Subgroups — C₂×C₆₇×#1841 #30700026

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1841	C₂	1	8	yes	maximal, direct product factor
3	C₂×#344×C₆₇	C₂	1	2	yes	maximal
4	C₂×#339×C₆₇	C₂	1	2	yes	maximal
5	C₂×#338×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₆₇	C₄	1	4	yes	complemented
13	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
14	C₂×#97×C₆₇	C₄	1	2	yes	complemented
15	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
16	#344×C₆₇	C₂²	1	8	yes	—
17	#339×C₆₇	C₂²	1	8	yes	—
18	#338×C₆₇	C₂²	1	4	yes	—
19	H(3,ℤ/4)×C₆₇	C₂²	1	4	yes	—
20	C₂³×C₄×C₆₇	—	2	1	no	—
21	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
34	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
35	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	#117×C₆₇	—	2	2	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	2	no	—
42	#117×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
53	#97×C₆₇	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
55	C₂⁴×C₆₇	#41	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	2	1	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	#41	1	2	yes	—
66	C₂²×C₄×C₆₇	#41	1	2	yes	—
67	C₂²×C₄×C₆₇	#41	1	2	yes	—
68	C₂²×C₄×C₆₇	#41	1	2	yes	—
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	2	4	no	—
74	M₁₆×C₆₇	—	2	4	no	—
75	M₁₆×C₆₇	—	2	4	no	—
76	M₁₆×C₆₇	—	2	4	no	—
77	M₁₆×C₆₇	—	2	4	no	—
78	M₁₆×C₆₇	—	4	2	no	—
79	M₁₆×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	4	2	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	4	2	no	—
91	C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	4	2	no	—
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	#117	1	1	yes	characteristic
103	C₂³×C₆₇	#117	1	1	yes	characteristic
104	C₂³×C₆₇	#126	1	2	yes	—
105	C₂³×C₆₇	#126	1	2	yes	—
106	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	8	4	no	—
115	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#117	1	2	yes	—
119	C₂×C₄×C₆₇	#117	1	2	yes	—
120	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#126	1	2	yes	—
124	C₂×C₄×C₆₇	#126	1	2	yes	—
125	C₂×C₄×C₆₇	#126	1	2	yes	—
126	C₂×C₄×C₆₇	#126	1	2	yes	—
127	C₂×C₄×C₆₇	#126	1	2	yes	—
128	C₂×C₄×C₆₇	#126	1	2	yes	—
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₈×C₆₇	—	2	4	no	—
134	C₈×C₆₇	—	2	4	no	—
135	C₈×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	4	no	—
138	C₂²×C₆₇	—	4	1	no	—
139	C₂²×C₆₇	—	4	1	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	#383	1	1	yes	characteristic
142	C₂²×C₆₇	#383	1	2	yes	—
143	C₂²×C₆₇	#383	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×#126	1	2	yes	—
146	C₄×C₆₇	—	2	4	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	—	8	4	no	—
149	C₄×C₆₇	#383	1	1	yes	characteristic
150	C₄×C₆₇	#383	1	2	yes	—
151	C₄×C₆₇	#383	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₆₇	C₂×#126	1	2	yes	—
155	C₂×#1841	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	2	2	no	—
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	#1841	1	2	yes	direct product factor
161	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
162	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
163	#1841	C₂×C₆₇	1	8	yes	direct product factor
164	C₂×#344	C₂×C₆₇	1	2	yes	—
165	C₂×#339	C₂×C₆₇	1	2	yes	—
166	C₂×#338	C₂×C₆₇	1	1	yes	characteristic
167	C₂×H(3,ℤ/4)	C₂×C₆₇	1	1	yes	characteristic
168	C₆₇	C₂×#1841	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
170	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
171	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×#117	C₂²×C₆₇	1	2	yes	—
173	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
174	C₄×M₁₆	C₄×C₆₇	1	4	yes	complemented
175	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
176	C₂×#97	C₄×C₆₇	1	2	yes	complemented
