Subgroups — C₆₇×#21253 #30700504

Order 17152.

1562 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3449	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#104×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1980	C₂	1	8	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	#537×C₆₇	—	2	2	no	—
10	#536×C₆₇	C₄	1	4	yes	complemented
11	#535×C₆₇	C₄	1	4	yes	complemented
12	#534×C₆₇	—	2	2	no	—
13	#532×C₆₇	C₂²	1	4	yes	—
14	#525×C₆₇	—	2	4	no	—
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#421×C₆₇	—	2	8	no	—
19	C₂×#104×C₆₇	—	2	8	no	—
20	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
21	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
22	#405×C₆₇	—	2	4	no	—
23	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
24	#389×C₆₇	—	2	2	no	—
25	#385×C₆₇	—	2	2	no	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	4	no	—
30	C₂×#41×C₆₇	—	2	4	no	—
31	C₂×#41×C₆₇	—	2	4	no	—
32	C₂×#41×C₆₇	—	2	4	no	—
33	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×#41×C₆₇	C₂³	1	4	yes	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	—	2	1	no	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	4	no	—
43	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
44	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
47	#127×C₆₇	C₂×C₄	1	4	yes	complemented
48	#124×C₆₇	C₂×C₄	1	8	yes	complemented
49	#123×C₆₇	—	2	8	no	—
50	C₂²⋊Q₈×C₆₇	—	2	8	no	—
51	#121×C₆₇	—	2	8	no	—
52	#121×C₆₇	—	2	8	no	—
53	#120×C₆₇	—	2	8	no	—
54	C₄×D₄×C₆₇	—	2	8	no	—
55	#117×C₆₇	—	2	4	no	—
56	#117×C₆₇	C₂×C₄	1	4	yes	—
57	#117×C₆₇	D₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
63	C₂×C₄²×C₆₇	—	2	4	no	—
64	C₂×C₄²×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	#104×C₆₇	—	2	16	no	—
69	#104×C₆₇	—	2	16	no	—
70	#97×C₆₇	—	2	4	no	—
71	C₄×C₈×C₆₇	—	2	4	no	—
72	#95×C₆₇	—	2	4	no	—
73	C₂⁴×C₆₇	—	2	2	no	—
74	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
75	#41×C₆₇	—	2	8	no	—
76	#41×C₆₇	—	2	8	no	—
77	#41×C₆₇	—	2	4	no	—
78	#41×C₆₇	—	2	4	no	—
79	#41×C₆₇	—	2	4	no	—
80	#41×C₆₇	—	2	4	no	—
81	#41×C₆₇	—	4	4	no	—
82	#41×C₆₇	—	4	4	no	—
83	#41×C₆₇	—	4	4	no	—
84	#41×C₆₇	—	4	4	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
89	C₂×D₄×C₆₇	—	2	4	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	8	no	—
93	C₂×D₄×C₆₇	—	2	4	no	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	4	4	no	—
96	C₂×D₄×C₆₇	—	4	4	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	4	4	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	—	4	4	no	—
106	C₂²×C₄×C₆₇	—	4	4	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₆₇	#41	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	M₁₆×C₆₇	—	2	8	no	—
115	M₁₆×C₆₇	—	2	4	no	—
116	M₁₆×C₆₇	C₂×D₄	1	8	yes	—
117	C₂×C₈×C₆₇	—	2	8	no	—
118	C₂×C₈×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	8	no	—
120	C₄⋊C₄×C₆₇	—	4	8	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
123	C₂²⋊C₄×C₆₇	—	2	16	no	—
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
126	C₄²×C₆₇	—	2	4	no	—
127	C₄²×C₆₇	—	2	4	no	—
128	C₄²×C₆₇	—	4	2	no	—
129	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
130	C₄²×C₆₇	C₂×D₄	1	4	yes	—
131	C₂³×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	8	4	no	—
135	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂³×C₆₇	#124	1	2	yes	—
137	C₂³×C₆₇	#127	1	2	yes	—
138	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	4	no	—
140	Q₈×C₆₇	—	2	4	no	—
141	Q₈×C₆₇	—	4	4	no	—
142	D₄×C₆₇	—	2	4	no	—
143	D₄×C₆₇	—	2	4	no	—
144	D₄×C₆₇	—	4	4	no	—
145	D₄×C₆₇	—	4	4	no	—
146	D₄×C₆₇	—	4	8	no	—
147	D₄×C₆₇	—	4	4	no	—
148	D₄×C₆₇	—	8	8	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	8	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
156	C₂×C₄×C₆₇	—	4	4	no	—
157	C₂×C₄×C₆₇	—	4	4	no	—
158	C₂×C₄×C₆₇	—	4	8	no	—
159	C₂×C₄×C₆₇	—	4	4	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
