Subgroups — C₂×#22559964 #30853334

Order 17152.

930 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863351	C₂	1	2	yes	maximal
2	#22559964	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863153	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863152	C₂	1	2	yes	maximal
6	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863351	C₄	1	8	yes	complemented
10	#16863351	C₂²	1	8	yes	—
11	C₂×#14523953	C₄	1	4	yes	complemented
12	C₂×C₈×Dic₆₇	C₄	1	4	yes	complemented
13	C₆₇⋊#338	C₂²	1	4	yes	—
14	#16863153	C₂²	1	4	yes	—
15	#16863152	C₂²	1	8	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
19	#405×C₆₇	C₄	1	4	yes	complemented
20	#405×C₆₇	C₂²	1	4	yes	—
21	C₂×#97×C₆₇	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	C₂×C₄	1	4	yes	—
28	C₂×#11761612	C₂³	1	4	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
30	#14524002	—	2	2	no	—
31	#14524002	—	2	2	no	—
32	#14524002	C₂×C₄	1	4	yes	—
33	#14524002	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
37	#14523953	C₂×C₄	1	16	yes	complemented
38	C₈×Dic₆₇	C₂×C₄	1	16	yes	complemented
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	#117×C₆₇	—	2	1	no	—
53	#117×C₆₇	—	2	1	no	—
54	#117×C₆₇	C₂×C₄	1	2	yes	—
55	#117×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
59	#97×C₆₇	C₂×C₄	1	8	yes	complemented
60	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₂²×Dic₆₇	—	2	4	no	—
62	#11761612	—	2	4	no	—
63	#11761612	—	2	4	no	—
64	#11761612	—	2	4	no	—
65	#11761612	—	2	4	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	2	4	no	—
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₂²⋊Dic₆₇	—	2	4	no	—
72	C₂²⋊Dic₆₇	—	2	4	no	—
73	C₄×Dic₆₇	—	2	4	no	—
74	C₄×Dic₆₇	C₂²×C₄	1	8	yes	—
75	C₂⁴×C₆₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	C₄²	1	2	yes	—
92	C₂×C₈×C₆₇	C₄²	1	4	yes	—
93	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₄²	1	4	yes	—
95	C₂×C₈×C₆₇	C₄²	1	2	yes	—
96	C₂×C₈×C₆₇	C₄²	1	2	yes	—
97	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	4	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	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	#117	1	2	yes	—
111	C₂³×C₆₇	#117	1	2	yes	—
112	C₂³×C₆₇	#117	1	1	yes	characteristic
113	C₂³×C₆₇	#117	1	1	yes	characteristic
114	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#117	1	2	yes	—
121	C₂×C₄×C₆₇	#117	1	2	yes	—
122	C₂×C₄×C₆₇	#117	1	2	yes	—
123	C₂×C₄×C₆₇	#117	1	2	yes	—
124	C₂×C₄×C₆₇	#117	1	2	yes	—
125	C₂×C₄×C₆₇	#117	1	2	yes	—
126	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#117	1	2	yes	—
129	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
135	C₂×Dic₆₇	—	2	8	no	—
136	C₂×Dic₆₇	—	4	8	no	—
137	C₈×C₆₇	—	2	4	no	—
138	C₈×C₆₇	C₂×C₄²	1	4	yes	—
139	C₈×C₆₇	C₂×C₄²	1	2	yes	—
140	C₈×C₆₇	C₂×C₄²	1	2	yes	—
141	C₆₇⋊C₈	—	2	8	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	—	4	1	no	—
146	C₂²×C₆₇	—	4	1	no	—
147	C₂²×C₆₇	#376	1	2	yes	—
148	C₂²×C₆₇	#376	1	1	yes	characteristic
149	C₂²×C₆₇	#376	1	1	yes	characteristic
150	C₂²×C₆₇	C₂×#117	1	2	yes	—
151	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
152	C₄×C₆₇	—	4	4	no	—
153	C₄×C₆₇	#376	1	1	yes	characteristic
154	C₄×C₆₇	#376	1	2	yes	—
155	C₄×C₆₇	#376	1	1	yes	characteristic
156	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
157	C₄×C₆₇	C₂×#117	1	2	yes	—
158	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
159	Dic₆₇	—	4	8	no	—
160	C₂×#1747	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	2	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	#1747	1	2	yes	complemented
166	C₂×C₆₇	C₂×#376	1	1	yes	characteristic
167	C₂×#405	—	67	2	no	—
168	C₂×#405	D₆₇	1	1	yes	characteristic
169	#1747	—	67	8	no	—
170	C₂×#338	—	67	1	no	—
171	C₂×H(3,ℤ/4)	—	67	1	no	—
172	C₂×H(3,ℤ/4)	—	67	2	no	—
173	C₆₇	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	67	2	no	—
175	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
176	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
177	C₂×#117	—	67	2	no	—
178	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
179	#405	—	67	8	no	—
180	#405	—	67	8	no	—
181	#405	Dic₆₇	1	4	yes	complemented
182	#405	C₂×D₆₇	1	4	yes	—
183	C₂×#97	—	67	4	no	—
184	C₂×#97	Dic₆₇	1	2	yes	complemented
185	C₂×C₄×C₈	—	67	4	no	—
186	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
187	#338	—	67	4	no	—
188	H(3,ℤ/4)	—	67	4	no	—
189	H(3,ℤ/4)	—	67	8	no	—
190	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	67	4	no	—
194	C₂×M₁₆	—	67	4	no	—
195	C₂×M₁₆	—	134	2	no	—
196	C₂×M₁₆	—	134	2	no	—
197	C₂×M₁₆	—	134	2	no	—
198	C₂×M₁₆	—	134	2	no	—
199	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
200	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
201	C₂²×C₈	—	67	4	no	—
202	C₂²×C₈	C₄×D₆₇	1	2	yes	—
203	C₂²×C₈	C₄×D₆₇	1	4	yes	—
204	C₂²×C₈	C₄×D₆₇	1	2	yes	—
