Subgroups — #31010405

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863812	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569925	C₂	1	2	yes	maximal, complemented
3	#22562336	C₂	1	4	yes	maximal, complemented
4	#22562331	C₂	1	2	yes	maximal, complemented
5	#22562326	C₂	1	2	yes	maximal, complemented
6	C₂×#16863226	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561958	C₂	1	2	yes	maximal, complemented
8	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#16864194	—	2	2	no	—
10	C₂×#14524049	C₂²	1	1	yes	characteristic
11	#16863812	—	2	1	no	—
12	#16863812	—	2	1	no	—
13	#16863812	C₂²	1	2	yes	—
14	#16863812	C₂²	1	2	yes	—
15	C₂×#14523983	C₂²	1	1	yes	characteristic
16	C₂×#14523982	—	2	1	no	—
17	C₂×C₁₆×D₆₇	—	2	1	no	—
18	#16863668	C₂²	1	4	yes	—
19	#16863663	C₂²	1	4	yes	—
20	#16863656	—	2	4	no	—
21	C₂×#14523979	C₂²	1	2	yes	—
22	#16863380	—	2	2	no	—
23	#16863226	C₂²	1	2	yes	complemented
24	#16863226	C₂²	1	2	yes	complemented
25	#16863212	C₂²	1	4	yes	complemented
26	#16863209	C₂²	1	4	yes	complemented
27	C₂×#135×C₆₇	C₂²	1	2	yes	—
28	#508×C₆₇	—	2	1	no	—
29	#508×C₆₇	—	2	1	no	—
30	#508×C₆₇	—	2	1	no	—
31	#508×C₆₇	—	2	1	no	—
32	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#11761628	D₄	1	1	yes	characteristic
37	#14524074	—	2	4	no	—
38	#14524073	—	4	2	no	—
39	#14524069	—	2	4	no	—
40	#14524068	—	4	2	no	—
41	#14524063	—	4	4	no	—
42	#14524049	—	2	1	no	—
43	#14524049	—	2	1	no	—
44	#14524049	D₄	1	2	yes	—
45	#14524049	C₂³	1	2	yes	—
46	C₂×#11761612	—	2	2	no	—
47	C₂×#11761612	D₄	1	1	yes	characteristic
48	C₂×#11761611	—	2	1	no	—
49	C₂×C₈×D₆₇	—	2	1	no	—
50	#14523983	—	2	1	no	—
51	#14523983	C₂³	1	2	yes	—
52	#14523982	—	2	2	no	—
53	#14523982	—	2	2	no	—
54	C₂×C₆₇⋊C₁₆	—	2	1	no	—
55	C₁₆×D₆₇	—	2	2	no	—
56	C₁₆×D₆₇	—	2	2	no	—
57	#14523979	—	2	2	no	—
58	#14523979	C₂³	1	4	yes	—
59	#14523962	—	2	4	no	—
60	#14523960	—	2	4	no	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	#135×C₆₇	—	2	2	no	—
63	#135×C₆₇	—	2	2	no	—
64	#135×C₆₇	—	2	2	no	—
65	#135×C₆₇	—	2	2	no	—
66	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
67	C₂×SD₁₆×C₆₇	—	2	2	no	—
68	C₂×D₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	Q₃₂×C₆₇	—	2	1	no	—
71	Q₃₂×C₆₇	—	2	1	no	—
72	SD₃₂×C₆₇	—	2	2	no	—
73	SD₃₂×C₆₇	—	2	2	no	—
74	D₁₆×C₆₇	—	2	1	no	—
75	D₁₆×C₆₇	—	2	1	no	—
76	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	#11761628	—	2	1	no	—
83	#11761628	—	2	1	no	—
84	#11761628	D₈	1	2	yes	—
85	#11761628	C₂×D₄	1	2	yes	—
86	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
87	C₂×D₂₆₈	D₈	1	1	yes	characteristic
88	C₂×C₂²⋊D₆₇	—	2	1	no	—
89	C₂×C₄×D₆₇	—	2	1	no	—
90	#11761618	—	4	4	no	—
91	#11761616	—	4	4	no	—
92	#11761615	—	4	4	no	—
93	#11761613	—	4	4	no	—
94	#11761612	—	2	1	no	—
95	#11761612	—	2	4	no	—
96	#11761612	—	4	2	no	—
97	#11761612	C₂×D₄	1	2	yes	—
98	#11761611	—	2	2	no	—
99	#11761611	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	—	4	2	no	—
102	C₈×D₆₇	—	2	2	no	—
103	C₈×D₆₇	—	2	2	no	—
104	C₆₇⋊C₁₆	—	2	2	no	—
105	#41×C₆₇	—	4	2	no	—
106	#41×C₆₇	—	4	2	no	—
107	#41×C₆₇	—	4	2	no	—
108	#41×C₆₇	—	4	2	no	—
109	C₂×Q₈×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	4	2	no	—
112	C₂²×C₄×C₆₇	—	4	2	no	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	Q₁₆×C₆₇	—	2	2	no	—
115	Q₁₆×C₆₇	—	2	2	no	—
116	SD₁₆×C₆₇	—	4	2	no	—
117	SD₁₆×C₆₇	—	4	2	no	—
118	D₈×C₆₇	—	2	2	no	—
119	D₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
124	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	#135	1	1	yes	characteristic
132	C₂²×D₆₇	—	2	1	no	—