162	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
165	C₂×C₄×C₆₇	#121	1	4	yes	—
166	C₂×C₄×C₆₇	#121	1	4	yes	—
167	C₂×C₄×C₆₇	#124	1	2	yes	—
168	C₂×C₄×C₆₇	#127	1	2	yes	—
169	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
173	C₈×C₆₇	—	2	8	no	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	2	4	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	4	8	no	—
179	C₂²×C₆₇	—	4	4	no	—
180	C₂²×C₆₇	—	8	8	no	—
181	C₂²×C₆₇	#390	1	4	yes	—
182	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
184	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
185	C₄×C₆₇	—	2	8	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	4	4	no	—
188	C₄×C₆₇	—	8	2	no	—
189	C₄×C₆₇	#390	1	4	yes	—
190	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
191	C₄×C₆₇	C₂×#121	1	2	yes	—
192	#21253	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	2	1	no	—
194	C₂×C₆₇	—	2	4	no	—
195	C₂×C₆₇	—	2	1	no	—
196	C₂×C₆₇	—	8	4	no	—
197	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
198	C₂×C₆₇	#2901	1	2	yes	—
199	#3449	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#421	C₂×C₆₇	1	2	yes	complemented
201	C₂²×#104	C₂×C₆₇	1	2	yes	complemented
202	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#1980	C₂×C₆₇	1	8	yes	complemented
205	C₆₇	#21253	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×#41	C₂²×C₆₇	1	2	yes	—
207	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
208	#537	—	2	2	no	—
209	#536	C₄×C₆₇	1	4	yes	complemented
210	#535	C₄×C₆₇	1	4	yes	complemented
211	#534	—	2	2	no	—
212	#532	C₂²×C₆₇	1	4	yes	—
213	#525	—	2	4	no	—
214	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
217	#421	—	2	8	no	—
218	C₂×#104	—	2	8	no	—
219	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
220	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
221	#405	—	2	4	no	—
222	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
223	#389	—	2	2	no	—
224	#385	—	2	2	no	—
225	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
226	C₂×#41	—	2	2	no	—
227	C₂×#41	—	2	2	no	—
228	C₂×#41	—	2	4	no	—
229	C₂×#41	—	2	4	no	—
230	C₂×#41	—	2	4	no	—
231	C₂×#41	—	2	4	no	—
232	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
233	C₂×#41	C₂³×C₆₇	1	4	yes	—
234	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
235	C₂²×D₄	—	2	2	no	—
236	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
237	C₂³×C₄	—	2	2	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	2	4	no	—
242	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
243	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
244	C₂²×C₈	—	2	2	no	—
245	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
246	#127	C₂×C₄×C₆₇	1	4	yes	complemented
247	#124	C₂×C₄×C₆₇	1	8	yes	complemented
248	#123	—	2	8	no	—
249	C₂²⋊Q₈	—	2	8	no	—
250	#121	—	2	8	no	—
251	#121	—	2	8	no	—
252	#120	—	2	8	no	—
253	C₄×D₄	—	2	8	no	—
254	#117	—	2	4	no	—
255	#117	C₂×C₄×C₆₇	1	4	yes	—
256	#117	D₄×C₆₇	1	4	yes	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	4	no	—
259	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
260	C₂×C₂²⋊C₄	—	2	1	no	—
261	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
262	C₂×C₄²	—	2	4	no	—
263	C₂×C₄²	—	2	1	no	—
264	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
265	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
266	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
267	#104	—	2	16	no	—
268	#104	—	2	16	no	—
269	#97	—	2	4	no	—
270	C₄×C₈	—	2	4	no	—
271	#95	—	2	4	no	—
272	C₂⁴	—	2	2	no	—
273	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
274	#41	—	2	8	no	—
275	#41	—	2	8	no	—
276	#41	—	2	4	no	—
277	#41	—	2	4	no	—
278	#41	—	2	4	no	—
279	#41	—	2	4	no	—
280	#41	—	4	4	no	—
281	#41	—	4	4	no	—
282	#41	—	4	4	no	—
283	#41	—	4	4	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	4	no	—
286	C₂×Q₈	—	2	4	no	—
287	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
288	C₂×D₄	—	2	4	no	—
