Subgroups — C₂×#22558317 #30770753

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
3	#22558317	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	1	no	—
6	C₂×#14523983	—	2	2	no	—
7	C₂×#14523983	—	2	1	no	—
8	C₂×#14523983	C₄	1	4	yes	complemented
9	C₂×#14523983	C₂²	1	2	yes	—
10	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
11	C₂×#14523979	—	2	2	no	—
12	C₂×#14523979	—	2	1	no	—
13	C₂×#14523979	C₄	1	4	yes	complemented
14	C₂×#14523979	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×#107×C₆₇	—	2	1	no	—
19	C₂×#107×C₆₇	—	2	1	no	—
20	C₂×#107×C₆₇	C₄	1	2	yes	—
21	C₂×#107×C₆₇	C₄	1	2	yes	—
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	C₂×#11761612	—	2	4	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	4	1	no	—
26	C₂×#11761612	—	4	2	no	—
27	C₂²×C₆₇⋊C₈	—	4	1	no	—
28	#14523983	—	2	4	no	—
29	#14523983	—	2	2	no	—
30	#14523983	C₂×C₄	1	4	yes	complemented
31	C₂×C₆₇⋊C₁₆	—	2	4	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	#14523979	—	2	4	no	—
34	#14523979	—	2	2	no	—
35	#14523979	C₂×C₄	1	4	yes	complemented
36	C₂³×C₄×C₆₇	—	4	1	no	—
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
54	#107×C₆₇	—	2	2	no	—
55	#107×C₆₇	—	2	2	no	—
56	#107×C₆₇	—	2	2	no	—
57	#107×C₆₇	C₂×C₄	1	4	yes	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	4	no	—
60	#11761612	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	4	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₆₇⋊C₁₆	—	2	4	no	—
64	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	—	8	4	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₄²	1	2	yes	—
80	C₂×C₈×C₆₇	C₄²	1	2	yes	—
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	4	2	no	—
99	C₄⋊C₄×C₆₇	—	4	2	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	#102	1	1	yes	characteristic
104	C₂³×C₆₇	#102	1	1	yes	characteristic
105	C₂³×C₆₇	#103	1	2	yes	—
106	C₂³×C₆₇	#106	1	2	yes	—
107	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	8	4	no	—
110	C₂×C₄×C₆₇	—	8	4	no	—
111	C₂×C₄×C₆₇	—	8	4	no	—
112	C₂×C₄×C₆₇	#95	1	2	yes	—
113	C₂×C₄×C₆₇	#95	1	2	yes	—
114	C₂×C₄×C₆₇	#102	1	2	yes	—
115	C₂×C₄×C₆₇	#102	1	2	yes	—
116	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#103	1	2	yes	—
119	C₂×C₄×C₆₇	#103	1	2	yes	—
120	C₂×C₄×C₆₇	#103	1	2	yes	—
121	C₂×C₄×C₆₇	#106	1	2	yes	—
122	C₂×C₄×C₆₇	#107	1	2	yes	—
123	C₂×C₄×C₆₇	#107	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
130	C₈×C₆₇	#95	1	2	yes	—
131	C₈×C₆₇	#95	1	2	yes	—
132	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₆₇⋊C₈	—	4	4	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	4	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	#340	1	2	yes	—
144	C₂²×C₆₇	#340	1	2	yes	—
145	C₂²×C₆₇	#361	1	2	yes	—
146	C₂²×C₆₇	#362	1	2	yes	—
147	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
150	C₄×C₆₇	—	8	4	no	—
151	C₄×C₆₇	#340	1	2	yes	—
152	C₄×C₆₇	#340	1	2	yes	—
153	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
154	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
156	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
157	C₂×#1387	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	—	2	1	no	—
161	C₂×C₆₇	#1387	1	4	yes	complemented
162	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
164	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
165	C₂²×M₃₂	—	67	1	no	—
166	C₂²×#108	—	67	1	no	—
167	C₂²×#107	D₆₇	1	1	yes	characteristic
168	#1387	—	67	4	no	—
169	C₆₇	C₂×#1387	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	—	134	1	no	—
171	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	—	2	1	no	—
173	C₂×M₃₂	—	67	4	no	—
174	C₂×M₃₂	—	67	2	no	—
175	C₂×M₃₂	—	134	2	no	—
176	C₂×M₃₂	—	134	1	no	—
177	C₂²×C₁₆	—	134	1	no	—
178	C₂×#108	—	67	4	no	—
179	C₂×#108	—	67	2	no	—
180	C₂×#108	—	134	2	no	—
181	C₂×#108	—	134	1	no	—
182	C₂×#107	—	2	1	no	—
183	C₂×#107	—	2	1	no	—
184	C₂×#107	—	2	1	no	—
185	C₂×#107	Dic₆₇	1	2	yes	—
186	C₂×#107	Dic₆₇	1	2	yes	—
187	C₂×#107	C₂×D₆₇	1	2	yes	—
188	C₂³×C₄	—	4	1	no	—
189	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂×M₁₆	—	268	1	no	—
191	C₂×M₁₆	—	268	2	no	—
192	C₂×M₁₆	—	134	4	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂²×C₈	—	268	1	no	—
195	C₂²×C₈	Dic₁₃₄	1	2	yes	—
