Subgroups — #31009082

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573312	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573311	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863511	C₂	1	1	yes	maximal, characteristic
4	#22562048	C₂	1	4	yes	maximal, complemented
5	#22562043	C₂	1	4	yes	maximal, complemented
6	C₂×#16863216	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561836	C₂	1	1	yes	maximal, characteristic, complemented
8	M₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524063	C₄	1	1	yes	characteristic
11	C₂×#14524049	C₂²	1	1	yes	characteristic
12	#16863814	C₂²	1	4	yes	complemented
13	#16863813	—	2	2	no	—
14	#16863813	C₂²	1	4	yes	complemented
15	#16863812	—	2	2	no	—
16	C₂×#14523983	—	2	2	no	—
17	C₂×#14523983	C₂²	1	1	yes	characteristic
18	C₂×#14523983	C₂²	1	1	yes	characteristic
19	M₃₂×D₆₇	—	2	2	no	—
20	#16863511	C₄	1	2	yes	—
21	#16863511	C₄	1	2	yes	—
22	#16863511	C₂²	1	2	yes	—
23	#16863511	C₂²	1	2	yes	—
24	C₂×#14523978	C₄	1	1	yes	characteristic
25	C₂×#14523970	C₄	1	1	yes	characteristic
26	C₂×#14523969	C₄	1	1	yes	characteristic
27	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
28	#16863216	C₂²	1	2	yes	complemented
29	#16863216	C₂²	1	2	yes	complemented
30	#16863179	C₂²	1	4	yes	complemented
31	#16863171	C₂²	1	2	yes	complemented
32	C₁₆×Dic₆₇	C₂²	1	2	yes	complemented
33	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
36	#350×C₆₇	C₂²	1	4	yes	complemented
37	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
38	#14524063	C₈	1	2	yes	—
39	#14524063	C₈	1	2	yes	—
40	#14524063	C₂×C₄	1	2	yes	—
41	#14524063	C₂×C₄	1	2	yes	—
42	C₂×#11761618	C₈	1	1	yes	characteristic
43	C₂×#11761616	C₈	1	1	yes	characteristic
44	C₂×#11761615	C₈	1	1	yes	characteristic
45	C₂×#11761613	C₈	1	1	yes	characteristic
46	#14524049	—	2	1	no	—
47	#14524049	—	2	1	no	—
48	#14524049	C₂×C₄	1	2	yes	—
49	#14524049	C₂³	1	2	yes	—
50	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
51	C₂×#11761611	—	2	1	no	—
52	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₆₇	—	2	1	no	—
54	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
55	#14523983	—	2	4	no	—
56	#14523983	—	2	1	no	—
57	#14523983	D₄	1	2	yes	—
58	#14523983	D₄	1	2	yes	—
59	#14523983	C₂³	1	2	yes	—
60	#14523982	—	2	4	no	—
61	C₂×C₆₇⋊C₁₆	—	2	4	no	—
62	C₂×C₆₇⋊C₁₆	—	2	1	no	—
63	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
64	C₁₆×D₆₇	—	2	4	no	—
65	#14523978	C₂×C₄	1	2	yes	—
66	#14523978	C₂×C₄	1	2	yes	—
67	#14523970	C₂×C₄	1	2	yes	—
68	#14523970	C₂×C₄	1	2	yes	—
69	#14523969	C₂×C₄	1	2	yes	—
70	#14523969	C₂×C₄	1	2	yes	—
71	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
74	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
75	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
76	#131×C₆₇	—	2	1	no	—
77	#131×C₆₇	—	2	1	no	—
78	#131×C₆₇	C₂×C₄	1	2	yes	—
79	#131×C₆₇	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₆₇	—	2	1	no	—
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	M₃₂×C₆₇	—	2	1	no	—
85	M₃₂×C₆₇	—	2	1	no	—
86	M₃₂×C₆₇	—	2	1	no	—
87	M₃₂×C₆₇	C₂³	1	2	yes	—
88	C₂×C₁₆×C₆₇	—	2	1	no	—
89	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
90	#11761628	—	2	1	no	—
91	#11761628	—	2	1	no	—
92	#11761628	C₂×C₈	1	2	yes	—
93	#11761628	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