205	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
206	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
208	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
210	#117	—	2	1	no	—
211	#117	—	2	1	no	—
212	#117	—	67	4	no	—
213	#117	—	67	4	no	—
214	#117	—	134	2	no	—
215	#117	—	134	2	no	—
216	#117	C₂×Dic₆₇	1	2	yes	—
217	#117	C₂²×D₆₇	1	2	yes	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	134	2	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	134	2	no	—
222	C₂×C₄²	—	67	4	no	—
223	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
224	#97	—	67	16	no	—
225	#97	C₂×Dic₆₇	1	8	yes	complemented
226	C₄×C₈	—	67	16	no	—
227	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
228	C₂⁴	#11761628	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
235	C₂²×C₄	#11761628	1	1	yes	characteristic
236	C₂²×C₄	#11761628	1	1	yes	characteristic
237	C₂²×C₄	#11761628	1	1	yes	characteristic
238	C₂²×C₄	#11761628	1	2	yes	—
239	C₂²×C₄	#11761628	1	2	yes	—
240	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	134	4	no	—
244	M₁₆	—	134	4	no	—
245	M₁₆	—	134	4	no	—
246	M₁₆	—	134	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	67	8	no	—
250	C₂×C₈	—	134	4	no	—
251	C₂×C₈	—	134	4	no	—
252	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
253	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
254	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
255	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
256	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
257	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
258	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
260	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
261	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
262	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
263	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	134	4	no	—
267	C₄⋊C₄	—	134	4	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	134	4	no	—
271	C₂²⋊C₄	—	134	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	67	8	no	—
274	C₄²	—	134	4	no	—
275	C₄²	C₂²×Dic₆₇	1	4	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	#14524000	1	2	yes	—
278	C₂³	#14524000	1	2	yes	—
279	C₂³	#14524002	1	1	yes	characteristic
280	C₂³	#14524002	1	1	yes	characteristic
281	C₂³	C₂×#11761628	1	1	yes	characteristic
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	268	8	no	—
285	C₂×C₄	—	134	8	no	—
286	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
289	C₂×C₄	#14524000	1	2	yes	—
290	C₂×C₄	#14524000	1	2	yes	—
291	C₂×C₄	#14524000	1	2	yes	—
292	C₂×C₄	#14524000	1	2	yes	—
293	C₂×C₄	#14524000	1	2	yes	—
294	C₂×C₄	#14524000	1	2	yes	—
295	C₂×C₄	#14524002	1	1	yes	characteristic
296	C₂×C₄	#14524002	1	1	yes	characteristic
297	C₂×C₄	#14524002	1	1	yes	characteristic
298	C₂×C₄	#14524002	1	1	yes	characteristic
299	C₂×C₄	#14524002	1	2	yes	—
300	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
301	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
303	C₂×C₄	C₂×#11761628	1	2	yes	—
304	C₈	—	2	4	no	—
305	C₈	—	134	8	no	—
306	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
307	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
308	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	#16863242	1	2	yes	—
315	C₂²	#16863242	1	1	yes	characteristic
316	C₂²	#16863242	1	1	yes	characteristic
317	C₂²	C₂×#14524000	1	2	yes	—
318	C₂²	C₂×#14524002	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	268	8	no	—
321	C₄	#16863242	1	2	yes	—
322	C₄	#16863242	1	1	yes	characteristic
323	C₄	#16863242	1	1	yes	characteristic
324	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
325	C₄	C₂×#14524000	1	2	yes	—
326	C₄	C₂×#14524002	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	#22559964	1	2	yes	direct product factor
332	C₂	C₂×#16863242	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	C₂×#1747
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22559964	C₂	direct (N × H)
9. #16863351	C₄	semidirect
11. C₂×#14523953	C₄	semidirect
12. C₂×C₈×Dic₆₇	C₄	semidirect
19. #405×C₆₇	C₄	semidirect
21. C₂×#97×C₆₇	C₄	semidirect
22. C₂×C₄×C₈×C₆₇	C₄	semidirect
37. #14523953	C₂×C₄	semidirect
38. C₈×Dic₆₇	C₂×C₄	semidirect
59. #97×C₆₇	C₂×C₄	semidirect
60. C₄×C₈×C₆₇	C₂×C₄	semidirect
165. C₂×C₆₇	#1747	semidirect
173. C₆₇	C₂×#1747	semidirect
181. #405	Dic₆₇	semidirect
184. C₂×#97	Dic₆₇	semidirect
186. C₂×C₄×C₈	Dic₆₇	semidirect
225. #97	C₂×Dic₆₇	semidirect
227. C₄×C₈	C₂×Dic₆₇	semidirect
331. C₂	#22559964	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×#16863351	2
#22559964	2
C₂×C₆₇⋊#338	2
C₂×#16863153	2
C₂×#16863152	2
C₂×#405×C₆₇	2
C₂×#1747	67

Subgroups — C2×#22559964 #30853334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559964 #30853334