196	C₂²×C₈	C₄×D₆₇	1	2	yes	—
197	C₂²×C₈	C₄×D₆₇	1	2	yes	—
198	C₂²×C₈	D₂₆₈	1	2	yes	—
199	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₄⋊C₄	—	4	1	no	—
211	M₃₂	—	67	4	no	—
212	M₃₂	—	134	4	no	—
213	M₃₂	—	134	2	no	—
214	C₂×C₁₆	—	134	4	no	—
215	C₂×C₁₆	—	134	2	no	—
216	#108	—	67	4	no	—
217	#108	—	134	4	no	—
218	#108	—	134	2	no	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	C₂×Dic₆₇	1	4	yes	—
223	C₂⁴	#11761616	1	1	yes	characteristic
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	8	4	no	—
228	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
229	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
232	C₂²×C₄	#11761613	1	1	yes	characteristic
233	C₂²×C₄	#11761613	1	1	yes	characteristic
234	C₂²×C₄	#11761615	1	2	yes	—
235	C₂²×C₄	#11761616	1	1	yes	characteristic
236	C₂²×C₄	#11761618	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	M₁₆	—	268	4	no	—
239	M₁₆	—	268	2	no	—
240	M₁₆	—	134	4	no	—
241	C₂×C₈	—	268	4	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
244	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
245	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
246	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
247	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
251	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
252	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
253	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
254	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
260	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₁₆	—	134	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	#14523964	1	1	yes	characteristic
269	C₂³	#14523964	1	1	yes	characteristic
270	C₂³	#14523966	1	2	yes	—
271	C₂³	#14523974	1	2	yes	—
272	C₂³	C₂×#11761616	1	1	yes	characteristic
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	C₆₇⋊#95	1	2	yes	—
278	C₂×C₄	C₆₇⋊#95	1	2	yes	—
279	C₂×C₄	#14523963	1	2	yes	—
280	C₂×C₄	#14523963	1	2	yes	—
281	C₂×C₄	#14523964	1	1	yes	characteristic
282	C₂×C₄	#14523964	1	1	yes	characteristic
283	C₂×C₄	#14523966	1	2	yes	—
284	C₂×C₄	#14523967	1	2	yes	—
285	C₂×C₄	#14523967	1	2	yes	—
286	C₂×C₄	#14523974	1	2	yes	—
287	C₂×C₄	#14523976	1	2	yes	—
288	C₂×C₄	#14523976	1	2	yes	—
289	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
293	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
294	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
295	C₈	—	268	4	no	—
296	C₈	C₆₇⋊#95	1	2	yes	—
297	C₈	C₆₇⋊#95	1	2	yes	—
298	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
299	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
300	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
301	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
302	C₂²	—	2	4	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16863158	1	2	yes	—
309	C₂²	#16863158	1	2	yes	—
310	C₂²	#16863209	1	2	yes	—
311	C₂²	#16863212	1	2	yes	—
312	C₂²	C₂×#14523964	1	1	yes	characteristic
313	C₂²	C₂×#14523966	1	1	yes	characteristic
314	C₂²	C₂×#14523974	1	1	yes	characteristic
315	C₄	—	8	4	no	—
316	C₄	#16863158	1	2	yes	—
317	C₄	#16863158	1	2	yes	—
318	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
319	C₄	C₂×#14523963	1	1	yes	characteristic
320	C₄	C₂×#14523967	1	1	yes	characteristic
321	C₄	C₂×#14523976	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	#22558317	1	4	yes	direct product factor
326	C₂	C₂×#16863158	1	1	yes	characteristic
327	C₂	C₂×#16863209	1	1	yes	characteristic
328	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14523964
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle	C₂³×C₆₇	#102
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	C₂×#1387
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #22558317	C₂	direct (N × H)
8. C₂×#14523983	C₄	semidirect
13. C₂×#14523979	C₄	semidirect
30. #14523983	C₂×C₄	semidirect
35. #14523979	C₂×C₄	semidirect
161. C₂×C₆₇	#1387	semidirect
169. C₆₇	C₂×#1387	semidirect
325. C₂	#22558317	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂²×#14523979	2
#22558317	2
C₂²×#107×C₆₇	2
C₂×#1387	67

Subgroups — C2×#22558317 #30770753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558317 #30770753