Subgroups — C₂×#22560091 #30853840

Order 17152.

1102 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863354	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22560091	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863157	C₂	1	2	yes	maximal
6	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14524002	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
10	#16863434	C₄	1	4	yes	complemented
11	#16863434	C₂²	1	4	yes	—
12	C₂×#14523977	C₄	1	2	yes	complemented
13	C₂×#14523975	C₄	1	2	yes	complemented
14	#16863354	—	2	4	no	—
15	#16863354	C₂²	1	8	yes	—
16	C₂×#14523953	—	2	2	no	—
17	C₂×C₈×Dic₆₇	—	2	2	no	—
18	C₂×C₈×Dic₆₇	C₄	1	8	yes	complemented
19	C₂×C₈×Dic₆₇	C₂²	1	4	yes	—
20	#16863157	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#108×C₆₇	C₄	1	8	yes	complemented
24	C₂×#108×C₆₇	C₂²	1	4	yes	—
25	C₂³×Dic₆₇	—	2	1	no	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	#14524002	—	2	2	no	—
33	#14524002	—	2	2	no	—
34	#14524002	C₂×C₄	1	4	yes	—
35	#14524002	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
38	C₂×C₄×Dic₆₇	—	2	2	no	—
39	C₂×C₄×Dic₆₇	—	2	2	no	—
40	C₂×C₄×Dic₆₇	—	2	1	no	—
41	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
43	#14523977	C₂×C₄	1	8	yes	complemented
44	#14523975	C₂×C₄	1	8	yes	complemented
45	#14523953	—	2	8	no	—
46	C₈×Dic₆₇	—	2	8	no	—
47	C₈×Dic₆₇	C₂×C₄	1	16	yes	complemented
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
52	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	#108×C₆₇	C₂×C₄	1	16	yes	complemented
61	C₂²×Dic₆₇	—	2	8	no	—
62	C₂²×Dic₆₇	—	2	4	no	—
63	C₂²×Dic₆₇	—	2	2	no	—
64	C₂²×Dic₆₇	—	2	4	no	—
65	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
66	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
67	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
69	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
70	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
71	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
72	C₄⋊Dic₆₇	—	2	4	no	—
73	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
74	C₂²⋊Dic₆₇	—	2	4	no	—
75	C₂²⋊Dic₆₇	—	2	4	no	—
76	C₄×Dic₆₇	—	2	4	no	—
77	C₄×Dic₆₇	—	2	4	no	—
78	C₄×Dic₆₇	—	2	4	no	—
79	C₄×Dic₆₇	—	2	2	no	—
80	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
81	C₂⁴×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₆₇	#41	1	2	yes	—
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₆₇	—	2	8	no	—
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂×C₈×C₆₇	—	2	8	no	—
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	C₄²	1	4	yes	—
98	C₂×C₈×C₆₇	C₄²	1	4	yes	—
99	C₂×C₈×C₆₇	C₄²	1	8	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
104	C₂³×C₆₇	—	2	4	no	—
105	C₂³×C₆₇	#117	1	2	yes	—
106	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
109	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₇	#117	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₆₇	—	2	8	no	—
129	C₂×Dic₆₇	—	2	16	no	—
130	C₂×Dic₆₇	—	2	4	no	—
131	C₂×Dic₆₇	—	2	8	no	—
132	C₂×Dic₆₇	—	4	8	no	—
133	C₈×C₆₇	—	2	8	no	—
134	C₈×C₆₇	C₂×C₄²	1	4	yes	—
135	C₈×C₆₇	C₂×C₄²	1	4	yes	—
136	C₆₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
137	C₆₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	—	2	4	no	—
142	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
144	C₂²×C₆₇	#443	1	4	yes	—
145	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
149	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
150	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
151	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
154	Dic₆₇	—	2	8	no	—
155	Dic₆₇	—	4	8	no	—
156	C₂×#1772	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	#1772	1	4	yes	complemented
161	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₆₇	C₂×#443	1	2	yes	—
163	C₂²×#108	D₆₇	1	1	yes	characteristic
164	C₂×#427	—	67	1	no	—
165	C₂×#405	—	67	2	no	—
166	C₂²×C₄×C₈	—	67	1	no	—
167	#1772	—	67	8	no	—
168	C₂×#339	—	67	2	no	—
169	C₆₇	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
171	C₂³×C₈	—	67	1	no	—
172	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
173	C₂×#117	—	67	2	no	—
174	C₂²×C₄²	—	67	1	no	—
175	C₂×#108	Dic₆₇	1	8	yes	complemented
176	C₂×#108	C₂×D₆₇	1	4	yes	—
177	#427	—	67	4	no	—
178	#427	—	67	4	no	—
179	C₂×#107	—	67	2	no	—
180	C₂×#106	—	67	2	no	—
181	#405	—	67	8	no	—
182	#405	—	134	4	no	—
183	C₂×#97	—	134	2	no	—
184	C₂×C₄×C₈	—	67	8	no	—
185	C₂×C₄×C₈	—	67	4	no	—
186	C₂×C₄×C₈	—	134	2	no	—
187	#339	—	67	8	no	—
188	C₂³×C₄	—	134	1	no	—
189	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
193	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	67	4	no	—
196	C₂²×C₈	—	67	2	no	—
197	C₂²×C₈	—	67	1	no	—
