Subgroups — C₂×#22560957 #30859240

Order 17152.

974 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863422	C₂	1	2	yes	maximal
4	#22560957	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863156	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	2	yes	—
9	C₂×#14523979	—	2	4	no	—
10	C₂×#14523979	C₂²	1	4	yes	—
11	#16863434	C₄	1	4	yes	complemented
12	#16863434	C₂²	1	4	yes	—
13	C₂×#14523977	C₄	1	2	yes	complemented
14	C₂×#14523975	C₄	1	2	yes	complemented
15	#16863422	—	2	4	no	—
16	#16863422	C₂²	1	8	yes	—
17	C₂×Dic₆₇⋊C₈	—	2	4	no	—
18	#16863156	C₂²	1	8	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	—	2	4	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	4	no	—
26	C₂×#11761612	—	2	4	no	—
27	C₂×#11761612	C₂³	1	4	yes	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	#14524002	C₂×C₄	1	4	yes	—
32	#14524002	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄×Dic₆₇	—	2	2	no	—
36	#14523979	—	2	8	no	—
37	#14523977	C₂×C₄	1	8	yes	complemented
38	#14523975	C₂×C₄	1	8	yes	complemented
39	Dic₆₇⋊C₈	—	2	16	no	—
40	C₂³×C₄×C₆₇	—	2	1	no	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	D₄	1	2	yes	—
48	C₂²×C₈×C₆₇	D₄	1	2	yes	—
49	C₂²×C₈×C₆₇	D₄	1	4	yes	—
50	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	—	2	1	no	—
55	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
57	C₄×C₈×C₆₇	—	2	8	no	—
58	C₂²×Dic₆₇	—	2	4	no	—
59	#11761612	—	2	4	no	—
60	#11761612	—	2	4	no	—
61	#11761612	—	4	4	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	4	4	no	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
66	C₂²⋊Dic₆₇	—	2	4	no	—
67	C₂²⋊Dic₆₇	—	2	4	no	—
68	C₄×Dic₆₇	—	2	4	no	—
69	C₄×Dic₆₇	—	2	4	no	—
70	C₂⁴×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
89	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
90	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
96	C₄²×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
99	C₂³×C₆₇	—	2	4	no	—
100	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
101	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₆₇	#125	1	2	yes	—
104	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₆₇	—	2	8	no	—
106	C₂×C₄×C₆₇	—	2	16	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	#121	1	2	yes	—
115	C₂×C₄×C₆₇	#121	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	#125	1	2	yes	—
118	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	8	no	—
127	C₂×Dic₆₇	—	4	8	no	—
128	C₈×C₆₇	—	2	4	no	—
129	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₆₇⋊C₈	—	4	4	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	—	2	4	no	—
136	C₂²×C₆₇	#389	1	2	yes	—
137	C₂²×C₆₇	#389	1	2	yes	—
138	C₂²×C₆₇	#443	1	4	yes	—
139	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
142	C₄×C₆₇	—	2	8	no	—
143	C₄×C₆₇	#389	1	2	yes	—
144	C₄×C₆₇	#389	1	2	yes	—
145	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
148	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	Dic₆₇	—	4	8	no	—
150	C₂×#1945	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	2	2	no	—
154	C₂×C₆₇	#1945	1	4	yes	complemented
155	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
156	C₂×C₆₇	C₂×#443	1	2	yes	—
157	C₂²×#108	—	67	1	no	—
158	C₂×#427	—	67	1	no	—
159	C₂×#424	—	67	2	no	—
160	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
161	#1945	—	67	8	no	—
162	C₂×#339	—	67	2	no	—
163	C₆₇	C₂×#1945	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	67	2	no	—
165	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
166	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
167	C₂×#117	—	67	2	no	—
168	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
169	C₂×#108	—	67	4	no	—
170	C₂×#108	—	134	4	no	—
171	#427	—	67	4	no	—
172	#427	—	67	4	no	—
173	C₂×#107	—	67	2	no	—
174	C₂×#106	—	67	2	no	—
175	#424	—	67	8	no	—
176	#424	—	134	4	no	—
177	C₂×C₄⋊C₈	—	134	4	no	—
178	C₂×C₄×C₈	—	2	4	no	—
179	C₂×C₄×C₈	C₂×D₆₇	1	4	yes	—
180	#339	—	67	8	no	—
181	C₂³×C₄	—	2	1	no	—
182	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
183	C₂×M₁₆	—	67	4	no	—
184	C₂×M₁₆	—	134	4	no	—
185	C₂×M₁₆	—	134	4	no	—
186	C₂×M₁₆	—	134	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₂²×C₈	Dic₁₃₄	1	4	yes	—