95	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
96	C₂×C₂²⋊D₆₇	—	2	1	no	—
97	C₂²×Dic₆₇	—	2	1	no	—
98	C₂×C₄×D₆₇	—	2	1	no	—
99	#11761618	C₂×C₈	1	2	yes	—
100	#11761618	C₂×C₈	1	2	yes	—
101	#11761616	C₂×C₈	1	2	yes	—
102	#11761616	C₂×C₈	1	2	yes	—
103	#11761615	C₂×C₈	1	2	yes	—
104	#11761615	C₂×C₈	1	2	yes	—
105	#11761613	C₂×C₈	1	2	yes	—
106	#11761613	C₂×C₈	1	2	yes	—
107	#11761612	—	2	1	no	—
108	#11761612	C₂²×C₄	1	2	yes	—
109	#11761611	—	2	2	no	—
110	#11761611	—	2	2	no	—
111	C₂×C₆₇⋊C₈	—	2	1	no	—
112	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
113	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
114	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
115	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
116	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
118	C₈×D₆₇	—	2	2	no	—
119	C₈×D₆₇	—	2	2	no	—
120	C₄×Dic₆₇	—	2	1	no	—
121	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₆₇⋊C₁₆	—	2	2	no	—
124	C₆₇⋊C₁₆	C₂×D₄	1	4	yes	—
125	#41×C₆₇	—	2	1	no	—
126	#41×C₆₇	—	2	1	no	—
127	#41×C₆₇	C₂×C₈	1	2	yes	—
128	#41×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	M₁₆×C₆₇	—	2	1	no	—
135	M₁₆×C₆₇	—	2	2	no	—
136	M₁₆×C₆₇	—	2	2	no	—
137	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
145	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
146	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₁₆×C₆₇	—	2	2	no	—
148	C₁₆×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	1	no	—
150	C₂³×C₆₇	#131	1	1	yes	characteristic
151	C₂²×D₆₇	—	2	1	no	—
152	Q₈×C₆₇	—	2	1	no	—
153	Q₈×C₆₇	C₂²×C₈	1	2	yes	—
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	2	1	no	—
157	D₄×C₆₇	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂²⋊D₆₇	—	2	2	no	—
168	C₂²⋊D₆₇	—	2	2	no	—
169	C₂×Dic₆₇	—	2	1	no	—
170	C₂×Dic₆₇	—	2	1	no	—
171	C₂×Dic₆₇	—	2	1	no	—
172	C₂×Dic₆₇	—	4	2	no	—
173	D₂₆₈	—	2	1	no	—
174	D₂₆₈	C₂²×C₈	1	2	yes	—
175	C₄×D₆₇	—	2	2	no	—
176	C₄×D₆₇	—	2	2	no	—
177	Dic₁₃₄	—	2	1	no	—
178	Dic₁₃₄	C₂²×C₈	1	2	yes	—
179	C₈×C₆₇	—	2	2	no	—
180	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
181	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₆₇⋊C₈	—	2	2	no	—
185	C₆₇⋊C₈	C₂²×C₈	1	1	yes	characteristic
186	C₆₇⋊C₈	C₂²×C₈	1	1	yes	characteristic
187	C₆₇⋊C₈	C₂²×C₈	1	1	yes	characteristic
188	C₆₇⋊C₈	C₂²×C₈	1	1	yes	characteristic
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	—	4	1	no	—
191	C₂²×C₆₇	—	8	1	no	—
192	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
193	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
195	C₂×D₆₇	—	2	2	no	—
196	C₂×D₆₇	—	8	1	no	—
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
199	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
200	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
201	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
202	Dic₆₇	—	2	2	no	—
203	Dic₆₇	—	2	2	no	—
204	Dic₆₇	—	4	1	no	—
205	#33443	—	67	1	no	maximal, Sylow
206	C₂×C₆₇	—	2	1	no	—
207	C₂×C₆₇	—	2	1	no	—
208	C₂×C₆₇	—	2	1	no	—
209	C₂×C₆₇	—	4	2	no	—
210	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
211	D₆₇	—	4	2	no	—
212	#3407	—	67	1	no	—
213	#3407	—	67	1	no	—
214	C₂×#443	—	67	1	no	—
215	#2171	—	67	4	no	—
216	#2170	—	67	4	no	—
217	C₂×#364	—	67	1	no	—
218	#2117	—	67	1	no	—
219	#2117	D₆₇	1	1	yes	characteristic, complemented
