Subgroups — #30764717

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863155	C₂	1	2	yes	maximal
3	#22559906	C₂	1	2	yes	maximal
4	C₂×#16863150	C₂	1	1	yes	maximal, characteristic
5	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	#16863161	—	2	2	no	—
10	#16863155	—	2	4	no	—
11	#16863151	—	2	4	no	—
12	#16863150	—	2	2	no	—
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	#427×C₆₇	—	2	2	no	—
16	#427×C₆₇	—	2	2	no	—
17	C₂×#107×C₆₇	C₄	1	1	yes	characteristic, complemented
18	C₂×#107×C₆₇	C₄	1	1	yes	characteristic, complemented
19	C₂×#106×C₆₇	C₄	1	2	yes	complemented
20	C₂³×Dic₆₇	—	2	1	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	Q₈	1	4	yes	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
33	C₂×C₄×Dic₆₇	—	2	1	no	—
34	C₂×C₄×Dic₆₇	—	2	2	no	—
35	C₂×C₄×Dic₆₇	—	2	2	no	—
36	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	#117×C₆₇	—	2	2	no	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	#117×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	#107×C₆₇	—	2	2	no	—
53	#107×C₆₇	—	2	2	no	—
54	#106×C₆₇	—	2	4	no	—
55	C₂²×Dic₆₇	—	2	4	no	—
56	C₂²×Dic₆₇	—	2	2	no	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	C₂²×Dic₆₇	—	4	2	no	—
59	#11761612	—	2	4	no	—
60	#11761612	—	2	4	no	—
61	#11761612	—	4	2	no	—
62	#11761612	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	2	2	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₂×C₆₇⋊C₈	—	4	1	no	—
67	C₂×C₆₇⋊C₈	—	4	2	no	—
68	C₂×C₆₇⋊C₈	—	4	1	no	—
69	C₂×C₆₇⋊C₈	—	4	2	no	—
70	C₂×C₆₇⋊C₈	—	4	2	no	—
71	C₄×Dic₆₇	—	2	4	no	—
72	C₄×Dic₆₇	—	4	1	no	—
73	C₄×Dic₆₇	—	4	1	no	—
74	C₂⁴×C₆₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
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	#41	1	2	yes	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₂²⋊C₄×C₆₇	—	4	2	no	—
99	C₂²⋊C₄×C₆₇	—	4	2	no	—
100	C₄²×C₆₇	—	4	2	no	—
101	C₄²×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₆₇	#121	1	1	yes	characteristic
106	C₂³×C₆₇	#124	1	1	yes	characteristic
107	C₂³×C₆₇	#125	1	2	yes	—
108	C₂³×C₆₇	#126	1	2	yes	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#123	1	2	yes	—
124	C₂×C₄×C₆₇	#123	1	2	yes	—
125	C₂×C₄×C₆₇	#123	1	2	yes	—
126	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#125	1	2	yes	—
128	C₂×C₄×C₆₇	#126	1	2	yes	—
129	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×Dic₆₇	—	2	4	no	—
132	C₂×Dic₆₇	—	4	8	no	—
133	C₂×Dic₆₇	—	4	2	no	—
134	C₂×Dic₆₇	—	4	2	no	—
135	C₈×C₆₇	—	2	2	no	—
136	C₈×C₆₇	—	2	2	no	—
137	C₆₇⋊C₈	—	4	2	no	—
138	C₆₇⋊C₈	—	4	4	no	—
139	C₂²×C₆₇	—	2	4	no	—
140	C₂²×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	#387	1	2	yes	—
146	C₂²×C₆₇	#390	1	1	yes	characteristic
147	C₂²×C₆₇	#397	1	1	yes	characteristic
148	C₂²×C₆₇	#398	1	1	yes	characteristic
149	C₂²×C₆₇	#400	1	2	yes	—
150	C₂²×C₆₇	#443	1	2	yes	—
151	C₂²×C₆₇	#444	1	2	yes	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#381	1	1	yes	characteristic
155	C₄×C₆₇	#387	1	2	yes	—
156	C₄×C₆₇	#391	1	1	yes	characteristic
157	C₄×C₆₇	#397	1	1	yes	characteristic
158	C₄×C₆₇	#398	1	1	yes	characteristic
159	C₄×C₆₇	#399	1	2	yes	—
160	Dic₆₇	—	4	4	no	—
161	#11376	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	2	no	—
165	C₂×C₆₇	#1445	1	1	yes	characteristic
166	C₂×C₆₇	#1922	1	4	yes	—
167	C₂×C₆₇	#1974	1	1	yes	characteristic
168	C₂×C₆₇	#1975	1	1	yes	characteristic
169	C₂×#427	D₆₇	1	1	yes	characteristic
170	C₂×#341	—	67	1	no	—
171	C₂×#339	—	67	2	no	—
172	#1729	—	67	2	no	—
173	C₂×#336	—	67	1	no	—
174	C₆₇	#11376	1	1	yes	characteristic, complemented, Sylow
175	C₂²×M₁₆	—	67	2	no	—
176	C₂³×C₈	—	67	1	no	—
177	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
178	C₂×#117	C₂×D₆₇	1	2	yes	—
179	C₂²×C₄²	—	67	1	no	—
180	#427	—	2	2	no	—
181	#427	—	2	2	no	—
182	C₂×#107	Dic₆₇	1	1	yes	characteristic, complemented
183	C₂×#107	Dic₆₇	1	1	yes	characteristic, complemented
184	C₂×#106	Dic₆₇	1	2	yes	complemented