191	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₄×D₆₇	1	2	yes	—
194	C₂²×C₈	D₂₆₈	1	2	yes	—
195	C₂²×C₈	D₂₆₈	1	2	yes	—
196	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
197	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
198	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
199	#117	—	67	4	no	—
200	#117	—	67	4	no	—
201	#117	—	134	2	no	—
202	#117	—	134	2	no	—
203	C₂×C₄⋊C₄	—	67	4	no	—
204	C₂×C₂²⋊C₄	—	134	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	134	2	no	—
208	C₂×C₄²	Dic₁₃₄	1	4	yes	—
209	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
210	#108	—	134	8	no	—
211	#107	—	67	8	no	—
212	#106	—	67	8	no	—
213	C₄⋊C₈	—	134	16	no	—
214	C₄×C₈	—	2	8	no	—
215	C₂⁴	#11761628	1	1	yes	characteristic
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	134	4	no	—
220	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
223	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
227	C₂²×C₄	#11761628	1	1	yes	characteristic
228	C₂²×C₄	#11761628	1	2	yes	—
229	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
230	M₁₆	—	268	4	no	—
231	M₁₆	—	134	4	no	—
232	M₁₆	—	134	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	268	4	no	—
238	C₂×C₈	—	134	4	no	—
239	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
240	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
241	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
242	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
245	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
246	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
247	C₄⋊C₄	—	67	8	no	—
248	C₄⋊C₄	—	134	4	no	—
249	C₂²⋊C₄	—	134	4	no	—
250	C₂²⋊C₄	—	134	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	134	4	no	—
254	C₄²	—	134	4	no	—
255	C₄²	C₂×Dic₁₃₄	1	4	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
258	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
259	C₂³	#14524022	1	2	yes	—
260	C₂³	#14524035	1	2	yes	—
261	C₂³	C₂×#11761628	1	1	yes	characteristic
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	268	8	no	—
266	C₂×C₄	—	134	8	no	—
267	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
268	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
269	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
270	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
271	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
272	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
273	C₂×C₄	#14524017	1	2	yes	—
274	C₂×C₄	#14524017	1	2	yes	—
275	C₂×C₄	#14524022	1	2	yes	—
276	C₂×C₄	#14524035	1	2	yes	—
277	C₂×C₄	#14524043	1	2	yes	—
278	C₂×C₄	#14524043	1	2	yes	—
279	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
282	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	—	268	4	no	—
287	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
288	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	#16863294	1	2	yes	—
294	C₂²	#16863294	1	2	yes	—
295	C₂²	#16863514	1	4	yes	—
296	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
297	C₂²	C₂×#14524022	1	1	yes	characteristic
298	C₂²	C₂×#14524035	1	1	yes	characteristic
299	C₄	—	2	8	no	—
300	C₄	—	268	8	no	—
301	C₄	#16863294	1	2	yes	—
302	C₄	#16863294	1	2	yes	—
303	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
304	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
305	C₄	C₂×#14524017	1	1	yes	characteristic
306	C₄	C₂×#14524043	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	#22560957	1	4	yes	direct product factor
311	C₂	C₂×#16863294	1	1	yes	characteristic
312	C₂	C₂×#16863514	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#1945
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #22560957	C₂	direct (N × H)
11. #16863434	C₄	semidirect
13. C₂×#14523977	C₄	semidirect
14. C₂×#14523975	C₄	semidirect
37. #14523977	C₂×C₄	semidirect
38. #14523975	C₂×C₄	semidirect
154. C₂×C₆₇	#1945	semidirect
163. C₆₇	C₂×#1945	semidirect
310. C₂	#22560957	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×#14523979	2
C₂×#16863434	2
C₂×#16863422	2
#22560957	2
C₂×#16863156	2
C₂²×C₄×C₈×C₆₇	2
C₂×#1945	67

Subgroups — C2×#22560957 #30859240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560957 #30859240