198	C₂²×C₈	—	67	1	no	—
199	C₂²×C₈	—	67	4	no	—
200	C₂²×C₈	—	67	2	no	—
201	C₂²×C₈	C₄×D₆₇	1	4	yes	—
202	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₄×D₆₇	1	2	yes	—
205	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
206	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
207	#117	—	67	4	no	—
208	#117	—	67	4	no	—
209	#117	—	134	2	no	—
210	#117	—	134	2	no	—
211	C₂×C₄⋊C₄	—	67	4	no	—
212	C₂×C₂²⋊C₄	—	134	2	no	—
213	C₂×C₄²	—	67	4	no	—
214	C₂×C₄²	—	67	2	no	—
215	C₂×C₄²	—	134	2	no	—
216	C₂×C₄²	—	134	2	no	—
217	C₂×C₄²	—	134	1	no	—
218	#108	C₂×Dic₆₇	1	16	yes	complemented
219	#107	—	67	8	no	—
220	#106	—	67	8	no	—
221	#97	—	134	8	no	—
222	C₄×C₈	—	67	16	no	—
223	C₄×C₈	—	134	8	no	—
224	C₂⁴	#11761629	1	1	yes	characteristic
225	C₂²×C₄	—	134	4	no	—
226	C₂²×C₄	—	134	8	no	—
227	C₂²×C₄	—	134	4	no	—
228	C₂²×C₄	—	134	2	no	—
229	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
233	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
235	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
236	C₂²×C₄	#11761627	1	2	yes	—
237	C₂²×C₄	#11761629	1	1	yes	characteristic
238	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
239	M₁₆	—	2	8	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	C₂²×Dic₆₇	1	8	yes	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	67	8	no	—
245	C₂×C₈	—	67	4	no	—
246	C₂×C₈	—	67	4	no	—
247	C₂×C₈	—	67	4	no	—
248	C₂×C₈	—	67	2	no	—
249	C₂×C₈	—	67	2	no	—
250	C₂×C₈	—	67	4	no	—
251	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
252	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
253	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
254	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
255	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
257	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
258	C₄⋊C₄	—	67	8	no	—
259	C₄⋊C₄	—	134	4	no	—
260	C₂²⋊C₄	—	134	4	no	—
261	C₂²⋊C₄	—	134	4	no	—
262	C₄²	—	67	4	no	—
263	C₄²	—	134	4	no	—
264	C₄²	—	134	4	no	—
265	C₄²	—	134	4	no	—
266	C₄²	—	134	2	no	—
267	C₂³	—	2	4	no	—
268	C₂³	#14524001	1	2	yes	—
269	C₂³	#14524005	1	1	yes	characteristic
270	C₂³	#14524005	1	1	yes	characteristic
271	C₂³	Q₈×Dic₆₇	1	2	yes	—
272	C₂³	C₂×#11761629	1	1	yes	characteristic
273	C₂×C₄	—	268	8	no	—
274	C₂×C₄	—	134	8	no	—
275	C₂×C₄	—	134	16	no	—
276	C₂×C₄	—	134	8	no	—
277	C₂×C₄	—	134	4	no	—
278	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
280	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
281	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
282	C₂×C₄	#14524001	1	2	yes	—
283	C₂×C₄	#14524005	1	1	yes	characteristic
284	C₂×C₄	#14524005	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
286	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
287	C₂×C₄	#14524008	1	2	yes	—
288	C₂×C₄	#14524008	1	2	yes	—
289	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
296	C₈	—	2	8	no	—
297	C₈	—	67	4	no	—
298	C₈	—	67	4	no	—
299	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
300	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
306	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
307	C₂²	#16863512	1	4	yes	—
308	C₂²	C₂×#14524001	1	1	yes	characteristic
309	C₂²	C₂×#14524005	1	1	yes	characteristic
310	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
311	C₄	—	268	8	no	—
312	C₄	—	134	8	no	—
313	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
314	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
315	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
316	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
317	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
318	C₄	C₂×#14524008	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	#22560091	1	4	yes	direct product factor
323	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
324	C₂	C₂×#16863512	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	C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂×#1772
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. #22560091	C₂	direct (N × H)
10. #16863434	C₄	semidirect
12. C₂×#14523977	C₄	semidirect
13. C₂×#14523975	C₄	semidirect
18. C₂×C₈×Dic₆₇	C₄	semidirect
23. C₂×#108×C₆₇	C₄	semidirect
43. #14523977	C₂×C₄	semidirect
44. #14523975	C₂×C₄	semidirect
47. C₈×Dic₆₇	C₂×C₄	semidirect
60. #108×C₆₇	C₂×C₄	semidirect
160. C₂×C₆₇	#1772	semidirect
169. C₆₇	C₂×#1772	semidirect
175. C₂×#108	Dic₆₇	semidirect
218. #108	C₂×Dic₆₇	semidirect
322. C₂	#22560091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863434	2
C₂×#16863354	2
C₂²×C₈×Dic₆₇	2
#22560091	2
C₂×#16863157	2
C₂²×#108×C₆₇	2
C₂×#1772	67

Subgroups — C2×#22560091 #30853840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560091 #30853840