220	C₄×M₃₂	—	67	1	no	—
221	C₆₇	#33443	1	1	yes	characteristic, complemented, Sylow
222	C₂×#135	—	67	1	no	—
223	C₂×#131	—	67	1	no	—
224	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
225	#504	—	67	4	no	—
226	#504	—	67	4	no	—
227	#504	—	134	2	no	—
228	#504	—	134	2	no	—
229	C₂×M₃₂	—	67	1	no	—
230	C₂×M₃₂	—	67	1	no	—
231	C₂×M₃₂	—	134	2	no	—
232	C₂×M₃₂	—	134	2	no	—
233	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
234	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
235	#443	—	67	2	no	—
236	#443	—	67	2	no	—
237	#443	—	67	2	no	—
238	#443	—	67	2	no	—
239	C₂×#108	—	67	1	no	—
240	C₂×#104	—	67	1	no	—
241	C₂×#104	—	67	1	no	—
242	C₂×C₄×C₈	—	67	1	no	—
243	#364	—	67	2	no	—
244	#364	—	67	2	no	—
245	#350	—	67	4	no	—
246	#350	C₂×D₆₇	1	4	yes	complemented
247	#346	—	67	2	no	—
248	C₄×C₁₆	—	67	2	no	—
249	C₂×#41	—	67	1	no	—
250	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
251	#135	—	67	2	no	—
252	#135	—	67	2	no	—
253	#135	—	67	2	no	—
254	#135	—	67	2	no	—
255	C₂×Q₁₆	—	67	1	no	—
256	C₂×SD₁₆	—	67	1	no	—
257	C₂×SD₁₆	—	67	1	no	—
258	C₂×D₈	—	67	1	no	—
259	#131	—	2	1	no	—
260	#131	—	2	1	no	—
261	#131	—	67	2	no	—
262	#131	—	67	2	no	—
263	#131	—	134	1	no	—
264	#131	—	134	1	no	—
265	#131	C₄×D₆₇	1	2	yes	—
266	#131	C₂²×D₆₇	1	2	yes	—
267	C₂×M₁₆	—	2	1	no	—
268	C₂×M₁₆	—	67	1	no	—
269	C₂×M₁₆	—	134	1	no	—
270	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	67	1	no	—
273	C₂²×C₈	—	134	1	no	—
274	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
275	C₂×C₄²	—	67	1	no	—
276	M₃₂	—	2	1	no	—
277	M₃₂	—	2	1	no	—
278	M₃₂	—	2	1	no	—
279	M₃₂	—	67	2	no	—
280	M₃₂	—	67	2	no	—
281	M₃₂	—	67	2	no	—
282	M₃₂	—	134	4	no	—
283	M₃₂	—	134	4	no	—
284	M₃₂	—	134	1	no	—
285	M₃₂	C₂²×D₆₇	1	2	yes	—
286	C₂×C₁₆	—	2	1	no	—
287	C₂×C₁₆	—	67	2	no	—
288	C₂×C₁₆	—	134	4	no	—
289	C₂×C₁₆	—	134	4	no	—
290	C₂×C₁₆	—	134	1	no	—
291	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
292	#108	—	67	2	no	—
293	#108	—	67	2	no	—
294	#104	—	67	2	no	—
295	#104	—	67	2	no	—
296	#104	—	67	2	no	—
297	#104	—	67	2	no	—
298	C₄×C₈	—	67	1	no	—
299	C₄×C₈	—	67	1	no	—
300	C₄×C₈	—	67	1	no	—
301	C₄×C₈	—	67	1	no	—
302	#41	—	2	1	no	—
303	#41	—	2	1	no	—
304	#41	—	67	2	no	—
305	#41	—	67	2	no	—
306	#41	—	134	1	no	—
307	#41	—	134	1	no	—
308	#41	C₈×D₆₇	1	2	yes	—
309	#41	C₂×C₄×D₆₇	1	2	yes	—
310	C₂×Q₈	—	67	1	no	—
311	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	67	1	no	—
314	C₂×D₄	—	134	1	no	—
315	C₂×D₄	C₈×D₆₇	1	1	yes	characteristic
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	134	1	no	—
318	C₂²×C₄	—	134	1	no	—
319	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
320	Q₁₆	—	67	2	no	—
321	Q₁₆	—	67	2	no	—
322	SD₁₆	—	67	2	no	—
323	SD₁₆	—	67	2	no	—
324	SD₁₆	—	67	2	no	—
325	SD₁₆	—	67	2	no	—
326	D₈	—	67	2	no	—
327	D₈	—	67	2	no	—
328	M₁₆	—	2	1	no	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	2	2	no	—
331	M₁₆	—	67	2	no	—
332	M₁₆	—	134	1	no	—
333	M₁₆	—	134	2	no	—
334	M₁₆	—	134	2	no	—
335	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	67	1	no	—
340	C₂×C₈	—	67	1	no	—
341	C₂×C₈	—	67	1	no	—
342	C₂×C₈	—	67	1	no	—
343	C₂×C₈	—	67	1	no	—
344	C₂×C₈	—	67	1	no	—
345	C₂×C₈	—	134	1	no	—
346	C₂×C₈	—	134	2	no	—
347	C₂×C₈	—	134	2	no	—
348	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
