Subgroups — #30745404

Order 17152.

818 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558050	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	1	no	—
6	#16863449	—	2	2	no	—
7	#16863449	C₄	1	4	yes	complemented
8	#16863420	—	2	2	no	—
9	#16863420	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	—	2	1	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×#95×C₆₇	—	2	1	no	—
23	C₂×#11761612	—	2	4	no	—
24	C₂×#11761612	—	4	2	no	—
25	C₂×#11761612	—	4	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	complemented
32	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
33	C₄×C₆₇⋊C₈	—	2	2	no	—
34	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	4	2	no	—
45	C₂×C₄²×C₆₇	—	4	2	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	2	yes	—
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#95×C₆₇	—	2	2	no	—
55	#95×C₆₇	—	2	2	no	—
56	#11761612	—	4	8	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	4	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	8	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	D₈	1	2	yes	—
79	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
81	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₆₇	—	2	8	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	M₁₆	1	8	yes	complemented
88	C₄²×C₆₇	—	4	4	no	—
89	C₄²×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	C₄²	1	2	yes	—
92	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
95	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
96	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
97	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	#95	1	2	yes	—
104	C₂³×C₆₇	#102	1	2	yes	—
105	C₂³×C₆₇	#103	1	2	yes	—
106	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	8	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	16	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	#95	1	2	yes	—
119	C₂×C₄×C₆₇	#95	1	2	yes	—
120	C₂×C₄×C₆₇	#95	1	2	yes	—
121	C₂×C₄×C₆₇	#102	1	4	yes	—
122	C₂×C₄×C₆₇	#102	1	2	yes	—
123	C₂×C₄×C₆₇	#103	1	4	yes	—
124	C₂×C₄×C₆₇	#103	1	2	yes	—
125	C₂×C₄×C₆₇	#106	1	4	yes	—
126	C₂×C₄×C₆₇	#107	1	4	yes	—
127	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
138	C₆₇⋊C₈	—	4	4	no	—
139	C₆₇⋊C₈	—	4	4	no	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	4	no	—
144	C₂²×C₆₇	#340	1	4	yes	—
145	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
146	C₂²×C₆₇	#407	1	2	yes	—
147	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
149	C₂²×C₆₇	#421	1	2	yes	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	4	8	no	—
155	C₄×C₆₇	#340	1	4	yes	—
156	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
157	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
158	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
159	C₄×C₆₇	#423	1	2	yes	—
160	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
161	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
162	#7901	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	4	no	—
164	C₂×C₆₇	#1486	1	2	yes	—
165	C₂×C₆₇	#1487	1	2	yes	—
166	C₂×C₆₇	#1728	1	1	yes	characteristic
167	C₂×C₆₇	#1733	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
169	C₂×#431	—	67	1	no	—
170	C₂×#423	—	67	1	no	—
171	C₂×C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
172	#1279	—	67	4	no	—
173	C₆₇	#7901	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	134	1	no	—
175	C₂²×C₄⋊C₄	Dic₆₇	1	2	yes	—
176	C₂²×C₄²	—	2	1	no	—
177	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
178	#431	—	67	4	no	—
179	#431	—	134	2	no	—
180	#423	—	67	4	no	—
181	#423	—	134	2	no	—
182	C₂×C₄⋊C₈	—	67	2	no	—
183	C₂×C₄⋊C₈	—	134	1	no	—
184	C₂×C₄⋊C₈	—	134	1	no	—
185	C₂×C₂²⋊C₈	—	134	1	no	—
186	C₂×C₄×C₈	—	67	2	no	—
187	C₄×C₄⋊C₄	—	2	1	no	—
188	C₄×C₄⋊C₄	—	2	1	no	—
189	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
190	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
191	C₂×#95	—	2	1	no	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
