Subgroups — C₆₇×#22690 #30701941

Order 17152.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2951	C₂	1	2	yes	maximal
3	C₆₇×#2912	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1952	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1860	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1815	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₇	C₂²	1	2	yes	—
11	#520×C₆₇	C₂²	1	4	yes	—
12	#443×C₆₇	C₄	1	8	yes	complemented
13	#440×C₆₇	—	2	4	no	—
14	C₄×Q₁₆×C₆₇	—	2	2	no	—
15	C₄×D₈×C₆₇	—	2	2	no	—
16	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
18	#428×C₆₇	—	2	4	no	—
19	#428×C₆₇	C₂²	1	4	yes	—
20	#427×C₆₇	—	2	2	no	—
21	#427×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	C₂×#106×C₆₇	—	2	2	no	—
24	C₂×#106×C₆₇	C₂²	1	2	yes	—
25	C₂×#104×C₆₇	C₂²	1	2	yes	—
26	#417×C₆₇	C₂²	1	4	yes	—
27	#405×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#382×C₆₇	C₂²	1	2	yes	—
30	#343×C₆₇	C₂²	1	4	yes	—
31	#339×C₆₇	C₂²	1	2	yes	—
32	C₂×#41×C₆₇	C₂³	1	2	yes	—
33	#135×C₆₇	C₂×C₄	1	4	yes	—
34	#135×C₆₇	Q₈	1	4	yes	—
35	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
36	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
37	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
38	#131×C₆₇	—	2	4	no	—
39	#131×C₆₇	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	—	2	4	no	—
47	C₄×D₄×C₆₇	—	2	4	no	—
48	C₄×D₄×C₆₇	—	4	4	no	—
49	#117×C₆₇	—	2	4	no	—
50	#117×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#108×C₆₇	—	2	2	no	—
55	#108×C₆₇	C₂×C₄	1	4	yes	complemented
56	#107×C₆₇	—	2	2	no	—
57	#107×C₆₇	—	2	2	no	—
58	#107×C₆₇	—	2	4	no	—
59	#107×C₆₇	D₄	1	2	yes	—
60	#106×C₆₇	—	2	2	no	—
61	#106×C₆₇	—	2	2	no	—
62	#106×C₆₇	—	2	4	no	—
63	#106×C₆₇	D₄	1	1	yes	characteristic
64	#106×C₆₇	D₄	1	1	yes	characteristic
65	#104×C₆₇	C₂×C₄	1	8	yes	complemented
66	#103×C₆₇	—	2	4	no	—
67	#102×C₆₇	—	2	4	no	—
68	#97×C₆₇	—	2	2	no	—
69	C₄×C₈×C₆₇	—	2	2	no	—
70	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
71	#95×C₆₇	—	4	2	no	—
72	#41×C₆₇	—	2	2	no	—
73	#41×C₆₇	—	2	2	no	—
74	#41×C₆₇	C₂²×C₄	1	4	yes	—
75	#41×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	4	4	no	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
85	SD₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
86	D₈×C₆₇	C₄⋊C₄	1	4	yes	—
87	M₁₆×C₆₇	—	2	2	no	—
88	M₁₆×C₆₇	—	2	8	no	—
89	M₁₆×C₆₇	—	2	2	no	—
90	M₁₆×C₆₇	—	2	2	no	—
91	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	8	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₆₇	#41	1	2	yes	—
103	C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	4	4	no	—
106	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₆₇	—	2	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	4	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	—	4	4	no	—
113	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂³×C₆₇	—	2	2	no	—
115	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
118	D₄×C₆₇	—	2	4	no	—
119	D₄×C₆₇	—	2	4	no	—
120	D₄×C₆₇	—	2	2	no	—
121	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	8	4	no	—
131	C₂×C₄×C₆₇	—	16	4	no	—
132	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈×C₆₇	—	2	4	no	—
139	C₈×C₆₇	—	4	2	no	—
140	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₆₇	C₄×D₄	1	2	yes	—
143	C₈×C₆₇	C₄×D₄	1	2	yes	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	4	no	—
146	C₂²×C₆₇	—	4	4	no	—
147	C₂²×C₆₇	#520	1	1	yes	characteristic
148	C₂²×C₆₇	D₄²	1	1	yes	characteristic
149	C₂²×C₆₇	#554	1	1	yes	characteristic
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	8	4	no	—
155	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₆₇	#547	1	1	yes	characteristic
158	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
159	#22690	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₆₇	—	2	1	no	—
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	—	4	4	no	—
163	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₆₇	#3290	1	2	yes	—
165	C₂×#443	C₂×C₆₇	1	1	yes	characteristic
166	#2951	C₂×C₆₇	1	2	yes	—
167	#2912	C₂×C₆₇	1	2	yes	complemented
168	#1952	C₂×C₆₇	1	2	yes	complemented
169	#1860	C₂×C₆₇	1	4	yes	complemented
170	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
171	#1815	C₂×C₆₇	1	2	yes	complemented
172	#1772	C₂×C₆₇	1	1	yes	characteristic, complemented
173	C₆₇	#22690	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#131	C₂²×C₆₇	1	2	yes	—
