Subgroups — C₂×#22562003 #31008886

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#22562003	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863225	C₂	1	4	yes	maximal, complemented
5	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	—	2	1	no	—
7	C₂×#14523983	C₄	1	2	yes	—
8	C₂×#14523983	C₄	1	2	yes	—
9	C₂×#14523983	C₄	1	4	yes	—
10	C₂×#14523983	C₂²	1	4	yes	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	C₂×#14523979	—	2	2	no	—
13	C₂×#14523979	—	2	1	no	—
14	C₂×#14523979	C₂²	1	2	yes	—
15	C₂×#14523979	C₂²	1	4	yes	—
16	#16863225	C₂²	1	16	yes	complemented
17	C₂²×M₁₆×C₆₇	—	2	1	no	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#108×C₆₇	—	2	2	no	—
20	C₂×#108×C₆₇	—	2	1	no	—
21	C₂×#108×C₆₇	C₂²	1	2	yes	—
22	C₂×#108×C₆₇	C₂²	1	4	yes	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	4	no	—
25	C₂×#11761612	—	4	1	no	—
26	C₂×#11761612	—	4	2	no	—
27	C₂²×C₆₇⋊C₈	—	4	1	no	—
28	#14523983	C₂×C₄	1	8	yes	—
29	#14523983	C₂×C₄	1	8	yes	—
30	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
33	#14523979	—	2	2	no	—
34	#14523979	—	2	4	no	—
35	#14523979	C₂³	1	4	yes	—
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	4	no	—
39	C₂×M₁₆×C₆₇	—	4	1	no	—
40	C₂×M₁₆×C₆₇	—	4	2	no	—
41	C₂²×C₈×C₆₇	—	4	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	D₄	1	2	yes	—
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	#108×C₆₇	—	2	2	no	—
53	#108×C₆₇	—	2	4	no	—
54	#108×C₆₇	C₂³	1	4	yes	—
55	#11761612	—	2	4	no	—
56	#11761612	—	4	4	no	—
57	#11761612	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	4	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	—
61	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	D₈	1	2	yes	—
68	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	4	4	no	—
74	M₁₆×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	4	4	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	#107	1	1	yes	characteristic
97	C₂³×C₆₇	#107	1	1	yes	characteristic
98	C₂³×C₆₇	#107	1	2	yes	—
99	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	#107	1	2	yes	—
102	C₂×C₄×C₆₇	#107	1	2	yes	—
103	C₂×C₄×C₆₇	#107	1	2	yes	—
104	C₂×C₄×C₆₇	#107	1	2	yes	—
105	C₂×C₄×C₆₇	#107	1	2	yes	—
106	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₆₇	—	4	4	no	—
120	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
126	C₆₇⋊C₈	—	4	4	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	C₂×#107	1	2	yes	—
133	C₂²×C₆₇	C₂×#107	1	2	yes	—
134	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
136	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
137	C₄×C₆₇	C₂×#107	1	2	yes	—
138	C₄×C₆₇	C₂×#107	1	2	yes	—
139	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
141	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
143	C₂×#2161	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	#2161	1	2	yes	complemented
149	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
150	C₂²×M₃₂	—	67	1	no	—
151	C₂²×#108	—	67	1	no	—
152	C₂²×#108	D₆₇	1	1	yes	characteristic
153	#2161	—	67	8	no	—
154	C₂×#368	—	67	4	no	—
155	C₆₇	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	2	1	no	—
157	C₂²×M₁₆	—	134	1	no	—
158	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂×M₃₂	—	67	2	no	—
160	C₂×M₃₂	—	67	2	no	—
161	C₂×M₃₂	—	67	4	no	—
162	C₂×M₃₂	—	67	4	no	—
163	C₂²×C₁₆	—	67	2	no	—
164	C₂×#108	—	2	2	no	—
165	C₂×#108	—	2	1	no	—
166	C₂×#108	—	67	2	no	—
167	C₂×#108	—	67	4	no	—
168	C₂×#108	—	134	2	no	—
169	C₂×#108	—	134	1	no	—
170	C₂×#108	C₂×D₆₇	1	2	yes	—
171	C₂×#108	C₂×D₆₇	1	4	yes	—
172	#368	—	67	16	no	—
173	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	4	2	no	—
177	C₂×M₁₆	—	4	1	no	—
178	C₂×M₁₆	—	268	2	no	—
179	C₂×M₁₆	—	268	1	no	—
180	C₂×M₁₆	—	134	4	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂²×C₈	—	4	1	no	—
183	C₂²×C₈	—	268	1	no	—
184	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
185	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
