Subgroups — #30854432

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863513	C₂	1	1	yes	maximal, characteristic
2	#135×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22568635	C₂	1	2	yes	maximal, complemented
4	#108×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560091	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560054	C₂	1	2	yes	maximal, complemented
8	#22560051	C₂	1	2	yes	maximal, complemented
9	#22560021	C₂	1	2	yes	maximal, complemented
10	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₂²	1	2	yes	—
13	#41×Dic₆₇	C₂²	1	2	yes	—
14	#16863513	C₄	1	8	yes	complemented
15	Q₁₆×Dic₆₇	C₄	1	2	yes	complemented
16	SD₁₆×Dic₆₇	C₄	1	4	yes	complemented
17	D₈×Dic₆₇	C₄	1	2	yes	complemented
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	#16863434	C₂²	1	1	yes	characteristic
20	C₂×#14523970	C₂²	1	2	yes	—
21	#16863392	C₂²	1	2	yes	—
22	#16863354	—	2	2	no	—
23	#16863354	C₂²	1	2	yes	—
24	#16863351	—	2	2	no	—
25	#16863351	C₂²	1	2	yes	—
26	M₁₆×Dic₆₇	—	2	2	no	—
27	M₁₆×Dic₆₇	C₂²	1	2	yes	—
28	C₂×#14523954	C₂²	1	1	yes	characteristic
29	C₂×#14523953	—	2	2	no	—
30	C₂×#14523953	C₂²	1	1	yes	characteristic
31	C₂×#14523953	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
34	#16863242	C₂²	1	1	yes	characteristic
35	#16863160	C₂²	1	2	yes	—
36	#16863157	C₂²	1	2	yes	—
37	#16863156	C₂²	1	2	yes	—
38	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#131×C₆₇	C₂²	1	2	yes	—
40	#443×C₆₇	C₄	1	8	yes	complemented
41	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#104×C₆₇	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#14524050	—	2	4	no	—
45	#14524050	C₂×C₄	1	8	yes	complemented
46	C₂×#11761612	—	2	2	no	—
47	C₂×#11761612	C₂³	1	2	yes	—
48	C₂²×C₆₇⋊C₈	—	2	2	no	—
49	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	Q₈×Dic₆₇	C₂×C₄	1	4	yes	—
52	D₄×Dic₆₇	—	2	4	no	—
53	D₄×Dic₆₇	C₂×C₄	1	4	yes	—
54	#14524002	—	2	2	no	—
55	#14524002	C₂³	1	2	yes	—
56	C₂×C₄×Dic₆₇	—	2	2	no	—
57	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
59	#14523979	C₂×C₄	1	4	yes	complemented
60	#14523977	C₂×C₄	1	2	yes	—
61	#14523975	C₂×C₄	1	2	yes	—
62	#14523970	C₂×C₄	1	8	yes	complemented
63	#14523967	C₂×C₄	1	4	yes	—
64	#14523964	C₂×C₄	1	4	yes	—
65	#14523954	—	2	2	no	—
66	#14523953	—	2	2	no	—
67	#14523953	—	2	2	no	—
68	#14523953	—	2	2	no	—
69	#14523953	—	2	2	no	—
70	#14523953	D₄	1	1	yes	characteristic
71	#14523953	D₄	1	1	yes	characteristic
72	#14523953	D₄	1	1	yes	characteristic
73	#14523953	D₄	1	1	yes	characteristic
74	C₈×Dic₆₇	—	2	2	no	—
75	C₈×Dic₆₇	—	2	2	no	—
76	C₈×Dic₆₇	—	2	2	no	—
77	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
78	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
79	C₈×Dic₆₇	C₂×C₄	1	4	yes	complemented
80	C₆₇⋊#95	—	2	2	no	—
81	C₂×#41×C₆₇	C₂³	1	2	yes	—
82	#135×C₆₇	C₂×C₄	1	4	yes	—
83	#135×C₆₇	C₂×C₄	1	4	yes	—
84	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
86	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	#131×C₆₇	—	2	4	no	—
88	#131×C₆₇	C₂×C₄	1	8	yes	complemented
89	C₂×M₁₆×C₆₇	—	2	2	no	—