349	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
350	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
351	C₂×C₈	#11761627	1	1	yes	characteristic
352	C₂×C₈	#11761627	1	1	yes	characteristic
353	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
354	C₄²	—	67	1	no	—
355	C₄²	—	67	1	no	—
356	C₄²	—	134	1	no	—
357	C₁₆	—	2	2	no	—
358	C₁₆	—	2	2	no	—
359	C₁₆	—	67	4	no	—
360	C₁₆	—	134	2	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	134	1	no	—
363	C₂³	#14524051	1	1	yes	characteristic
364	Q₈	—	2	1	no	—
365	Q₈	—	67	2	no	—
366	Q₈	—	134	1	no	—
367	Q₈	C₂×C₈×D₆₇	1	2	yes	—
368	D₄	—	2	2	no	—
369	D₄	—	2	2	no	—
370	D₄	—	2	1	no	—
371	D₄	—	67	2	no	—
372	D₄	—	134	2	no	—
373	D₄	—	134	2	no	—
374	D₄	—	134	1	no	—
375	D₄	C₂×C₈×D₆₇	1	2	yes	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	268	2	no	—
380	C₂×C₄	—	134	1	no	—
381	C₂×C₄	—	134	1	no	—
382	C₂×C₄	—	134	1	no	—
383	C₂×C₄	—	134	2	no	—
384	C₂×C₄	—	134	2	no	—
385	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
386	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
387	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
388	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
389	C₂×C₄	#14524051	1	1	yes	characteristic
390	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	67	1	no	—
393	C₈	—	67	1	no	—
394	C₈	—	67	1	no	—
395	C₈	—	67	1	no	—
396	C₈	—	134	2	no	—
397	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
398	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
399	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
400	C₈	C₂×#11761627	1	1	yes	characteristic
401	C₂²	—	2	2	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	8	1	no	—
404	C₂²	—	536	1	no	—
405	C₂²	—	134	2	no	—
406	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
407	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
408	C₂²	C₂×#14524051	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	268	1	no	—
411	C₄	—	134	2	no	—
412	C₄	—	134	2	no	—
413	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
414	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
415	C₄	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
416	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	4	2	no	—
421	C₂	—	268	2	no	—
422	C₂	C₂×(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2117	C₂
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	C₂×#14524063	C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2117	D₆₇
hypocenter	C₆₇	#33443
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2117

Complements

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

N	H ≅ G/N	type
1. #22573312	C₂	semidirect
2. #22573311	C₂	semidirect
4. #22562048	C₂	semidirect
5. #22562043	C₂	semidirect
6. C₂×#16863216	C₂	semidirect
7. #22561836	C₂	semidirect
8. M₃₂×Dic₆₇	C₂	semidirect
9. C₆₇×#2117	C₂	semidirect
12. #16863814	C₂²	semidirect
14. #16863813	C₂²	semidirect
28. #16863216	C₂²	semidirect
29. #16863216	C₂²	semidirect
30. #16863179	C₂²	semidirect
31. #16863171	C₂²	semidirect
32. C₁₆×Dic₆₇	C₂²	semidirect
36. #350×C₆₇	C₂²	semidirect
219. #2117	D₆₇	semidirect
221. C₆₇	#33443	semidirect
246. #350	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573312	2
#22573311	2
C₂×#16863511	2
#22562048	2
#22562043	2
C₂×#16863216	2
#22561836	2
M₃₂×Dic₆₇	2
C₆₇×#2117	2
#33443	67