Subgroups — C₆₇×#21949 #30701200

Order 17152.

1010 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#407×C₆₇	C₂	1	2	yes	maximal
2	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1855	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1811	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂×#108×C₆₇	C₂²	1	8	yes	complemented
9	#407×C₆₇	—	2	2	no	—
10	#407×C₆₇	—	2	2	no	—
11	#407×C₆₇	C₄	1	2	yes	—
12	#407×C₆₇	C₄	1	4	yes	—
13	#407×C₆₇	C₄	1	2	yes	—
14	#407×C₆₇	C₂²	1	4	yes	—
15	#407×C₆₇	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
17	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
19	#341×C₆₇	C₂²	1	8	yes	complemented
20	C₂⁵×C₆₇	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	—	4	4	no	—
22	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	D₄	1	2	yes	—
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
29	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₆₇	—	2	4	no	—
33	C₂×M₁₆×C₆₇	—	4	4	no	—
34	C₂×M₁₆×C₆₇	—	4	8	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
40	#108×C₆₇	—	4	8	no	—
41	C₂²⋊C₈×C₆₇	—	2	4	no	—
42	C₂²⋊C₈×C₆₇	—	2	4	no	—
43	C₂²⋊C₈×C₆₇	—	2	4	no	—
44	C₂²⋊C₈×C₆₇	—	2	4	no	—
45	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
47	C₂⁴×C₆₇	—	4	2	no	—
48	C₂⁴×C₆₇	—	4	4	no	—
49	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
51	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	8	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	—	4	4	no	—
58	C₂²×C₄×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	—	4	4	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₆₇	—	2	16	no	—
83	M₁₆×C₆₇	—	4	4	no	—
84	M₁₆×C₆₇	—	4	4	no	—
85	M₁₆×C₆₇	—	4	8	no	—
86	C₂×C₈×C₆₇	—	2	8	no	—
87	C₂×C₈×C₆₇	—	2	8	no	—
88	C₂×C₈×C₆₇	—	4	8	no	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
90	C₂³×C₆₇	—	2	4	no	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	—	2	4	no	—
93	C₂³×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	—	4	1	no	—
96	C₂³×C₆₇	—	4	2	no	—
97	C₂³×C₆₇	—	4	1	no	—
98	C₂³×C₆₇	—	4	2	no	—
99	C₂³×C₆₇	—	4	2	no	—
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	16	no	—
103	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
132	C₈×C₆₇	—	2	8	no	—
133	C₈×C₆₇	—	4	8	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	4	8	no	—
145	C₂²×C₆₇	—	4	4	no	—
146	C₂²×C₆₇	—	4	4	no	—
147	C₂²×C₆₇	—	4	8	no	—
148	C₂²×C₆₇	—	4	4	no	—
149	C₂²×C₆₇	#430	1	4	yes	—
150	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₆₇	#543	1	2	yes	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	#518	1	2	yes	—
157	C₄×C₆₇	#534	1	2	yes	—
158	#21949	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	—	2	4	no	—
161	C₂×C₆₇	—	4	1	no	—
162	C₂×C₆₇	—	4	2	no	—
163	C₂×C₆₇	—	4	1	no	—
164	C₂×C₆₇	#2315	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#430	1	2	yes	—
166	C₂×#407	C₂×C₆₇	1	2	yes	—
167	C₂×#407	C₂×C₆₇	1	1	yes	characteristic
168	#1855	C₂×C₆₇	1	8	yes	complemented
169	#1811	C₂×C₆₇	1	4	yes	complemented
170	C₆₇	#21949	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
173	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
174	C₂×#108	C₂²×C₆₇	1	8	yes	complemented
175	#407	—	2	2	no	—
176	#407	—	2	2	no	—
177	#407	C₄×C₆₇	1	2	yes	—
178	#407	C₄×C₆₇	1	4	yes	—
179	#407	C₄×C₆₇	1	2	yes	—
180	#407	C₂²×C₆₇	1	4	yes	—
181	#407	C₂²×C₆₇	1	4	yes	—
182	C₂×C₂²⋊C₈	—	2	2	no	—
183	C₂×C₂²⋊C₈	—	2	2	no	—
184	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
185	#341	C₂²×C₆₇	1	8	yes	complemented
186	C₂⁵	Q₈×C₆₇	1	1	yes	characteristic
187	C₂³×C₄	—	4	4	no	—
188	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
190	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
192	C₂³×C₄	D₄×C₆₇	1	2	yes	—
193	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
194	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
195	C₂³×C₄	Q₈×C₆₇	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
197	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	4	4	no	—
200	C₂×M₁₆	—	4	8	no	—
201	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
202	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
203	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
204	C₂²×C₈	—	2	4	no	—
205	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
206	#108	—	4	8	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
212	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
213	C₂⁴	—	4	2	no	—
214	C₂⁴	—	4	4	no	—
215	C₂⁴	C₄⋊C₄×C₆₇	1	1	yes	characteristic
216	C₂⁴	C₄⋊C₄×C₆₇	1	2	yes	—
217	C₂⁴	C₄⋊C₄×C₆₇	1	1	yes	characteristic
218	C₂⁴	C₂×Q₈×C₆₇	1	2	yes	—
219	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
232	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
233	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
235	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
245	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
248	M₁₆	—	2	16	no	—
249	M₁₆	—	4	4	no	—
250	M₁₆	—	4	4	no	—
251	M₁₆	—	4	8	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	—	4	8	no	—
255	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	16	no	—
269	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
270	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
271	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
272	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
273	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
298	C₈	—	2	8	no	—
299	C₈	—	4	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	4	8	no	—
314	C₂²	—	4	4	no	—
315	C₂²	#430×C₆₇	1	4	yes	—
316	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
317	C₂²	#543×C₆₇	1	2	yes	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	#518×C₆₇	1	2	yes	—
323	C₄	#534×C₆₇	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	2	4	no	—
326	C₂	—	4	1	no	—
327	C₂	—	4	2	no	—
328	C₂	—	4	1	no	—
329	C₂	C₆₇×#2315	1	1	yes	characteristic
330	C₂	C₂×#430×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₆₇	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1855	C₂	semidirect
4. C₆₇×#1811	C₂	semidirect
8. C₂×#108×C₆₇	C₂²	semidirect
19. #341×C₆₇	C₂²	semidirect
158. #21949	C₆₇	direct (N × H)
168. #1855	C₂×C₆₇	semidirect
169. #1811	C₂×C₆₇	semidirect
170. C₆₇	#21949	direct (N × H)
174. C₂×#108	C₂²×C₆₇	semidirect
185. #341	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#407×C₆₇	2
C₆₇×#1855	2
C₆₇×#1811	2
#21949	67

Subgroups — C67×#21949 #30701200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21949 #30701200