Subgroups — #30761579

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863166	C₂	1	2	yes	maximal
3	C₂×#16863155	C₂	1	2	yes	maximal
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂²×#11761612	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₆₇	—	2	4	no	—
10	C₂×#14523953	C₄	1	2	yes	complemented
11	C₂×C₈×Dic₆₇	C₄	1	2	yes	complemented
12	#16863166	—	2	4	no	—
13	#16863155	C₄	1	8	yes	complemented
14	C₆₇⋊#338	—	2	2	no	—
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	H(3,ℤ/4)×C₆₇	—	2	2	no	—
18	C₂³×Dic₆₇	—	2	1	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	1	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	1	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	D₄	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₆₇	—	2	2	no	—
32	C₂×C₄×Dic₆₇	—	2	1	no	—
33	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
36	#14523953	—	2	4	no	—
37	C₈×Dic₆₇	—	2	4	no	—
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
42	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	#117×C₆₇	—	2	2	no	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	#117×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	—	2	2	no	—
51	C₂²×Dic₆₇	—	2	1	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	4	4	no	—
55	#11761612	—	2	4	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	2	4	no	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	2	no	—
60	#11761612	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	—
67	C₄×Dic₆₇	—	2	1	no	—
68	C₄×Dic₆₇	—	2	1	no	—
69	C₄×Dic₆₇	—	2	4	no	—
70	C₄×Dic₆₇	C₄²	1	2	yes	—
71	C₄×Dic₆₇	C₄²	1	2	yes	—
72	C₂⁴×C₆₇	#41	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	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×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	M₁₆×C₆₇	—	2	4	no	—
85	M₁₆×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	C₄⋊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	4	no	—
94	C₄²×C₆₇	—	4	2	no	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	#117	1	1	yes	characteristic
97	C₂³×C₆₇	C₄×D₄	1	2	yes	—
98	C₂³×C₆₇	#124	1	1	yes	characteristic
99	C₂³×C₆₇	#125	1	1	yes	characteristic
100	C₂³×C₆₇	#126	1	2	yes	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	8	4	no	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#117	1	2	yes	—
108	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	#121	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	#123	1	2	yes	—
117	C₂×C₄×C₆₇	#123	1	2	yes	—
118	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#126	1	2	yes	—
121	C₂×Dic₆₇	—	2	4	no	—
122	C₂×Dic₆₇	—	2	2	no	—
123	C₂×Dic₆₇	—	2	2	no	—
124	C₂×Dic₆₇	—	4	8	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₈×C₆₇	—	2	4	no	—
127	C₆₇⋊C₈	—	2	4	no	—
128	C₆₇⋊C₈	—	4	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	—	4	1	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₆₇	#382	1	1	yes	characteristic
136	C₂²×C₆₇	#383	1	1	yes	characteristic
137	C₂²×C₆₇	#387	1	2	yes	—
138	C₂²×C₆₇	#388	1	2	yes	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	#376	1	1	yes	characteristic
141	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₆₇	#380	1	1	yes	characteristic
143	C₄×C₆₇	#381	1	1	yes	characteristic
144	C₄×C₆₇	#387	1	2	yes	—
145	C₄×C₆₇	#388	1	2	yes	—
146	Dic₆₇	—	2	4	no	—
147	Dic₆₇	—	4	2	no	—
148	#10825	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	#1434	1	1	yes	characteristic
154	C₂×C₆₇	#1841	1	2	yes	—
155	C₂×C₄×M₁₆	—	67	1	no	—
156	C₂×#344	—	67	2	no	—
157	C₂×#339	—	67	2	no	—
158	C₂×#338	—	67	1	no	—
159	C₂×H(3,ℤ/4)	D₆₇	1	1	yes	characteristic
160	C₆₇	#10825	1	1	yes	characteristic, complemented, Sylow
161	C₂²×M₁₆	—	67	1	no	—
162	C₂²×M₁₆	—	67	2	no	—
163	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#117	C₂×D₆₇	1	2	yes	—
165	C₂²×C₄²	—	67	1	no	—
166	C₄×M₁₆	—	134	4	no	—
167	C₂×#97	—	67	2	no	—
168	C₂×C₄×C₈	—	67	2	no	—
169	#344	—	134	4	no	—
170	#339	—	67	8	no	—
171	#338	—	134	2	no	—
172	H(3,ℤ/4)	—	2	2	no	—
173	C₂³×C₄	—	134	1	no	—
