Subgroups — C₂×#22560117 #30853934

Order 17152.

1046 conjugacy classes of subgroups.

The subgroup lattice has 304 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×M₁₆×Dic₆₇	C₂	1	2	yes	maximal
3	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
4	#22560117	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863160	C₂	1	2	yes	maximal
6	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
10	C₂×#14523979	C₄	1	8	yes	complemented
11	C₂×#14523979	C₂²	1	4	yes	—
12	M₁₆×Dic₆₇	—	2	4	no	—
13	M₁₆×Dic₆₇	C₂²	1	8	yes	—
14	C₂×#14523953	—	2	2	no	—
15	C₂×#14523953	C₄	1	4	yes	—
16	C₂×#14523953	C₄	1	4	yes	—
17	C₂×#14523953	C₂²	1	4	yes	—
18	C₂×C₈×Dic₆₇	—	2	2	no	—
19	#16863160	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₆₇	C₄	1	8	yes	complemented
23	C₂×#108×C₆₇	C₂²	1	4	yes	—
24	C₂³×Dic₆₇	Q₈	1	2	yes	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	4	no	—
27	C₂×#11761612	C₂×C₄	1	8	yes	—
28	C₂×#11761612	C₂³	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
41	#14523979	C₂×C₄	1	16	yes	complemented
42	#14523953	—	2	8	no	—
43	#14523953	C₂×C₄	1	16	yes	—
44	C₈×Dic₆₇	—	2	8	no	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
49	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	#108×C₆₇	C₂×C₄	1	16	yes	complemented
57	C₂²×Dic₆₇	—	4	4	no	—
58	C₂²×Dic₆₇	C₄⋊C₄	1	4	yes	—
59	C₂²×Dic₆₇	C₄⋊C₄	1	4	yes	—
60	C₂²×Dic₆₇	C₂×Q₈	1	4	yes	—
61	#11761612	—	2	8	no	—
62	#11761612	—	2	4	no	—
63	#11761612	C₂²×C₄	1	8	yes	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₂×C₆₇⋊C₈	—	2	8	no	—
67	C₂×C₆₇⋊C₈	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
71	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
72	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
74	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
75	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₆₇	—	2	8	no	—
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
89	C₂×C₈×C₆₇	—	2	8	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	C₄²	1	8	yes	—
92	C₂×C₈×C₆₇	C₄²	1	8	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
100	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
101	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#117	1	2	yes	—
107	C₂×C₄×C₆₇	#117	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₆₇	—	2	8	no	—
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
124	C₈×C₆₇	—	2	8	no	—
125	C₈×C₆₇	C₂×C₄²	1	8	yes	—
126	C₆₇⋊C₈	—	2	4	no	—
127	C₆₇⋊C₈	—	2	8	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
133	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
134	C₂²×C₆₇	#430	1	4	yes	—
135	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
139	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
140	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
142	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
144	Dic₆₇	—	2	8	no	—
145	C₂×#1778	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	#1778	1	4	yes	complemented
150	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#430	1	2	yes	—
152	C₂²×#108	—	67	1	no	—
153	C₂²×#108	D₆₇	1	1	yes	characteristic
154	C₂×C₄×M₁₆	—	67	2	no	—
155	C₂²×#97	—	67	1	no	—
156	#1778	—	67	8	no	—
157	C₂×#341	—	67	2	no	—
158	C₆₇	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	67	2	no	—
160	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
161	C₂³×C₈	—	67	1	no	—
162	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
163	C₂²×C₄²	—	67	1	no	—
164	C₂×#108	—	67	8	no	—
165	C₂×#108	—	67	4	no	—
166	C₂×#108	Dic₆₇	1	8	yes	complemented
167	C₂×#108	C₂×D₆₇	1	4	yes	—
168	C₄×M₁₆	—	67	8	no	—
169	C₄×M₁₆	—	134	4	no	—
170	C₂×#97	—	67	4	no	—
171	C₂×#97	—	67	4	no	—
172	C₂×#97	—	67	4	no	—
173	C₂×#97	—	134	2	no	—
174	C₂×C₄×C₈	—	134	2	no	—
175	#341	—	67	8	no	—
176	C₂³×C₄	—	67	2	no	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	67	8	no	—
181	C₂×M₁₆	—	67	4	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	4	no	—
184	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