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	8	2	no	—
138	D₄×C₆₇	—	8	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	—	8	2	no	—
142	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₆₇	—	2	2	no	—
149	C₂²⋊D₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	2	1	no	—
151	D₂₆₈	—	2	1	no	—
152	D₂₆₈	C₂×D₈	1	2	yes	—
153	C₄×D₆₇	—	2	2	no	—
154	C₄×D₆₇	—	2	2	no	—
155	Dic₁₃₄	—	2	1	no	—
156	Dic₁₃₄	C₂×D₈	1	2	yes	—
157	C₈×C₆₇	#121	1	1	yes	characteristic
158	C₈×C₆₇	#121	1	1	yes	characteristic
159	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₆₇⋊C₈	—	2	2	no	—
162	C₆₇⋊C₈	—	4	4	no	—
163	C₂²×C₆₇	—	4	1	no	—
164	C₂²×C₆₇	—	8	2	no	—
165	C₂²×C₆₇	—	16	2	no	—
166	C₂²×C₆₇	#466	1	1	yes	characteristic
167	C₂²×C₆₇	#466	1	1	yes	characteristic
168	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
169	C₂×D₆₇	—	2	2	no	—
170	C₂×D₆₇	—	4	2	no	—
171	C₄×C₆₇	—	8	2	no	—
172	C₄×C₆₇	#466	1	1	yes	characteristic
173	C₄×C₆₇	#466	1	1	yes	characteristic
174	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
175	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
176	Dic₆₇	—	2	2	no	—
177	#33617	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	—	2	1	no	—
179	C₂×C₆₇	—	2	1	no	—
180	C₂×C₆₇	—	2	1	no	—
181	C₂×C₆₇	—	16	2	no	—
182	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
183	D₆₇	—	4	2	no	—
184	C₂×#508	D₆₇	1	1	yes	characteristic, complemented
185	C₂×#504	—	67	1	no	—
186	#3067	—	67	2	no	—
187	#2223	—	67	4	no	—
188	#2222	—	67	2	no	—
189	#2221	—	67	2	no	—
190	C₂×#368	—	67	1	no	—
191	#2148	—	67	2	no	—
192	C₆₇	#33617	1	1	yes	characteristic, complemented, Sylow
193	#575	—	134	2	no	—
194	C₂×#135	C₂×D₆₇	1	2	yes	—
195	C₂×#131	—	67	1	no	—
196	#508	—	2	1	no	—
197	#508	—	2	1	no	—
198	#508	—	2	1	no	—
199	#508	—	2	1	no	—
200	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic, complemented
201	C₂×SD₃₂	C₂×D₆₇	1	2	yes	complemented
202	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
203	#504	—	67	2	no	—
204	#504	—	67	2	no	—
205	#504	—	134	1	no	—
206	#504	—	134	1	no	—
207	C₂×M₃₂	—	67	1	no	—
208	C₂×M₃₂	—	134	1	no	—
209	C₂²×C₁₆	—	134	1	no	—
210	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
211	#473	—	67	4	no	—
212	#472	—	67	4	no	—
213	#471	—	134	4	no	—
214	C₂×#108	—	67	2	no	—
215	#413	—	134	2	no	—
216	#368	—	67	2	no	—
217	#368	—	67	2	no	—
218	#362	—	67	4	no	—
219	#361	—	67	4	no	—
220	C₂×#41	—	2	2	no	—
221	C₂×#41	—	67	1	no	—
222	#137	—	268	2	no	—
223	#137	—	134	4	no	—
224	AGL(1,ℤ/8)	—	268	2	no	—
225	AGL(1,ℤ/8)	—	134	4	no	—
226	#135	—	2	2	no	—
227	#135	—	2	2	no	—
228	#135	—	2	2	no	—
229	#135	—	2	2	no	—
230	#135	—	268	4	no	—
231	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
232	C₂×SD₁₆	—	2	2	no	—
233	C₂×D₈	C₂²×D₆₇	1	2	yes	—
234	#131	—	67	2	no	—
235	#131	—	67	2	no	—
236	#131	—	134	1	no	—
237	#131	—	134	1	no	—
238	C₂×M₁₆	—	67	1	no	—
239	C₂×M₁₆	—	134	2	no	—
240	C₂×M₁₆	—	134	1	no	—
241	C₂²×C₈	—	134	1	no	—
242	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
243	Q₃₂	—	2	1	no	—
244	Q₃₂	—	2	1	no	—
245	SD₃₂	—	2	2	no	—
246	SD₃₂	—	2	2	no	—
247	D₁₆	—	2	1	no	—
248	D₁₆	—	2	1	no	—
249	M₃₂	—	67	2	no	—
250	M₃₂	—	134	1	no	—
251	M₃₂	—	134	2	no	—
252	M₃₂	—	134	2	no	—
253	C₂×C₁₆	—	134	1	no	—
254	C₂×C₁₆	—	134	2	no	—
255	C₂×C₁₆	—	134	2	no	—
256	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
259	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
262	#108	—	67	4	no	—
263	#108	—	134	2	no	—
264	#101	—	134	4	no	—
265	#100	—	134	4	no	—
266	#41	—	4	2	no	—
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	#41	—	4	2	no	—