90	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂²×C₈×C₆₇	—	2	2	no	—
92	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	#108×C₆₇	C₂×C₄	1	4	yes	complemented
96	#104×C₆₇	C₂×C₄	1	8	yes	complemented
97	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
98	C₂²×Dic₆₇	—	2	1	no	—
99	C₂²×Dic₆₇	—	2	4	no	—
100	C₂²×Dic₆₇	#41	1	2	yes	—
101	#11761612	—	2	2	no	—
102	#11761612	—	2	8	no	—
103	#11761612	C₂²×C₄	1	4	yes	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	1	no	—
106	C₂×C₆₇⋊C₈	—	2	1	no	—
107	C₂×C₆₇⋊C₈	—	2	8	no	—
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₆₇⋊C₈	#41	1	2	yes	—
113	C₄⋊Dic₆₇	—	2	4	no	—
114	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
115	C₂²⋊Dic₆₇	—	2	2	no	—
116	C₂²⋊Dic₆₇	—	2	4	no	—
117	C₄×Dic₆₇	—	2	2	no	—
118	C₄×Dic₆₇	—	2	1	no	—
119	C₄×Dic₆₇	—	2	4	no	—
120	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
121	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
122	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	C₂²×C₄	1	4	yes	—
127	#41×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₆₇	C₄²	1	4	yes	—
135	SD₁₆×C₆₇	C₄²	1	8	yes	—
136	D₈×C₆₇	C₄²	1	4	yes	—
137	M₁₆×C₆₇	—	2	2	no	—
138	M₁₆×C₆₇	—	2	8	no	—
139	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	8	no	—
142	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	#41	1	2	yes	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	C₂×C₄²	1	4	yes	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
171	C₂×Dic₆₇	—	2	4	no	—
172	C₂×Dic₆₇	—	2	2	no	—
173	C₂×Dic₆₇	—	4	2	no	—
174	C₂×Dic₆₇	—	4	2	no	—
175	C₂×Dic₆₇	—	8	4	no	—
176	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
177	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
178	C₈×C₆₇	—	2	4	no	—
179	C₈×C₆₇	C₂×C₄²	1	2	yes	—
180	C₈×C₆₇	C₂×C₄²	1	2	yes	—
181	C₈×C₆₇	C₄×D₄	1	2	yes	—
182	C₈×C₆₇	C₄×D₄	1	2	yes	—
183	C₆₇⋊C₈	—	2	4	no	—
184	C₆₇⋊C₈	—	2	2	no	—
185	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
186	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
192	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
193	C₄×C₆₇	—	2	4	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
196	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
199	Dic₆₇	—	2	2	no	—
200	Dic₆₇	—	2	2	no	—
201	Dic₆₇	—	4	2	no	—
202	Dic₆₇	—	4	4	no	—
203	#20514	—	67	1	no	maximal, Sylow
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	4	4	no	—
207	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
208	C₂×C₆₇	#2957	1	2	yes	—
209	C₂×#443	—	67	1	no	—
210	C₂×#443	D₆₇	1	1	yes	characteristic
211	C₄×#135	—	67	1	no	—
212	#2875	—	67	2	no	—
213	C₄×#108	—	67	1	no	—
214	#1772	—	67	1	no	—
215	#1772	—	67	1	no	—
216	#1765	—	67	2	no	—
217	#1765	—	67	2	no	—
218	#1759	—	67	2	no	—
219	#1726	—	67	1	no	—
220	C₆₇	#20514	1	1	yes	characteristic, complemented, Sylow
221	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
222	C₂×#131	—	67	2	no	—
223	C₂×#131	C₂×D₆₇	1	2	yes	—
224	C₄×#41	—	67	2	no	—
225	#443	—	67	8	no	—
226	#443	Dic₆₇	1	8	yes	complemented
227	C₄×Q₁₆	—	67	2	no	—
228	C₄×SD₁₆	—	67	4	no	—
229	C₄×D₈	—	67	2	no	—