174	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	67	4	no	—
178	C₂×M₁₆	—	67	4	no	—
179	C₂×M₁₆	—	134	2	no	—
180	C₂×M₁₆	—	134	2	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	1	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂×M₁₆	—	134	1	no	—
187	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
188	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
189	C₂²×C₈	—	67	2	no	—
190	C₂²×C₈	—	134	2	no	—
191	C₂²×C₈	C₄×D₆₇	1	2	yes	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	C₄×D₆₇	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	67	4	no	—
200	C₂×C₄²	—	67	1	no	—
201	C₂×C₄²	—	67	1	no	—
202	C₂×C₄²	—	134	2	no	—
203	C₂×C₄²	—	134	1	no	—
204	#97	—	134	4	no	—
205	C₄×C₈	—	134	4	no	—
206	C₂⁴	#11761627	1	1	yes	characteristic
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	268	4	no	—
209	C₂²×C₄	—	134	1	no	—
210	C₂²×C₄	—	134	4	no	—
211	C₂²×C₄	—	134	1	no	—
212	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
214	C₂²×C₄	D₄×D₆₇	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
217	C₂²×C₄	#11761627	1	2	yes	—
218	C₂²×C₄	#11761627	1	1	yes	characteristic
219	C₂²×C₄	#11761628	1	2	yes	—
220	C₂²×C₄	#11761629	1	1	yes	characteristic
221	C₂²×C₄	#11761629	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	268	2	no	—
225	M₁₆	—	268	2	no	—
226	M₁₆	—	134	4	no	—
227	M₁₆	—	134	4	no	—
228	M₁₆	—	134	4	no	—
229	M₁₆	—	134	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	268	2	no	—
233	C₂×C₈	—	268	2	no	—
234	C₂×C₈	—	67	4	no	—
235	C₂×C₈	—	134	4	no	—
236	C₂×C₈	—	134	2	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	67	2	no	—
246	C₄²	—	67	2	no	—
247	C₄²	—	134	1	no	—
248	C₄²	—	134	1	no	—
249	C₄²	—	134	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#14524001	1	1	yes	characteristic
252	C₂³	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
253	C₂³	#14524030	1	1	yes	characteristic
254	C₂³	#14524034	1	1	yes	characteristic
255	C₂³	#14524036	1	2	yes	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	268	2	no	—
259	C₂×C₄	—	268	8	no	—
260	C₂×C₄	—	134	2	no	—
261	C₂×C₄	—	134	4	no	—
262	C₂×C₄	—	134	2	no	—
263	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
266	C₂×C₄	#14523999	1	1	yes	characteristic
267	C₂×C₄	#14523999	1	2	yes	—
268	C₂×C₄	#14524000	1	1	yes	characteristic
269	C₂×C₄	#14524001	1	1	yes	characteristic
270	C₂×C₄	#14524001	1	2	yes	—
271	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
272	C₂×C₄	#14524005	1	2	yes	—
273	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
274	C₂×C₄	#14524015	1	2	yes	—
275	C₂×C₄	#14524020	1	2	yes	—
276	C₂×C₄	#14524025	1	2	yes	—
277	C₂×C₄	#14524028	1	2	yes	—
278	C₂×C₄	#14524032	1	1	yes	characteristic
279	C₂×C₄	#14524034	1	1	yes	characteristic
280	C₂×C₄	#14524038	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	268	4	no	—
283	C₈	—	134	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#16863243	1	1	yes	characteristic
290	C₂²	#16863261	1	1	yes	characteristic
291	C₂²	#16863263	1	1	yes	characteristic
292	C₂²	#16863283	1	2	yes	—
293	C₂²	#16863290	1	2	yes	—
294	C₄	—	8	4	no	—
295	C₄	—	268	2	no	—
296	C₄	—	134	4	no	—
297	C₄	#16863241	1	1	yes	characteristic
298	C₄	C₄×Dic₆₇⋊C₄	1	1	yes	characteristic
299	C₄	#16863254	1	1	yes	characteristic
300	C₄	#16863258	1	1	yes	characteristic
301	C₄	#16863279	1	2	yes	—
302	C₄	#16863286	1	2	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	#22558435	1	1	yes	characteristic
308	C₂	#22560356	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524000
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×H(3,ℤ/4)×C₆₇	C₂
socle	C₂²×C₆₇	#383
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₆₇
hypocenter	C₆₇	#10825
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
10. C₂×#14523953	C₄	semidirect
11. C₂×C₈×Dic₆₇	C₄	semidirect
13. #16863155	C₄	semidirect
160. C₆₇	#10825	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₆₇	2
C₂×#16863166	2
C₂×#16863155	2
C₂×C₆₇⋊#338	2
C₂×H(3,ℤ/4)×C₆₇	2
#10825	67