270	#41	—	67	2	no	—
271	#41	—	67	2	no	—
272	#41	—	134	1	no	—
273	#41	—	134	1	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	67	1	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	67	1	no	—
279	C₂×D₄	—	134	1	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	134	1	no	—
282	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	268	4	no	—
286	SD₁₆	—	4	2	no	—
287	SD₁₆	—	4	2	no	—
288	SD₁₆	—	268	4	no	—
289	SD₁₆	—	268	4	no	—
290	D₈	—	2	2	no	—
291	D₈	—	2	2	no	—
292	D₈	—	268	4	no	—
293	M₁₆	—	268	2	no	—
294	M₁₆	—	67	2	no	—
295	M₁₆	—	134	4	no	—
296	M₁₆	—	134	1	no	—
297	M₁₆	—	134	2	no	—
298	M₁₆	—	134	2	no	—
299	C₂×C₈	—	268	2	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	—	134	2	no	—
303	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
306	C₂×C₈	#11761627	1	1	yes	characteristic
307	C₂×C₈	#11761627	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
309	C₁₆	—	134	2	no	—
310	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
311	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
312	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
313	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
314	C₂³	—	4	2	no	—
315	C₂³	—	134	1	no	—
316	C₂³	#14524062	1	1	yes	characteristic
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	67	2	no	—
320	Q₈	—	134	1	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	67	2	no	—
326	D₄	—	134	2	no	—
327	D₄	—	134	2	no	—
328	D₄	—	134	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	134	1	no	—
333	C₂×C₄	—	134	2	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	#14524014	1	1	yes	characteristic
336	C₂×C₄	#14524014	1	1	yes	characteristic
337	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
338	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
339	C₂×C₄	#14524062	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
341	C₈	—	268	4	no	—
342	C₈	—	134	2	no	—
343	C₈	#14524014	1	1	yes	characteristic
344	C₈	#14524014	1	1	yes	characteristic
345	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
346	C₈	C₂×#11761627	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	268	2	no	—
350	C₂²	—	16	2	no	—
351	C₂²	—	134	2	no	—
352	C₂²	#16863629	1	1	yes	characteristic
353	C₂²	#16863629	1	1	yes	characteristic
354	C₂²	C₂×#14524062	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	134	2	no	—
357	C₄	#16863629	1	1	yes	characteristic
358	C₄	#16863629	1	1	yes	characteristic
359	C₄	C₂×#14524014	1	1	yes	characteristic
360	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	268	2	no	—
365	C₂	—	16	2	no	—
366	C₂	C₂×#16863629	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863629
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#508×C₆₇	C₂
socle	C₂×C₆₇	C₂×#466
hypercenter	C₂×#508	D₆₇
hypocenter	C₆₇	#33617
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#16863812	C₂	semidirect
2. #22569925	C₂	semidirect
3. #22562336	C₂	semidirect
4. #22562331	C₂	semidirect
5. #22562326	C₂	semidirect
6. C₂×#16863226	C₂	semidirect
7. #22561958	C₂	semidirect
8. C₂×#508×C₆₇	C₂	semidirect
23. #16863226	C₂²	semidirect
24. #16863226	C₂²	semidirect
25. #16863212	C₂²	semidirect
26. #16863209	C₂²	semidirect
32. C₂×Q₃₂×C₆₇	C₂²	semidirect
33. C₂×SD₃₂×C₆₇	C₂²	semidirect
34. C₂×D₁₆×C₆₇	C₂²	semidirect
184. C₂×#508	D₆₇	semidirect
192. C₆₇	#33617	semidirect
200. C₂×Q₃₂	C₂×D₆₇	semidirect
201. C₂×SD₃₂	C₂×D₆₇	semidirect
202. C₂×D₁₆	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863812	2
#22569925	2
#22562336	2
#22562331	2
#22562326	2
C₂×#16863226	2
#22561958	2
C₂×#508×C₆₇	2
#33617	67