₆₇	Q₈×Q₁₆	1	1	yes	characteristic
165	C₂×C₆₇	#3383	1	1	yes	characteristic
166	C₂×C₄×Q₁₆	—	67	2	no	—
167	C₂×C₄×Q₁₆	—	67	1	no	—
168	Q₈×C₄⋊C₄	—	67	2	no	—
169	#1942	—	67	2	no	—
170	#1863	—	67	4	no	—
171	#1845	D₆₇	1	1	yes	characteristic
172	#1803	—	67	2	no	—
173	C₈×C₄⋊C₄	—	67	1	no	—
174	C₆₇	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	67	1	no	—
176	C₂×C₄×Q₈	—	67	4	no	—
177	C₂×C₄×Q₈	—	67	2	no	—
178	C₄×Q₁₆	—	67	8	no	—
179	C₄×Q₁₆	—	134	8	no	—
180	C₂×#107	—	67	1	no	—
181	C₂×#107	C₂×D₆₇	1	2	yes	—
182	C₂×#107	C₂×D₆₇	1	2	yes	—
183	C₂×C₄⋊C₈	—	67	1	no	—
184	C₂×#103	—	67	2	no	—
185	C₂×#103	—	67	4	no	—
186	C₂×C₄×C₈	—	67	2	no	—
187	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
188	#389	—	67	4	no	—
189	#389	—	134	4	no	—
190	#384	—	134	2	no	—
191	#384	C₂×D₆₇	1	2	yes	—
192	C₄×C₄⋊C₄	—	67	1	no	—
193	C₄×C₄⋊C₄	—	134	2	no	—
194	#340	—	67	4	no	—
195	#336	—	67	2	no	—
196	C₂²×Q₈	—	67	2	no	—
197	C₂×Q₁₆	—	67	4	no	—
198	C₂×Q₁₆	—	67	4	no	—
199	C₂×Q₁₆	—	67	4	no	—
200	C₂²×C₈	—	67	2	no	—
201	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
203	C₄×Q₈	—	268	4	no	—
204	C₄×Q₈	—	67	8	no	—
205	C₄×Q₈	—	134	8	no	—
206	C₄×Q₈	—	134	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	67	1	no	—
210	C₂×C₄⋊C₄	—	67	2	no	—
211	C₂×C₄⋊C₄	—	67	1	no	—
212	C₂×C₄⋊C₄	—	134	4	no	—
213	C₂×C₄⋊C₄	—	134	4	no	—
214	C₂×C₄⋊C₄	—	134	2	no	—
215	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
216	C₂×C₄²	—	67	2	no	—
217	C₂×C₄²	—	134	4	no	—
218	C₂×C₄²	—	134	2	no	—
219	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	#107	—	4	2	no	—
222	#107	—	67	4	no	—
223	C₄⋊C₈	—	134	2	no	—
224	#103	—	67	8	no	—
225	#103	—	134	8	no	—
226	C₄×C₈	—	134	4	no	—
227	C₄×C₈	C₄×D₆₇	1	4	yes	—
228	#95	—	67	2	no	—
229	#95	—	134	4	no	—
230	C₂×Q₈	—	67	4	no	—
231	C₂×Q₈	—	67	4	no	—
232	C₂×Q₈	—	67	4	no	—
233	C₂×Q₈	—	134	2	no	—
234	C₂×Q₈	—	134	2	no	—
235	C₂×Q₈	—	134	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	67	2	no	—
238	C₂²×C₄	—	67	2	no	—
239	C₂²×C₄	—	134	2	no	—
240	C₂²×C₄	—	134	2	no	—
241	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
244	Q₁₆	—	67	16	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	134	2	no	—
248	C₂×C₈	—	134	2	no	—
249	C₂×C₈	—	134	2	no	—
250	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
252	C₂×C₈	D₄×D₆₇	1	2	yes	—
253	C₂×C₈	Q₈×D₆₇	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	268	8	no	—
259	C₄⋊C₄	—	536	4	no	—
260	C₄⋊C₄	—	67	4	no	—
261	C₄⋊C₄	—	67	4	no	—
262	C₄⋊C₄	—	134	8	no	—
263	C₄⋊C₄	—	134	2	no	—
264	C₄⋊C₄	—	134	2	no	—
265	C₄²	—	268	8	no	—
266	C₄²	—	134	8	no	—
267	C₄²	—	134	2	no	—
268	C₄²	—	134	2	no	—
269	C₄²	C₂×C₄×D₆₇	1	2	yes	—
270	C₄²	#11761629	1	2	yes	—
271	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
272	Q₈	—	67	8	no	—
273	Q₈	—	134	4	no	—
274	Q₈	—	134	4	no	—
275	Q₈	—	134	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	268	2	no	—
282	C₂×C₄	—	268	2	no	—
283	C₂×C₄	—	67	4	no	—
284	C₂×C₄	—	134	2	no	—
285	C₂×C₄	—	134	4	no	—
286	C₂×C₄	—	134	2	no	—
287	C₂×C₄	—	134	2	no	—
288	C₂×C₄	—	134	2	no	—
289	C₂×C₄	—	134	4	no	—
290	C₂×C₄	—	134	4	no	—
291	C₂×C₄	—	134	4	no	—
292	C₂×C₄	—	134	2	no	—
293	C₂×C₄	#14524005	1	2	yes	—
294	C₂×C₄	#14524005	1	2	yes	—
295	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
296	C₂×C₄	#14524062	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
301	C₂×C₄	#14524088	1	1	yes	characteristic
302	C₂×C₄	#14524091	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	268	2	no	—
305	C₈	C₄⋊C₄×D₆₇	1	4	yes	—
306	C₂²	C₂×#14524005	1	1	yes	characteristic
307	C₂²	#16864031	1	1	yes	characteristic
308	C₂²	#16864061	1	1	yes	characteristic
309	C₂²	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
310	C₂²	C₂×#14524062	1	1	yes	characteristic
311	C₂²	#16864197	1	1	yes	characteristic
312	C₂²	#16864209	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	8	4	no	—
315	C₄	—	268	4	no	—
316	C₄	—	268	2	no	—
317	C₄	—	134	8	no	—
318	C₄	—	134	4	no	—
319	C₄	#16863490	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
321	C₄	#16863877	1	1	yes	characteristic
322	C₄	#16864049	1	1	yes	characteristic
323	C₄	#16864107	1	1	yes	characteristic
324	C₂	#22563109	1	1	yes	characteristic
325	C₂	C₂×#16863490	1	1	yes	characteristic
326	C₂	#22569083	1	1	yes	characteristic
327	C₂	#22571959	1	1	yes	characteristic
328	C₂	#22571967	1	1	yes	characteristic
329	C₂	#22573092	1	1	yes	characteristic
330	C₂	#22573102	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1845	D₆₇
hypocenter	C₆₇	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₂₆₈	C₄	semidirect
74. Dic₂₆₈	C₄⋊C₄	semidirect
80. C₄⋊Dic₆₇	Q₁₆	semidirect
149. C₄×C₆₇	C₄×Q₁₆	semidirect
174. C₆₇	C₄⋊C₄×Q₁₆	semidirect
319. C₄	#16863490	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆₈	2
C₂×#16863490	2
#22563121	2
#22560940	2
#22560460	2
#22560211	2
C₈×C₄⋊Dic₆₇	2
C₆₇×#1845	2
C₄⋊C₄×Q₁₆	67