230	C₂×#108	—	67	1	no	—
231	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
232	#427	—	67	1	no	—
233	C₂×#104	—	67	2	no	—
234	C₂×#104	C₂×D₆₇	1	2	yes	—
235	#416	—	67	2	no	—
236	#405	—	67	2	no	—
237	#405	—	67	2	no	—
238	#405	—	134	2	no	—
239	#405	—	134	2	no	—
240	C₄×M₁₆	—	67	2	no	—
241	C₄×M₁₆	—	134	2	no	—
242	C₂×#97	—	67	1	no	—
243	C₂×#97	—	67	1	no	—
244	C₂×#97	—	67	1	no	—
245	C₂×#97	—	134	2	no	—
246	C₂×C₄×C₈	—	67	1	no	—
247	C₂×C₄×C₈	—	67	1	no	—
248	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
249	#376	—	67	1	no	—
250	#341	—	67	2	no	—
251	#339	—	67	2	no	—
252	#339	—	67	2	no	—
253	C₂×#41	C₂²×D₆₇	1	2	yes	—
254	#135	C₄×D₆₇	1	4	yes	—
255	#135	C₂×Dic₆₇	1	4	yes	—
256	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
257	C₂×SD₁₆	C₄×D₆₇	1	2	yes	—
258	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
259	#131	—	2	4	no	—
260	#131	—	67	8	no	—
261	#131	—	134	4	no	—
262	#131	C₂×Dic₆₇	1	8	yes	complemented
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	—	67	2	no	—
265	C₂×M₁₆	—	134	2	no	—
266	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	67	1	no	—
269	C₂²×C₈	—	67	1	no	—
270	C₂²×C₈	—	134	2	no	—
271	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
273	C₄×Q₈	—	67	4	no	—
274	C₄×D₄	—	67	4	no	—
275	C₄×D₄	—	134	4	no	—
276	#117	—	67	2	no	—
277	#117	—	134	2	no	—
278	C₂×C₄²	—	67	1	no	—
279	C₂×C₄²	—	67	1	no	—
280	C₂×C₄²	—	134	2	no	—
281	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
282	#108	—	67	4	no	—
283	#108	C₂×Dic₆₇	1	4	yes	complemented
284	#107	—	67	2	no	—
285	#106	—	67	2	no	—
286	#104	—	67	8	no	—
287	#104	C₂×Dic₆₇	1	8	yes	complemented
288	#103	—	67	4	no	—
289	#102	—	67	4	no	—
290	#97	—	67	1	no	—
291	#97	—	67	1	no	—
292	#97	—	67	1	no	—
293	#97	—	67	1	no	—
294	#97	—	134	2	no	—
295	#97	—	134	2	no	—
296	#97	—	134	2	no	—
297	#97	—	134	2	no	—
298	#97	—	134	2	no	—
299	C₄×C₈	—	67	2	no	—
300	C₄×C₈	—	67	2	no	—
301	C₄×C₈	—	67	4	no	—
302	C₄×C₈	—	134	2	no	—
303	C₄×C₈	—	134	2	no	—
304	C₄×C₈	—	134	2	no	—
305	C₄×C₈	C₂×Dic₆₇	1	4	yes	complemented
306	#95	—	134	2	no	—
307	#41	—	2	2	no	—
308	#41	—	2	2	no	—
309	#41	C₂×C₄×D₆₇	1	4	yes	—
310	#41	C₂²×Dic₆₇	1	4	yes	—
311	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	67	2	no	—
317	C₂²×C₄	—	134	1	no	—
318	C₂²×C₄	—	134	4	no	—
319	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
320	Q₁₆	C₄×Dic₆₇	1	4	yes	—
321	SD₁₆	C₄×Dic₆₇	1	8	yes	—
322	D₈	C₄×Dic₆₇	1	4	yes	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	8	no	—
325	M₁₆	—	67	4	no	—
326	M₁₆	—	134	8	no	—
327	M₁₆	—	134	2	no	—
328	M₁₆	C₂²×Dic₆₇	1	4	yes	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	8	no	—
331	C₂×C₈	—	67	2	no	—
332	C₂×C₈	—	67	1	no	—
333	C₂×C₈	—	67	1	no	—
334	C₂×C₈	—	67	2	no	—
335	C₂×C₈	—	67	2	no	—
336	C₂×C₈	—	134	8	no	—
337	C₂×C₈	—	134	2	no	—
338	C₂×C₈	—	134	1	no	—
339	C₂×C₈	—	134	1	no	—
340	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
342	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
343	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
344	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