185	#341	—	134	2	no	—
186	#339	—	134	4	no	—
187	#336	—	134	4	no	—
188	#336	—	134	2	no	—
189	C₂³×C₄	—	134	1	no	—
190	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
191	C₂×M₁₆	—	67	4	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	—	134	2	no	—
196	C₂×M₁₆	—	134	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	67	2	no	—
200	C₂²×C₈	—	134	2	no	—
201	C₂²×C₈	—	134	1	no	—
202	C₂²×C₈	—	134	1	no	—
203	C₂²×C₈	—	134	2	no	—
204	C₂²×C₈	—	134	2	no	—
205	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
214	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	67	2	no	—
218	C₂×C₄²	—	134	1	no	—
219	C₂×C₄²	—	134	2	no	—
220	C₂×C₄²	—	134	2	no	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#106	—	2	4	no	—
224	C₂⁴	#11761627	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	268	2	no	—
228	C₂²×C₄	—	268	2	no	—
229	C₂²×C₄	—	134	4	no	—
230	C₂²×C₄	—	134	2	no	—
231	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
237	C₂²×C₄	#11761627	1	1	yes	characteristic
238	C₂²×C₄	#11761628	1	2	yes	—
239	C₂²×C₄	#11761628	1	2	yes	—
240	C₂²×C₄	#11761629	1	2	yes	—
241	M₁₆	—	268	2	no	—
242	M₁₆	—	268	2	no	—
243	M₁₆	—	134	4	no	—
244	M₁₆	—	134	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	268	1	no	—
251	C₂×C₈	—	268	2	no	—
252	C₂×C₈	—	268	1	no	—
253	C₂×C₈	—	268	2	no	—
254	C₂×C₈	—	268	2	no	—
255	C₂×C₈	—	134	2	no	—
256	C₂×C₈	—	134	4	no	—
257	C₂×C₈	—	134	4	no	—
258	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
259	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	268	1	no	—
269	C₄²	—	268	1	no	—
270	C₄²	—	134	4	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
274	C₂³	#14524014	1	1	yes	characteristic
275	C₂³	#14524030	1	1	yes	characteristic
276	C₂³	#14524033	1	2	yes	—
277	C₂³	#14524038	1	2	yes	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	268	8	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	—	134	4	no	—
286	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
287	C₂×C₄	#14524002	1	2	yes	—
288	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
289	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
290	C₂×C₄	#14524014	1	1	yes	characteristic
291	C₂×C₄	#14524019	1	2	yes	—
292	C₂×C₄	#14524021	1	2	yes	—
293	C₂×C₄	#14524021	1	2	yes	—
294	C₂×C₄	#14524026	1	1	yes	characteristic
295	C₂×C₄	#14524026	1	1	yes	characteristic
296	C₂×C₄	#14524026	1	2	yes	—
297	C₂×C₄	#14524028	1	2	yes	—
298	C₂×C₄	#14524028	1	2	yes	—
299	C₂×C₄	#14524032	1	1	yes	characteristic
300	C₂×C₄	#14524033	1	2	yes	—
301	C₂×C₄	#14524038	1	2	yes	—
302	C₂×C₄	#14524039	1	1	yes	characteristic
303	C₂×C₄	#14524041	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	268	2	no	—
307	C₈	—	268	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	#16863282	1	2	yes	—
315	C₂²	#16863299	1	1	yes	characteristic
316	C₂²	#16863322	1	1	yes	characteristic
317	C₂²	#16863327	1	1	yes	characteristic
318	C₂²	#16863334	1	2	yes	—
319	C₂²	#16863513	1	2	yes	—
320	C₂²	#16863518	1	2	yes	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	268	4	no	—
324	C₄	#16863259	1	1	yes	characteristic
325	C₄	#16863282	1	2	yes	—
326	C₄	#16863303	1	1	yes	characteristic
327	C₄	#16863322	1	1	yes	characteristic
328	C₄	#16863327	1	1	yes	characteristic
329	C₄	#16863331	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	#22558477	1	1	yes	characteristic
334	C₂	#22560810	1	4	yes	—
335	C₂	#22561098	1	1	yes	characteristic
336	C₂	#22561103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#11376
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
17. C₂×#107×C₆₇	C₄	semidirect
18. C₂×#107×C₆₇	C₄	semidirect
19. C₂×#106×C₆₇	C₄	semidirect
174. C₆₇	#11376	semidirect
182. C₂×#107	Dic₆₇	semidirect
183. C₂×#107	Dic₆₇	semidirect
184. C₂×#106	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863161	2
C₂×#16863155	2
#22559906	2
C₂×#16863150	2
C₂×#427×C₆₇	2
#11376	67