289	C₂×D₄	—	2	4	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	4	no	—
293	C₂×D₄	—	2	4	no	—
294	C₂×D₄	—	4	4	no	—
295	C₂×D₄	—	4	4	no	—
296	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	4	4	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	4	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
311	C₂²×C₄	#41×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
313	M₁₆	—	2	8	no	—
314	M₁₆	—	2	4	no	—
315	M₁₆	C₂×D₄×C₆₇	1	8	yes	—
316	C₂×C₈	—	2	8	no	—
317	C₂×C₈	—	2	4	no	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	4	8	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
322	C₂²⋊C₄	—	2	16	no	—
323	C₂²⋊C₄	—	4	2	no	—
324	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
325	C₄²	—	2	4	no	—
326	C₄²	—	2	4	no	—
327	C₄²	—	4	2	no	—
328	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
329	C₄²	C₂×D₄×C₆₇	1	4	yes	—
330	C₂³	—	2	4	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	8	4	no	—
334	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₂³	#124×C₆₇	1	2	yes	—
336	C₂³	#127×C₆₇	1	2	yes	—
337	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
338	Q₈	—	2	4	no	—
339	Q₈	—	2	4	no	—
340	Q₈	—	4	4	no	—
341	D₄	—	2	4	no	—
342	D₄	—	2	4	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	4	no	—
345	D₄	—	4	8	no	—
346	D₄	—	4	4	no	—
347	D₄	—	8	8	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	8	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
361	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
362	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
363	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
364	C₂×C₄	#121×C₆₇	1	4	yes	—
365	C₂×C₄	#121×C₆₇	1	4	yes	—
366	C₂×C₄	#124×C₆₇	1	2	yes	—
367	C₂×C₄	#127×C₆₇	1	2	yes	—
368	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
370	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
372	C₈	—	2	8	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	8	no	—
379	C₂²	—	8	8	no	—
380	C₂²	#390×C₆₇	1	4	yes	—
381	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
382	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
383	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
384	C₄	—	2	8	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	4	no	—
387	C₄	—	8	2	no	—
388	C₄	#390×C₆₇	1	4	yes	—
389	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
390	C₄	C₂×#121×C₆₇	1	2	yes	—
391	C₂	—	2	1	no	—
392	C₂	—	2	4	no	—
393	C₂	—	2	1	no	—
394	C₂	—	8	4	no	—
395	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
396	C₂	C₆₇×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#421×C₆₇	C₂	semidirect
3. C₂²×#104×C₆₇	C₂	semidirect
4. C₆₇×#2873	C₂	semidirect
5. C₆₇×#2296	C₂	semidirect
6. C₆₇×#1980	C₂	semidirect
10. #536×C₆₇	C₄	semidirect
11. #535×C₆₇	C₄	semidirect
20. C₂×#104×C₆₇	C₂²	semidirect
21. C₂×#104×C₆₇	C₂²	semidirect
23. C₄×M₁₆×C₆₇	C₂²	semidirect
26. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
43. C₂×M₁₆×C₆₇	D₄	semidirect
46. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
47. #127×C₆₇	C₂×C₄	semidirect
48. #124×C₆₇	C₂×C₄	semidirect
192. #21253	C₆₇	direct (N × H)
200. C₂×#421	C₂×C₆₇	semidirect
201. C₂²×#104	C₂×C₆₇	semidirect
202. #2873	C₂×C₆₇	semidirect
203. #2296	C₂×C₆₇	semidirect
204. #1980	C₂×C₆₇	semidirect
205. C₆₇	#21253	direct (N × H)
209. #536	C₄×C₆₇	semidirect
210. #535	C₄×C₆₇	semidirect
219. C₂×#104	C₂²×C₆₇	semidirect
220. C₂×#104	C₂²×C₆₇	semidirect
222. C₄×M₁₆	C₂²×C₆₇	semidirect
225. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
242. C₂×M₁₆	D₄×C₆₇	semidirect
245. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
246. #127	C₂×C₄×C₆₇	semidirect
247. #124	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3449	2
C₂×#421×C₆₇	2
C₂²×#104×C₆₇	2
C₆₇×#2873	2
C₆₇×#2296	2
C₆₇×#1980	2
#21253	67

Subgroups — C67×#21253 #30700504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21253 #30700504