345	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
346	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
347	C₂×C₈	#11761627	1	2	yes	—
348	C₄⋊C₄	—	67	4	no	—
349	C₄⋊C₄	—	134	4	no	—
350	C₂²⋊C₄	—	134	4	no	—
351	C₂²⋊C₄	—	134	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	67	2	no	—
354	C₄²	—	67	1	no	—
355	C₄²	—	67	1	no	—
356	C₄²	—	67	2	no	—
357	C₄²	—	134	4	no	—
358	C₄²	—	134	2	no	—
359	C₄²	—	134	1	no	—
360	C₄²	C₂²×Dic₆₇	1	2	yes	—
361	C₂³	—	2	2	no	—
362	C₂³	#41×D₆₇	1	1	yes	characteristic
363	Q₈	—	2	2	no	—
364	Q₈	C₂×C₄×Dic₆₇	1	4	yes	—
365	D₄	—	2	4	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	2	no	—
368	D₄	C₂×C₄×Dic₆₇	1	4	yes	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	268	2	no	—
375	C₂×C₄	—	268	2	no	—
376	C₂×C₄	—	536	4	no	—
377	C₂×C₄	—	67	2	no	—
378	C₂×C₄	—	67	2	no	—
379	C₂×C₄	—	134	4	no	—
380	C₂×C₄	—	134	2	no	—
381	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
385	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
386	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
387	C₈	—	2	4	no	—
388	C₈	—	67	2	no	—
389	C₈	—	67	2	no	—
390	C₈	—	134	4	no	—
391	C₈	—	134	2	no	—
392	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
393	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
394	C₈	D₄×Dic₆₇	1	2	yes	—
395	C₈	D₄×Dic₆₇	1	2	yes	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
400	C₂²	C₄×#11761627	1	1	yes	characteristic
401	C₂²	#41×Dic₆₇	1	1	yes	characteristic
402	C₄	—	2	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	268	2	no	—
405	C₄	—	268	4	no	—
406	C₄	—	134	2	no	—
407	C₄	—	134	2	no	—
408	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
409	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
410	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
411	C₄	C₄×#11761627	1	1	yes	characteristic
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	4	4	no	—
415	C₂	C₄×D₄×Dic₆₇	1	1	yes	characteristic
416	C₂	#22569121	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	#20514
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
3. #22568635	C₂	semidirect
4. #108×Dic₆₇	C₂	semidirect
5. #22560091	C₂	semidirect
6. #22560088	C₂	semidirect
7. #22560054	C₂	semidirect
8. #22560051	C₂	semidirect
9. #22560021	C₂	semidirect
10. #22559895	C₂	semidirect
14. #16863513	C₄	semidirect
15. Q₁₆×Dic₆₇	C₄	semidirect
16. SD₁₆×Dic₆₇	C₄	semidirect
17. D₈×Dic₆₇	C₄	semidirect
40. #443×C₆₇	C₄	semidirect
45. #14524050	C₂×C₄	semidirect
59. #14523979	C₂×C₄	semidirect
62. #14523970	C₂×C₄	semidirect
79. C₈×Dic₆₇	C₂×C₄	semidirect
88. #131×C₆₇	C₂×C₄	semidirect
95. #108×C₆₇	C₂×C₄	semidirect
96. #104×C₆₇	C₂×C₄	semidirect
97. C₄×C₈×C₆₇	C₂×C₄	semidirect
220. C₆₇	#20514	semidirect
226. #443	Dic₆₇	semidirect
262. #131	C₂×Dic₆₇	semidirect
283. #108	C₂×Dic₆₇	semidirect
287. #104	C₂×Dic₆₇	semidirect
305. C₄×C₈	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863513	2
#135×Dic₆₇	2
#22568635	2
#108×Dic₆₇	2
#22560091	2
#22560088	2
#22560054	2
#22560051	2
#22560021	2
#22559895	2
C₂×#443×C₆₇	2
#20514	67