177	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	complemented
178	#344	C₂²×C₆₇	1	8	yes	—
179	#339	C₂²×C₆₇	1	8	yes	—
180	#338	C₂²×C₆₇	1	4	yes	—
181	H(3,ℤ/4)	C₂²×C₆₇	1	4	yes	—
182	C₂³×C₄	—	2	1	no	—
183	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
194	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
195	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
196	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
197	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
200	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	C₂³×C₆₇	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
214	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
215	#97	C₂×C₄×C₆₇	1	8	yes	complemented
216	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
217	C₂⁴	#41×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
226	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	#41×C₆₇	1	2	yes	—
228	C₂²×C₄	#41×C₆₇	1	2	yes	—
229	C₂²×C₄	#41×C₆₇	1	2	yes	—
230	C₂²×C₄	#41×C₆₇	1	2	yes	—
231	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
250	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
262	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
263	C₂³	—	4	2	no	—
264	C₂³	#117×C₆₇	1	1	yes	characteristic
265	C₂³	#117×C₆₇	1	1	yes	characteristic
266	C₂³	#126×C₆₇	1	2	yes	—
267	C₂³	#126×C₆₇	1	2	yes	—
268	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
278	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
279	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
280	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
281	C₂×C₄	#117×C₆₇	1	2	yes	—
282	C₂×C₄	#117×C₆₇	1	2	yes	—
283	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
284	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
285	C₂×C₄	#126×C₆₇	1	2	yes	—
286	C₂×C₄	#126×C₆₇	1	2	yes	—
287	C₂×C₄	#126×C₆₇	1	2	yes	—
288	C₂×C₄	#126×C₆₇	1	2	yes	—
289	C₂×C₄	#126×C₆₇	1	2	yes	—
290	C₂×C₄	#126×C₆₇	1	2	yes	—
291	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
293	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
294	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	4	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#383×C₆₇	1	1	yes	characteristic
304	C₂²	#383×C₆₇	1	2	yes	—
305	C₂²	#383×C₆₇	1	1	yes	characteristic
306	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
307	C₂²	C₂×#126×C₆₇	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	4	no	—
311	C₄	#383×C₆₇	1	1	yes	characteristic
312	C₄	#383×C₆₇	1	1	yes	characteristic
313	C₄	#383×C₆₇	1	2	yes	—
314	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
315	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
316	C₄	C₂×#126×C₆₇	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	C₆₇×#1841	1	2	yes	direct product factor
322	C₂	C₂×#383×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#117
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	#383
prime radical	C₂⁴×C₆₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1841	C₂	direct (N × H)
12. C₄×M₁₆×C₆₇	C₄	semidirect
14. C₂×#97×C₆₇	C₄	semidirect
15. C₂×C₄×C₈×C₆₇	C₄	semidirect
53. #97×C₆₇	C₂×C₄	semidirect
54. C₄×C₈×C₆₇	C₂×C₄	semidirect
155. C₂×#1841	C₆₇	direct (N × H)
160. C₂×C₆₇	#1841	direct (N × H)
163. #1841	C₂×C₆₇	direct (N × H)
168. C₆₇	C₂×#1841	direct (N × H)
174. C₄×M₁₆	C₄×C₆₇	semidirect
176. C₂×#97	C₄×C₆₇	semidirect
177. C₂×C₄×C₈	C₄×C₆₇	semidirect
215. #97	C₂×C₄×C₆₇	semidirect
216. C₄×C₈	C₂×C₄×C₆₇	semidirect
321. C₂	C₆₇×#1841	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×C₄×M₁₆×C₆₇	2
C₆₇×#1841	2
C₂×#344×C₆₇	2
C₂×#339×C₆₇	2
C₂×#338×C₆₇	2
C₂×H(3,ℤ/4)×C₆₇	2
C₂×#1841	67

Subgroups — C2×C67×#1841 #30700026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1841 #30700026