186	C₂²×C₈	Dic₁₃₄	1	2	yes	—
187	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₆₇	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
194	M₃₂	—	67	8	no	—
195	M₃₂	—	67	8	no	—
196	C₂×C₁₆	—	67	4	no	—
197	C₂×C₁₆	—	67	4	no	—
198	C₂×C₁₆	—	67	4	no	—
199	#108	—	2	2	no	—
200	#108	—	2	4	no	—
201	#108	—	67	4	no	—
202	#108	—	134	2	no	—
203	#108	—	134	4	no	—
204	#108	C₂²×D₆₇	1	4	yes	—
205	C₂⁴	#11761618	1	1	yes	characteristic
206	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
209	C₂²×C₄	#11761613	1	2	yes	—
210	C₂²×C₄	#11761613	1	1	yes	characteristic
211	C₂²×C₄	#11761613	1	1	yes	characteristic
212	C₂²×C₄	#11761618	1	1	yes	characteristic
213	C₂²×C₄	#11761618	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	268	4	no	—
220	M₁₆	—	268	2	no	—
221	M₁₆	—	134	4	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	268	4	no	—
225	C₂×C₈	—	268	2	no	—
226	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
227	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
228	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
229	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
230	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
231	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
232	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
239	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
240	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
241	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
242	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
243	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
244	C₁₆	—	67	8	no	—
245	C₂³	—	4	2	no	—
246	C₂³	#14523976	1	1	yes	characteristic
247	C₂³	#14523976	1	1	yes	characteristic
248	C₂³	#14523976	1	2	yes	—
249	C₂³	C₂×#11761618	1	1	yes	characteristic
250	C₂³	C₂×#11761618	1	2	yes	—
251	C₂×C₄	#14523976	1	2	yes	—
252	C₂×C₄	#14523976	1	2	yes	—
253	C₂×C₄	#14523976	1	2	yes	—
254	C₂×C₄	#14523976	1	1	yes	characteristic
255	C₂×C₄	#14523976	1	1	yes	characteristic
256	C₂×C₄	#14523976	1	2	yes	—
257	C₂×C₄	#14523976	1	2	yes	—
258	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
259	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
260	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×#11761613	1	2	yes	—
263	C₂×C₄	C₂×#11761613	1	2	yes	—
264	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
266	C₂×C₄	C₂×#11761618	1	2	yes	—
267	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	—	268	4	no	—
271	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
272	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
273	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
274	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
275	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
276	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂×#14523976	1	2	yes	—
283	C₂²	C₂×#14523976	1	2	yes	—
284	C₂²	C₂×#14523976	1	1	yes	characteristic
285	C₂²	C₂×#14523976	1	1	yes	characteristic
286	C₂²	C₂²×#11761618	1	1	yes	characteristic
287	C₄	C₂×#14523976	1	2	yes	—
288	C₄	C₂×#14523976	1	2	yes	—
289	C₄	C₂×#14523976	1	1	yes	characteristic
290	C₄	C₂×#14523976	1	1	yes	characteristic
291	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
292	C₄	C₂²×#11761613	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	#22562003	1	2	yes	direct product factor
298	C₂	C₂²×#14523976	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂×#2161
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #22562003	C₂	direct (N × H)
4. C₂×#16863225	C₂	semidirect
16. #16863225	C₂²	semidirect
148. C₂×C₆₇	#2161	semidirect
155. C₆₇	C₂×#2161	semidirect
297. C₂	#22562003	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
#22562003	2
C₂×#16863225	2
C₂²×#108×C₆₇	2
C₂×#2161	67

Subgroups — C2×#22562003 #31008886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562003 #31008886