185	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	67	2	no	—
188	C₂²×C₈	—	67	2	no	—
189	C₂²×C₈	—	67	2	no	—
190	C₂²×C₈	—	134	2	no	—
191	C₂²×C₈	—	134	2	no	—
192	C₂²×C₈	—	134	2	no	—
193	C₂²×C₈	C₄×D₆₇	1	2	yes	—
194	C₂²×C₈	C₄×D₆₇	1	2	yes	—
195	C₂²×C₈	C₄×D₆₇	1	4	yes	—
196	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
197	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
198	C₂×C₄²	—	67	2	no	—
199	C₂×C₄²	—	67	2	no	—
200	C₂×C₄²	—	67	2	no	—
201	C₂×C₄²	—	67	2	no	—
202	C₂×C₄²	—	67	2	no	—
203	C₂×C₄²	—	67	2	no	—
204	#108	—	67	16	no	—
205	#108	C₂×Dic₆₇	1	16	yes	complemented
206	#97	—	67	16	no	—
207	#97	—	134	8	no	—
208	C₄×C₈	—	134	8	no	—
209	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
210	C₂²×C₄	—	268	4	no	—
211	C₂²×C₄	—	67	4	no	—
212	C₂²×C₄	—	67	4	no	—
213	C₂²×C₄	—	67	4	no	—
214	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
218	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
221	C₂²×C₄	#11761627	1	2	yes	—
222	C₂²×C₄	#11761629	1	2	yes	—
223	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
224	M₁₆	—	2	8	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	67	8	no	—
227	M₁₆	—	134	8	no	—
228	M₁₆	—	134	4	no	—
229	M₁₆	C₂²×Dic₆₇	1	8	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	67	4	no	—
233	C₂×C₈	—	134	4	no	—
234	C₂×C₈	—	134	4	no	—
235	C₂×C₈	—	134	8	no	—
236	C₂×C₈	—	134	4	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	—	134	2	no	—
239	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
240	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
241	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
243	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
244	C₄²	—	67	4	no	—
245	C₄²	—	67	4	no	—
246	C₄²	—	67	4	no	—
247	C₄²	—	67	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
250	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
251	C₂³	#14524008	1	2	yes	—
252	C₂³	Q₈×Dic₆₇	1	2	yes	—
253	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
254	C₂×C₄	—	268	4	no	—
255	C₂×C₄	—	268	4	no	—
256	C₂×C₄	—	67	8	no	—
257	C₂×C₄	—	134	8	no	—
258	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
262	C₂×C₄	#14524001	1	2	yes	—
263	C₂×C₄	#14524001	1	2	yes	—
264	C₂×C₄	#14524005	1	2	yes	—
265	C₂×C₄	#14524005	1	2	yes	—
266	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
267	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
268	C₂×C₄	#14524008	1	2	yes	—
269	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
275	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₈	—	134	4	no	—
278	C₈	—	134	8	no	—
279	C₈	C₂×C₄×Dic₆₇	1	8	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
285	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
286	C₂²	#16863443	1	4	yes	—
287	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
288	C₂²	C₂×#14524008	1	1	yes	characteristic
289	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
290	C₄	—	134	8	no	—
291	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
292	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
293	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
294	C₄	C₂×#14524001	1	1	yes	characteristic
295	C₄	C₂×#14524005	1	1	yes	characteristic
296	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	#22560117	1	4	yes	direct product factor
301	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
302	C₂	C₂×#16863443	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂×#1778
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #22560117	C₂	direct (N × H)
10. C₂×#14523979	C₄	semidirect
22. C₂×#108×C₆₇	C₄	semidirect
41. #14523979	C₂×C₄	semidirect
56. #108×C₆₇	C₂×C₄	semidirect
149. C₂×C₆₇	#1778	semidirect
158. C₆₇	C₂×#1778	semidirect
166. C₂×#108	Dic₆₇	semidirect
205. #108	C₂×Dic₆₇	semidirect
300. C₂	#22560117	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×M₁₆×Dic₆₇	2
C₂²×#14523953	2
#22560117	2
C₂×#16863160	2
C₂²×#108×C₆₇	2
C₂×#1778	67

Subgroups — C2×#22560117 #30853934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560117 #30853934