194	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂×M₁₆	—	268	2	no	—
197	C₂×M₁₆	—	268	2	no	—
198	C₂×M₁₆	—	134	4	no	—
199	C₂²×C₈	—	134	2	no	—
200	C₂²×C₈	—	134	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	8	no	—
203	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	4	2	no	—
207	C₂×C₄²	—	4	2	no	—
208	C₂×C₄²	Dic₁₃₄	1	2	yes	—
209	C₂×C₄²	C₄×D₆₇	1	2	yes	—
210	C₂×C₄²	C₄×D₆₇	1	2	yes	—
211	C₂×C₄²	D₂₆₈	1	2	yes	—
212	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	C₄⋊C₈	—	67	4	no	—
217	C₄⋊C₈	—	134	4	no	—
218	C₄⋊C₈	—	134	4	no	—
219	C₄⋊C₈	—	134	2	no	—
220	C₂²⋊C₈	—	134	4	no	—
221	C₄×C₈	—	67	4	no	—
222	C₄×C₈	—	134	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
233	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
234	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
235	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
238	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
239	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
240	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
241	C₂²×C₄	#11761613	1	2	yes	—
242	C₂²×C₄	#11761615	1	2	yes	—
243	C₂²×C₄	#11761616	1	2	yes	—
244	C₂²×C₄	#11761618	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
248	M₁₆	—	268	8	no	—
249	C₂×C₈	—	268	4	no	—
250	C₂×C₈	—	268	2	no	—
251	C₂×C₈	—	268	2	no	—
252	C₂×C₈	—	134	4	no	—
253	C₂×C₈	—	134	4	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	#11761612	1	8	yes	complemented
257	C₄²	—	4	4	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₄×Dic₆₇	1	2	yes	—
261	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
262	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
263	C₄²	Dic₆₇⋊C₄	1	2	yes	—
264	C₄²	Dic₆₇⋊C₄	1	2	yes	—
265	C₄²	C₄⋊Dic₆₇	1	2	yes	—
266	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
268	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
269	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₆₇⋊#95	1	2	yes	—
273	C₂³	#14523964	1	2	yes	—
274	C₂³	#14523967	1	2	yes	—
275	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₆₇⋊#95	1	2	yes	—
288	C₂×C₄	C₆₇⋊#95	1	2	yes	—
289	C₂×C₄	C₆₇⋊#95	1	2	yes	—
290	C₂×C₄	#14523963	1	4	yes	—
291	C₂×C₄	#14523964	1	2	yes	—
292	C₂×C₄	#14523966	1	4	yes	—
293	C₂×C₄	#14523967	1	2	yes	—
294	C₂×C₄	#14523974	1	4	yes	—
295	C₂×C₄	#14523976	1	4	yes	—
296	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×#11761612	1	4	yes	—
303	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
304	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
307	C₈	—	268	4	no	—
308	C₈	—	268	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	#16863158	1	4	yes	—
314	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
315	C₂²	#16863360	1	2	yes	—
316	C₂²	C₂×#14523964	1	1	yes	characteristic
317	C₂²	C₂×#14523967	1	1	yes	characteristic
318	C₂²	#16863413	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	8	no	—
324	C₄	#16863158	1	4	yes	—
325	C₄	C₄×#11761612	1	2	yes	—
326	C₄	C₂×#14523963	1	1	yes	characteristic
327	C₄	C₂×#14523966	1	1	yes	characteristic
328	C₄	#16863420	1	2	yes	—
329	C₄	C₂×#14523974	1	1	yes	characteristic
330	C₄	C₂×#14523976	1	1	yes	characteristic
331	C₂	—	2	4	no	—
332	C₂	#22558634	1	2	yes	—
333	C₂	#22558638	1	2	yes	—
334	C₂	#22559903	1	1	yes	characteristic
335	C₂	#22559918	1	1	yes	characteristic
336	C₂	C₂×#16863158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×C₄⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#7901
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22558050	C₂	semidirect
7. #16863449	C₄	semidirect
9. #16863420	C₄	semidirect
31. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
34. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
87. C₄⋊C₄×C₆₇	M₁₆	semidirect
173. C₆₇	#7901	semidirect
256. C₄⋊C₄	#11761612	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
C₂×#16863420	2
#22558050	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#7901	67