176	#520	C₂²×C₆₇	1	4	yes	—
177	#443	C₄×C₆₇	1	8	yes	complemented
178	#440	—	2	4	no	—
179	C₄×Q₁₆	—	2	2	no	—
180	C₄×D₈	—	2	2	no	—
181	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
183	#428	—	2	4	no	—
184	#428	C₂²×C₆₇	1	4	yes	—
185	#427	—	2	2	no	—
186	#427	C₂²×C₆₇	1	1	yes	characteristic
187	C₂×#107	C₂²×C₆₇	1	2	yes	—
188	C₂×#106	—	2	2	no	—
189	C₂×#106	C₂²×C₆₇	1	2	yes	—
190	C₂×#104	C₂²×C₆₇	1	2	yes	—
191	#417	C₂²×C₆₇	1	4	yes	—
192	#405	C₂²×C₆₇	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
194	#382	C₂²×C₆₇	1	2	yes	—
195	#343	C₂²×C₆₇	1	4	yes	—
196	#339	C₂²×C₆₇	1	2	yes	—
197	C₂×#41	C₂³×C₆₇	1	2	yes	—
198	#135	C₂×C₄×C₆₇	1	4	yes	—
199	#135	Q₈×C₆₇	1	4	yes	—
200	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
201	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
202	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
203	#131	—	2	4	no	—
204	#131	C₂×C₄×C₆₇	1	8	yes	complemented
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
207	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
211	C₄×Q₈	—	2	4	no	—
212	C₄×D₄	—	2	4	no	—
213	C₄×D₄	—	4	4	no	—
214	#117	—	2	4	no	—
215	#117	C₂³×C₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
218	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
219	#108	—	2	2	no	—
220	#108	C₂×C₄×C₆₇	1	4	yes	complemented
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	4	no	—
224	#107	D₄×C₆₇	1	2	yes	—
225	#106	—	2	2	no	—
226	#106	—	2	2	no	—
227	#106	—	2	4	no	—
228	#106	D₄×C₆₇	1	1	yes	characteristic
229	#106	D₄×C₆₇	1	1	yes	characteristic
230	#104	C₂×C₄×C₆₇	1	8	yes	complemented
231	#103	—	2	4	no	—
232	#102	—	2	4	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
236	#95	—	4	2	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	C₂²×C₄×C₆₇	1	4	yes	—
240	#41	C₂×Q₈×C₆₇	1	4	yes	—
241	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
249	Q₁₆	C₄⋊C₄×C₆₇	1	4	yes	—
250	SD₁₆	C₄⋊C₄×C₆₇	1	8	yes	—
251	D₈	C₄⋊C₄×C₆₇	1	4	yes	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	8	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
261	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
267	C₂×C₈	#41×C₆₇	1	2	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
272	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	4	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	4	4	no	—
278	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
279	C₂³	—	2	2	no	—
280	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
283	D₄	—	2	4	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	16	4	no	—
297	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
302	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
306	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
307	C₈	C₄×D₄×C₆₇	1	2	yes	—
308	C₈	C₄×D₄×C₆₇	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	#520×C₆₇	1	1	yes	characteristic
313	C₂²	D₄²×C₆₇	1	1	yes	characteristic
314	C₂²	#554×C₆₇	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	4	no	—
320	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
321	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
322	C₄	#547×C₆₇	1	1	yes	characteristic
323	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	4	4	no	—
327	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
328	C₂	C₆₇×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2912	C₂	semidirect
4. C₆₇×#1952	C₂	semidirect
5. C₆₇×#1860	C₂	semidirect
6. C₆₇×#1846	C₂	semidirect
7. C₆₇×#1815	C₂	semidirect
8. C₆₇×#1772	C₂	semidirect
12. #443×C₆₇	C₄	semidirect
39. #131×C₆₇	C₂×C₄	semidirect
55. #108×C₆₇	C₂×C₄	semidirect
65. #104×C₆₇	C₂×C₄	semidirect
70. C₄×C₈×C₆₇	C₂×C₄	semidirect
159. #22690	C₆₇	direct (N × H)
167. #2912	C₂×C₆₇	semidirect
168. #1952	C₂×C₆₇	semidirect
169. #1860	C₂×C₆₇	semidirect
170. #1846	C₂×C₆₇	semidirect
171. #1815	C₂×C₆₇	semidirect
172. #1772	C₂×C₆₇	semidirect
173. C₆₇	#22690	direct (N × H)
177. #443	C₄×C₆₇	semidirect
204. #131	C₂×C₄×C₆₇	semidirect
220. #108	C₂×C₄×C₆₇	semidirect
230. #104	C₂×C₄×C₆₇	semidirect
235. C₄×C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443×C₆₇	2
C₆₇×#2951	2
C₆₇×#2912	2
C₆₇×#1952	2
C₆₇×#1860	2
C₆₇×#1846	2
C₆₇×#1815	2
C₆₇×#1772	2
#22690	67

Subgroups — C67×#22690 #30701941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22690 #30701941